Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson
Table of Contents 1.2 Graphs of Equations... 1 1.4 Functions... 9 1.5 Analyzing Graphs of Functions... 14 1.6 A Library of Parent Functions... 21 1.7 Transformations of Functions... 24 1.8 Combinations of Functions: Composite Functions... 30 1.9 Inverse Functions... 33 2.1 Quadratic Functions and Models... 37 2.2 Polynomial Functions of Higher Degree... 43 2.3 Polynomial and Synthetic Division... 49 2.4 Complex Numbers... 53 2.5 Zeros of Polynomial Functions... 57 2.6 Rational Functions... 63 3.1 Exponential Functions and Their Graphs... 70 3.2 Logarithmic Functions and Their Graphs... 76 3.3 Properties of Logarithms... 81 3.4 Exponential and Logarithmic Equations... 83 3.5 Exponential and Logarithmic Models... 85 4.5 Graphs of Sine and Cosine Functions... 89 4.6 Graphs of Other Trigonometric Functions... 93 4.7 Inverse Trigonometric Functions... 98 5.1 Using Fundamental Identities... 104 5.2 Verifying Trigonometric Identities... 108 5.3 Solving Trigonometric Equations... 111 5.4 Sum and Difference Formulas... 114 5.5 Multiple-Angle and Product-to-Sum Formulas... 117 6.3 Vectors in the Plane... 123 6.4 Vectors and Dot Products... 129 6.5 Trigonometric Form of a Complex Number... 132 7.1 Linear and Nonlinear Systems of Equations... 139 7.2 Two-Variable Linear Systems... 141 7.3 Multivariable Linear Systems... 144 7.4 Partial Fractions... 145 7.5 Systems of Inequalities... 151 8.1 Matrices and Systems of Equations... 154
8.2 Operations with Matrices... 164 8.4 The Determinant of a Square Matrix... 170 8.3 The Inverse of a Square Matrix... 174 8.5 Applications of Matrices and Determinants... 180 9.1 Sequences and Series... 184 9.2 Arithmetic Sequences and Partial Sums... 191 9.3 Geometric Sequences and Series... 196 9.4 Mathematical Induction... 202 9.5 The Binomial Theorem... 205 10.2 Introduction to Conics: Parabolas... 208 10.3 Ellipses... 214 10.4 Hyperbolas... 219 10.6 Parametric Equations... 224 10.7 Polar Coordinates... 228 10.8 Graphs of Polar Equations... 232
Section 1.2 1 1.2 Graphs of Equations Example 1 Sketch the graph of by plotting points. Intercepts of a Graph The -intercepts of the graph of an equation are the points at which the graph intersects or touches the -axis. The -intercepts of the graph of an equation are the points at which the graph intersects or touches the -axis. Identify the - and - intercepts of the graph sketched in Example 1: -intercept(s): -intercept(s): Finding Intercepts 1. To find -intercepts, 2. To find -intercepts,
2 Section 1.2 Example 2 Find the - and - intercepts of the graph of the equation. (a) (b) (c)
Section 1.2 3 Symmetry Knowing the symmetry of a graph before attempting to sketch it is helpful, because then you need only half as many solution points to sketch the graph. There are 3 basic types of symmetry a graph may have: -axis symmetry -axis symmetry origin symmetry Type of symmetry -axis Graphical Tests for Symmetry Whenever is on the graph, is also on the graph. Algebraic Tests for Symmetry Replacing with yields an equivalent equation. -axis Whenever is on the graph, is also on the graph. Replacing with yields an equivalent equation. Origin Whenever is on the graph, is also on the graph. Replacing with and with yields an equivalent equation.
4 Section 1.2 Example 3 Use the algebraic tests to check for symmetry with respect to both axes and the origin. (a) (b)
Section 1.2 5 Use the algebraic tests to check for symmetry with respect to both axes and the origin. (c) Recall the following formula: Distance Formula The distance between two points and is given by the formula
6 Section 1.2 Circles Center: Point on circle: Use the distance formula to derive the equation of a circle of radius and center : Standard Form of the Equation of a Circle The point lies on the circle of radius and center if and only if For a circle with center at the origin,, the standard form simplifies to:
Section 1.2 7 Example 4 Find the center and radius of the circle, and sketch its graph. Example 5 Write the standard form of the equation of the circle with the given characteristics. (a) Center: ; Radius: 3 (b) Center: ; Solution point:
8 Section 1.2 Write the standard form of the equation of the circle with the given characteristics. (c) Endpoints of a diameter:
Section 1.4 9 1.4 Functions Definition of Function A function from a set to a set is a rule of correspondence (or relation) that assigns to each element in the set exactly one element in the set. The set of inputs,, is the of the function. The set of outputs,, is the of the function. A function may be represented as a set mapping, as a set of ordered pairs, graphically, or as an equation. In algebra, it is most common to represent functions by equations. Can you see why? Set Mapping Set of Ordered Pairs Graphically Equation
10 Section 1.4 Example 1 Determine whether the relation represents as a function of. Circle your answer. Function Function Function Not a Function Not a Function Not a Function Example 2 Determine whether the relation represents as a function of. (a) (b) Note: If is a function of, then is the independent variable and is the dependent variable. Function Notation When an equation is used to represent a function, it is convenient to name the function so that it can be referenced easily. For example, the equation describes as a function of. Suppose we give the function the name. Then we can use the following function notation. Input Output Equation The symbol represents the -value of the function at. So we can write.
Section 1.4 11 Example 3 Let. Evaluate at each specified value of the independent variable and simplify. (a) (b) (c) A function defined by two or more equations over a specified domain is called a piecewise-defined function. Example 4 Evaluate at each specified value of the independent variable and simplify. (a) (b) (c) (d) As previously mentioned, the domain of a function is the set of all input values. The domain can be described explicitly or it can be implied by the expression used to define the function. The implied domain of a function is.
12 Section 1.4 Example 5 Find the domain of the function. (a) (b) (c) (d)
Section 1.4 13 Difference Quotients The expression is called a difference quotient and is important in calculus. The difference quotient can also take on the following form: Example 6 Find the difference quotient and simplify your answer.
14 Section 1.5 1.5 Analyzing Graphs of Functions Interval Notation for Sets: Inequality Interval Notation Graph x b (, b) x b (, b] ) b ] b x a ( a, ) x a [ a, ) a x b ( a, b) a x b ( a, b] ( a [ a ( ) a b ( ] a b a x b [ a, b) a x b [ a, b] [ a [ a ) b ] b All real numbers (, ) x a (, a ) ( a, ) )( a Note: The union symbol is used to indicate the union of disjoint sets. Example: To represent or in interval notation, we write
Section 1.5 15 A closed dot An open dot indicates a point is included in the graph of a function. indicates a point is excluded from the graph of a function. The use of dots at the extreme left and right points of a graph indicates that the graph does not extend beyond these points. If no such dots are shown, assume that the graph extends beyond these points. Example 1 Use the graph of the function to find the following. Use interval notation when appropriate. 4 3 2 1-6 -4-2 2 4 6-1 -2-3 -4 (a) Domain of (b) Range of f (c) (d) (e) Interval(s) for which
16 Section 1.5 Vertical Line Test Intersects in one point Intersects in more than one point Passes the test Function Fails the test Not a Function If a vertical line intersects a graph in more than one point, then the graph is not the graph of a function. Example 2 Determine whether each graph is that of a function. (a) (b) Zeros of a Function If the graph of a function of has an -intercept at, then is a zero of the function. The zeros of a function of are the -values for which. Example 3 Find the zeros of the function algebraically. (a) (b)
Section 1.5 17 Increasing, Decreasing, and Constant Functions A function f is increasing on an interval if, for any and in the interval, implies. A function f is decreasing on an interval if, for any and in the interval, implies. A function f is constant on an interval if, for any and in the interval,. Example 4 Determine the intervals over which the function is increasing, decreasing, or constant.
18 Section 1.5 Example 5 (a) Find the domain and range of g. (b) Determine the intervals of increasing, decreasing, or constant. (c) Determine the interval(s) for which.
Section 1.5 19 Even and Odd Functions A function f is even if, for every number x in its domain, the number is also in the domain and The graph of an even function is symmetric with respect to the y-axis. A function f is odd if, for every number x in its domain, the number is also in the domain and The graph of an odd function is symmetric with respect to the origin. To determine whether a function is even, odd, or possibly neither, replace by in the formula. Example 6 Determine whether the function is even, odd, or neither. Then describe the symmetry. (a)
20 Section 1.5 Determine whether the function is even, odd, or neither. Then describe the symmetry. (b) (c)
Section 1.6 21 1.6 A Library of Parent Functions Quick Overview of Lines Slope Formula: Point-Slope Form: Slope-Intercept Form: Vertical Lines: Horizontal Lines: A linear function is any function that can be expressed in the form The graph of a linear function is a line with slope and -intercept. The domain is all real numbers. Example 1 Write the linear function for which and. Then sketch the graph.
22 Section 1.6 Library of Parent Functions Constant Function Identity Function Absolute Value Function Quadratic Function Cubic Function Reciprocal Function Square Root Function Cube Root Function
Section 1.6 23 When functions are defined by more than one equation, they are called piecewise-defined functions. Example: Absolute Value Function To graph a piecewise-defined function, sketch each piece separately. Remember to use open and closed dots appropriately. Example 2 Graph the following piecewise-defined function.
24 Section 1.7 1.7 Transformations of Functions Vertical shift units upward Vertical shift units downward Horizontal shift units to the right Horizontal shift units to the left Vertical and Horizontal Shifts Let be a positive real number. Vertical and horizontal shifts in the graph of are represented as follows. 1. Vertical shift units upward: 2. Vertical shift units downward: 3. Vertical shift units to the right: 4. Vertical shift units to the left:
Section 1.7 25 Reflection in the x-axis Reflection in the y-axis Reflections in the Coordinate Axes Reflections in the coordinate axes of the graph of are represented as follows. 1. Reflection in the x-axis: 2. Reflection in the y-axis: ********************************************************************************** *Note: Always graph a function working with the expression inside to outside. 2 nd 1 st 3 rd 1. Shift left 2 units 2. Reflect across x-axis 3. Shift up 3 units Exception: Apply a horizontal reflection before a horizontal shift. (But only after you ve factored out the negative in front of!!) 1. Reflect across y-axis 2. Shift right 2 units 1 st 2 nd
26 Section 1.7 Vertical stretch (each value is multiplied by ) Vertical shrink (each value is multiplied by ) Horizontal stretch (each value is multiplied by ) Horizontal shrink (each value is multiplied by ) Vertical and Horizontal Stretches and Shrinks Let be a positive real number. Vertical and horizontal stretches and shrinks in the graph of are represented as follows. 1. Vertical stretch (each value is multiplied by ) 2. Vertical shrink (each value is multiplied by ) 3. Horizontal stretch (each value is multiplied by ) 4. Horizontal shrink (each value is multiplied by )
Section 1.7 27 Example 1 Use the graph of (a) to sketch each graph. List which transformations are performed. (b) Example 2 Use the graph of to write an equation for the function whose graph is shown. (a)
28 Section 1.7 Use the graph of to write an equation for the function whose graph is shown. (b) Example 3 (a) Identify the parent function. (b) Describe the sequence of transformations from to. (c) Sketch the graph of. (d) Use function notation to write in terms of.
Section 1.7 29 Example 4 (a) Identify the parent function. (b) Describe the sequence of transformations from to. (c) Sketch the graph of. (d) Use function notation to write in terms of. Example 5 Write an equation for the function that is described by the given characteristics: The shape of, but shifted nine units to the right and reflected in both the -axis and -axis.
30 Section 1.8 1.8 Combinations of Functions: Composite Functions Arithmetic Combinations of Functions Let and be two functions with overlapping domains. Then, for all common to both domains, the sum, difference, product, and quotient of and are defined as follows. 1. Sum: 2. Difference: 3. Product: 4. Quotient:, Example 1 Given and, find the following: a) b) c) d) and specify domain e) Evaluate
Section 1.8 31 Composition of Functions The composition of the function with the function is. The domain of is the set of all in the domain of such that is in the domain of. Example 2 Given and, find the following: a) b) c) Example 3 Given and, find and state the domain.
32 Section 1.8 Example 4 Given and, find and state the domain. Example 5 Find two functions and such that.
Section 1.9 33 1.9 Inverse Functions One-to-One Functions A function is one-to-one if each output value corresponds to exactly one input value. Example 1 Is the function one-to-one? (a) (b) The may be used to determine if a function is one-to-one.
34 Section 1.9 The inverse function of a function, denoted, is found by interchanging the first and second coordinates. A function has an inverse function if and only if is. Domain of Range of Since the inverse functions and have the effect of undoing each other, when you form the composition you obtain the identify function: This is how we verify two functions are inverses! Example 2 Use the table of values for to complete the table for. 34
Section 1.9 35 Example 3 (a) Show (or verify) that and are inverse functions algebraically. (b) Sketch the graphs of and on the same coordinate grid. What do you notice? How are the graphs of a function and its inverse function related?
36 Section 1.9 Finding an Inverse Function Algebraically 1. 2. 3. Example 4 Find the inverse function of and state the domain and range of and. Example 5 Determine whether the function has an inverse function. If it does, find the inverse function. (a) (b) 36
Section 2.1 37 2.1 Quadratic Functions and Models Definition Let be a nonnegative integer and let be real numbers with. The function given by is called a polynomial function of with degree. Special Cases of Polynomial Functions: Degree 0 1 2 3 4 Type We will focus on the quadratic function in this section. Definition Let and be real numbers with. The function given by is called a quadratic function. The shape of the graph of a quadratic function is called a. If the leading coefficient is positive ( If the leading coefficient is negative ( ), the graph opens. ), the graph opens. All parabolas are symmetric with respect to a line called the. The highest (or lowest) point on the graph is called the of the parabola.
38 Section 2.1 Example 1 Match the quadratic function with its graph. (a) (b) (c) +3 (d) (i) (ii) (iii) (iv)
Example 2 Sketch the graph of. Identify the vertex, axis of symmetry and - intercept(s). Section 2.1 39
40 Section 2.1 Standard Form of a Quadratic Function Example 3 Sketch the graph of. Identify the vertex, axis of symmetry and -intercept(s).
Section 2.1 41 Example 4 Write the standard form of the equation of the parabola whose vertex is the point. and passes through Example 5 Find a quadratic function whose graph has -intercepts and.
42 Section 2.1 In general, if, then completing the square yields the standard form: (Try it at home!) From this standard form, we get a formula for the vertex of the parabola: Many applications involve finding the maximum or minimum value of a quadratic function. The maximum or minimum value occurs at the vertex. Example 6 Find two positive real numbers whose product is a maximum if the sum of the first and three times the second is 42.
Section 2.2 43 2.2 Polynomial Functions of Higher Degree The graphs of polynomial functions are continuous with smooth rounded turns. This means there are no jumps, holes, or sharp turns! Identify which of the following could be the graph of a polynomial function. The polynomial functions that have the simplest graph are of the form ( is an integer) Polynomial functions of this form are called. If is even, the graph of is similar to the graph of. If is odd, the graph of is similar to the graph of. The greater the value of, the the graph near the origin. Example 1 Sketch the graph of each polynomial function. (a) (b)
44 Section 2.2 The Leading Coefficient Test (for determining End Behavior ) As moves without bound to the left or to the right, the graph of the polynomial function eventually rises or falls in the following manner. 1. When is odd: Leading coefficient is positive Leading coefficient is negative The graph to the left and to the right. The graph to the left and to the right. 2. When is even: Leading coefficient is positive Leading coefficient is negative The graph to the left and to the right. The graph to the left and to the right.
Section 2.2 45 The following function notation is often used to indicate end behavior of a graph: Falls to the left: Falls to the right: Rises to the left: Rises to the right: Example 2 Describe the right-hand and left-hand behavior of the graph. If is a polynomial function and is a real number, the following are equivalent (TFAE): 1. 2. 3. 4. Consider the graph of below. Observe the number of turning points, where the graph crosses the -axis, and where the graph touches the -axis. Factored form :
46 Section 2.2 For a polynomial function of degree, 1. has real zeros 2. The graph of has turning points. Repeated Zeros A factor yields a repeated zero of multiplicity. 1. If the multiplicity is odd, the graph the -axis at. 2. If the multiplicity is even, the graph the -axis at. Note: If the multiplicity, the graph will also at How to Sketch the Graph of a Polynomial Function 1. 2. 3.
Section 2.2 47 Example 3 Sketch the graph of the function by applying the Leading Coefficient Test, finding the zeros of the polynomial, and plotting any additional points as necessary. (a) (b)
48 Section 2.2 The Intermediate Value Theorem Let and be real numbers such that. If is a polynomial function such that, then, in the interval, takes on every value between and. Example 4 Use the Intermediate Value Theorem to approximate the real zero of given the following table of values.
Section 2.3 49 2.3 Polynomial and Synthetic Division Example 1 Use long division to divide. Example 2 Use long division to divide. If you need to review the long division process, also refer to Examples 1, 2 and 3 in the book.
50 Section 2.3 Synthetic Division is a consolidated algorithm (a.k.a. short-cut) for dividing a polynomial by (where is any constant). Illustrative Example Divide by. Solution: - - - - Example 3 Use synthetic division to divide. Example 4 Consider. (a) Divide by. Then compare your answer with the function value. (b) Divide by. Then compare your answer with the function value.
Section 2.3 51 From Example 4, we observe the following: The Remainder Theorem If a polynomial is divided by, the remainder is. Example 5 Use the Remainder Theorem and synthetic division to evaluate if Now, suppose is a zero of, so that. Then by the remainder theorem, when is divided by, the remainder is. But this implies. The Factor Theorem A polynomial has a factor if and only if. Example 6 Determine if is a factor of.
52 Section 2.3 Example 7 Use synthetic division to show that and are solutions of the equation Then use the result to factor the polynomial completely. List all real solutions of the equation.
Section 2.4 53 2.4 Complex Numbers History of Numbers as Solutions to Equations Integers rational numbers irrational numbers imaginary numbers Historical Note: Imaginary numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano (1501-1576), who called them fictitious. We now know the applications extend to engineering, physics, and applied mathematics. Cardano was a friend of Leonardo da Vinci. Definition The imaginary unit is the number whose square is. That is, Definition Complex numbers are numbers of the form, where and are real numbers. The number The number is called the. is called the.* (*Other books define the imaginary part to be, not ) *Note: Write instead of, and instead of. Powers of
54 Section 2.4 Example 1 Evaluate. Equality of Complex Numbers if and only if and Sum or Difference of Complex Numbers Example 2 Product of Complex Numbers To find the product of two complex numbers, follow the usual rules for multiplying two binomials: FOIL Example 3
Section 2.4 55 Definition and are called. Example 4 Multiply the number by its complex conjugate. We use conjugates to write the quotient of complex numbers in standard from Example 5 Write the quotient in standard form ( ). Definition If is a positive real number, the principal square root of, denoted by, is defined as
56 Section 2.4 Example 6 Write the complex number in standard form. Example 7 Perform the indicated operation and write the result in standard form. Example 8 Solve using the quadratic formula.
Section 2.5 57 2.5 Zeros of Polynomial Functions The Fundamental Theorem of Algebra (FTA) If is a polynomial with positive degree, then. Historical Note: The FTA was proved by arguably one of the greatest mathematicians/physicists of all time, Carl Friedrich Gauss (1777-1855). The proof is beyond the scope of this class. Now observe the following: If is a polynomial of degree, then by the FTA,. Then by Factor Theorem,. Thus, By the FTA again,. Then by Factor Theorem,. Thus, We can continue this process until is completely factored Linear Factorization Theorem If is an th-degree polynomial, then. ie. A polynomial of degree has exactly zeros. (The zeros may be real or complex, and they may be repeated) Example 1 List all the zeros of the function.
58 Section 2.5 Descartes Rule of Signs Suppose is a polynomial with real coefficients. 1) If the formula for has variations in sign, there are either, or etc. positive real zeros of. 2) If the formula for has variations in sign, there are either, or etc. negative real zeros of. *A variation in sign means that two consecutive coefficients have opposite signs. Example 2 Describe the possible real zeros of each function. (a) (b) Rational Zero Test Suppose is a polynomial with integer coefficients. Then the possible rational zeros are all numbers of the form where is a factor of and is a factor of. Example 3 List the possible rational zeros of the function.
Example 4 Find all the rational zeros of the function. Section 2.5 59
60 Section 2.5 Example 5 Find all solutions of the polynomial equation.
Section 2.5 61 Definitions A quadratic factor with no real zeros is said to be irreducible over the reals. Example: A quadratic factor with no rational zeros is said to be irreducible over the rationals. Example: Example 6 Write (a) as the product of factors that are irreducible over the rationals. (b) as the product of linear and quadratic factors that are irreducible over the reals. (c) in completely factored form. The Conjugate Pairs Theorem If the polynomial function has real coefficients, and is a zero ( ), then is also a zero. Example 7 Find a polynomial function with real coefficients that has the given zeros: and
62 Section 2.5 Example 8 Find all the zeros of given that is a zero. Then write the polynomial as a product of linear factors.
Section 2.6 63 Definition: A rational function can be written in the form 2.6 Rational Functions where and are polynomials and. Example 1 Find the domain of and discuss the behavior of using limit notation. Domain: 100 100 Limits:
64 Section 2.6 Definition of Vertical and Horizontal Asymptotes 1. The line is a vertical asymptote of the graph of if or as, either from the right or from the left. 2. The line is a horizontal asymptote of the graph of if as or. How to Find Vertical Asymptotes and Holes 1. Write the rational function in lowest terms by factoring and canceling common factors. 2. The graph of will have holes at any zeros which were common to both numerator and denominator and thus canceled out. 3. The graph of will have vertical asymptotes at the zeros of the simplified denominator. Example 2 Find any vertical asymptotes and/ or holes of the graph: (a) (b) (c)
Section 2.6 65 How to Find Horizontal/Slant Asymptotes: Let be the rational function given by 1. If, the line (the -axis) is a horizontal asymptote. 2. If, the line (the ratio of leading coefficients) is a horizontal asymptote. 3. If is exactly one more than, there is a slant asymptote which is found using long division: Example: Since as the the slant asymptote is. 4. If is two or more than, there is no horizontal or slant asymptote. Example 3 Find any horizontal/slant asymptotes of the graphs: (a) (b) (c)
66 Section 2.6 How to Graph a Rational Function Let, where and are polynomials. 1. Find and sketch the horizontal/slant asymptote, if any. 2. Simplify, if possible (factor and cancel common factors). Write for the zeros of any common factors that were canceled. There will be holes in the graph at these values. 3. Find and sketch the vertical asymptote(s), if any. 4. Find and plot the - and - intercepts. 5. Use the - intercepts and vertical asymptotes to divide the -axis into intervals. Evaluate at one test point in each interval, emphasizing if the value is or. Plot these points on the graph. 6. Use smooth curves to complete the sketch of the graph. Use the following guidelines: The graph must remain above the -axis for intervals where the test value was, and below the - axis for intervals where the text value was. The graph must never cross a vertical asymptote, but should go to near it. The graph must approach any horizontal or slant asymptote as. Don t forget to indicate any holes on the graph!
Section 2.6 67 Example 4 Graph
68 Section 2.6 Example 5 Graph
Section 2.6 69 Example 6 Graph
70 Section 3.1 3.1 Exponential Functions and Their Graphs An exponential function with base is denoted by where, and is any real number. 1. Graph by plotting points. x f( x) 2 x 3 1 1 3 2 0.125 3 2 8 2 1 0 1 2 3 3 2 8 2. Graph by plotting points. x 3 2 1 f( x) 2 1 2 3 x 3 2 8 1 0 1 2 3 3 1 1 1 0.125 3 2 2 8
Section 3.1 71 The graph of f ( x) a x x If a 1, then f ( x) a is increasing. If 0 a 1, then f ( x) a is decreasing. x (0,1) (1, a ) (0,1) (1, a ) Base e There is a certain irrational exponential base which occurs frequently in calculus applications, the number e. Definition: The number e is defined as n 1 10 1,000 1,000,000 1 e lim 1 n n 1 1 n 1 n 1 1 2 1 10 1 1 2.59374246 10 1,000 1 1 2.716923932 1,000 1 1 1,000,000 n 1,000,000 2.718280469 As n gets larger, the approximation for e gets better! For all intents and purposes, we can think of e as approximately 2.718. x Since e 1, the graph of f ( x) e is an increasing exponential function.
72 Section 3.1 Graphing Exponential Functions There are two techniques that can be used to sketch the graph of an exponential function. The following two examples illustrate these techniques. Example 1 Construct a table of values. Then sketch the graph of Example 2 Use transformations to sketch the graph of
Section 3.1 73 Principal Total amount borrowed (or invested) Compound Interest Compound Interest When interest due (or earned) at the end of a payment period is added to the principal, so that the interest computed at the end of the next payment period is based on this new higher principal amount (old principal + interest), interest is said to have been compounded. Annually 1 time per year Semiannually 2 times per year Quarterly 4 times per year Monthly 12 times per year Daily 365* times per year *In practice, most banks use a 360-day year Simple Interest Formula If a principal of P dollars is borrowed (or invested) for a period of t years at an annual interest rate r, expressed as a decimal, the interest I charged (or earned) is: Suppose we invest $100 in a Money Market account at an annual interest rate 6% for 1 year 1) If interest is compounded annually, then: At the end of 1 year, we will earn the following in simple interest: I Prt =100(0.06)(1) = $6 2) If, however, interest is compounded semiannually, then: After 6 months (1/2 year), we will earn the following in simple interest: 1 I Prt = 100(0.06) 2 = $3 Compounding #1 Assuming this earned interest remains in the Money Market account, the new higher principal amount is now: $100 $3 =$103 After another 6 months, we will earn the following in simple interest: 1 I Prt = 103(0.06) 2 = $3.09 Compounding #2 So TOTAL interest earned after 1 year is $3 $3.09 = $6.09
74 Section 3.1 So we earned more than 6% interest annually with semiannually compounding because the interest earned after the first compounding (6 months) made the principal HIGHER for the second compounding!! In fact, since we started with $100 and earned a total of $6.09 in interest after semiannually compounding, we effectively earned 6.09% simple interest after 1 year. We call 6.09% the effective interest rate. Compound Interest Formula The balance A in an account after t years due to a principal P invested at an annual interest rate r (in decimal form) compounded n times per year is: What happens to the balance as the number of times interest is compounded per year gets larger and larger?? In other words, what happens as n? n 1 r r Recall, e lim 1. In general, e lim 1 n n n n. Hence, nt n r r A P 1 P 1 P e n n t r t as n n Continuous Compounding Formula The balance A in an account after t years due to a principal P invested at an annual interest rate r (in decimal form) compounded continuously is :
Example 3 Find the balance in an account after 10 years if is invested at 3% interest compounded a) monthly Section 3.1 75 b) continuously
76 Section 3.2 3.2 Logarithmic Functions and Their Graphs Since the exponential function function: is one-to-one, we can compute the inverse Step 1: Replace by. Step 2: Interchange x and y. Step 3: Solve for y. It would be helpful to have a function that would give the power of such a function and give it the name logarithmic function! that produces x. We create Note: Since is positive for all y, we have the result that for, x must be positive! Logarithmic Function If and, then for we have the following: if and only if The function given by is called the logarithmic function with base. Memory Device: Logarithmic form into EXPONENTIAL form log b x y means Logarithmic Form Exponential Form Exponential form into LOGARITHMIC form means
Section 3.2 77 Example 1 Write each equation in exponential form. (a) (b) (c) Example 2 Write each equation in logarithmic form. (a) (b) (c) If the base of a logarithmic function is 10, we call this the common logarithmic function: *Note: If the base of a logarithm is not indicated, it is understood to be 10. If the base of a logarithmic function is e, we call this the natural logarithmic function: *Note: This function occurs so frequently in applications that it is given a special symbol, ln. Example 3 Evaluate the function at the indicated value of without using a calculator. (a) ; (b) ; Example 4 Use a calculator to evaluate at. Round your result to three decimal places.
78 Section 3.2 Properties of logarithms 1. 2. 3. and 4. If then. Example 5 Use the properties of logarithms to simplify the expression. (a) (b) (c) Graphs of Logarithmic Functions Example 6 The graph of is provided below. Sketch the graph of: Step 1. Replace with. Step 2. Rewrite in exponential form. Step 3. Plot points and graph. x y
Section 3.2 79 Example 7 The graph of is provided below. Sketch the graph of: Step 1. Replace with. Step 2. Rewrite in exponential form. Step 3. Plot points and graph. x y Note: All logarithmic functions pass through the points and and have the y-axis as a vertical asymptote. If, then is increasing. If, then is decreasing.
80 Section 3.2 We can apply transformations to graph logarithmic functions. Example 8 Sketch the graph of. Also, find the domain, -intercept, and vertical asymptote. Example 9 Sketch the graph of Also, find the domain, -intercept, and vertical asymptote.
Section 3.3 81 3.3 Properties of Logarithms Here are some more properties of logarithms (the derivations are in the book on page 278). Do they remind you of rules of exponents? 1. Product Property 2. Quotient Property 3. Power Property Example 1 Assume that x, y, and z are positive numbers. Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (a) (b) (c)
82 Section 3.3 Example 2 Assume that x, y, z, and b are positive numbers. Use the properties of logarithms to condense the expression to the logarithm of a single quantity. (a) (b) Most calculators only have ln and log keys. So how should we approximate calculator when a is neither e nor 10? of a value on a Change-of-Base Formula Change to exponential form: Derivation of Change-of-Base Formula: Let Take the of both sides : Apply Power Property: Solve for y : Example 3 Use the Change-of-Base Formula and a calculator to evaluate the following logarithm. Round your answer to three decimal places.
Section 3.4 83 3.4 Exponential and Logarithmic Equations Strategies for Solving Exponential and Logarithmic Equations 1. If possible, rewrite the original equation in a form that allows the use of the One-to-One Property of exponential or logarithmic functions. Example 1 Example 2 Solve: Solve: 2. To solve an exponential equation, take the log or ln of both sides. Note it may be necessary to isolate the exponential term first. Then apply the Power Property of logarithms. To solve a logarithmic equation, condense all logarithms into a single logarithm on one side of the equation, and then rewrite the equation in exponential form. Remember to check a logarithmic equation for extraneous solutions!! Example 3 Example 4 Solve: Solve:
84 Section 3.4 Example 5 Solve.
Section 3.5 85 3.5 Exponential and Logarithmic Models Example 1 An initial investment of $10,000 in a savings account in which interest is compounded continuously doubles in 12 years. What is the annual % rate? Example 2 Determine the time necessary for $100 to triple if it is invested at 4.5% compounded monthly.
86 Section 3.5 Law of Exponential (or Uninhibited) Growth/Decay Many natural phenomena have been found to follow the law that an amount according to the models (growth) or (decay) varies with time where is the growth or decay rate, and is the initial quantity at time. Doubling time = Time for a population to double. Half-life = Time for a radio-active substance to decay to half the original amount. Example 3 The radioactive isotope decays exponentially with a half-life of 1599 years. After 1000 years, only 1.5 g of a sample remain. What was the initial quantity?
Section 3.5 87 Example 4 The number of bacteria in a culture is modeled by where is the time in hours. If when, estimate the time required for the population to double in size.
88 Section 3.5 Other Applications Example 5 Find the exponential model whose graph passes through the points and. Example 6 Use the acidity model given by where acidity (ph) is a measure of the hydrogen ion concentration for a solution in which. of a solution, to compute
Section 4.5 89 4.5 Graphs of Sine and Cosine Functions One complete revolution of the unit circle is 2, after which values of sine and cosine begin to repeat. The sine and cosine functions are periodic functions with period. Symmetric about Function Symmetric about Function The amplitude of a periodic function is half the distance between the maximum and minimum values of the function. The functions and have amplitude and period. Explanation: Notice that when a is negative, the graph is reflected in the x-axis.
90 Section 4.5 Example 1 Find the period and amplitude. (a) (b) To sketch the graph of * 1. Method #1 Rewrite by factoring out of the parentheses. Use the horizontal shift units and period to find an interval of one period. -OR- Method #2 Solve the 3-part inequality to find an interval of one period. 2. Divide the period-interval into four equal parts to locate the 5 key points on the graph. 3. Either evaluate the function at the 5 key points on the graph, or use your knowledge of the parent function, reflections, the vertical shift units, and the amplitude to complete the sketch of the graph. *The process above may also be used to graph any transformation of.
Section 4.5 91 Example 2 Sketch the graph of. (Include two full periods.) Example 3 Sketch the graph of. (Include two full periods.)
92 Section 4.5 Example 4 After exercising for a few minutes, a person has a respiratory cycle for which the velocity of air flow is approximated by, where is the time (in seconds). (a) Find the time for one full respiratory cycle. (b) Find the number of cycles per minute. (c) Sketch the graph of the velocity function.
Section 4.6 93 4.6 Graphs of Other Trigonometric Functions Let s graph y tan x: x sin x cos x /2 1 0 sin x y tan x cos x undefined 1 0 /3 3 / 2 1/ 2 3 /4 2 / 2 2 / 2 1 2.5 /6 1/ 2 3 / 2 2 0 0 1 /6 1/ 2 3 / 2 1.5 1 /4 2 / 2 2 / 2 0.5 /3 3 / 2 1/ 2 /2 1 0 2 / 3 3 / 2 1/ 2 2 3 4 6 0.5 1 6 2 3 5 7 5 4 3 4 3 2 3 4 6 6 4 3 2 3 / 4 2 / 2 2 / 2 5 / 6 1/ 2 3 / 2 1.5 2 0 1 7 / 6 1/ 2 3 / 2 5 / 4 2 / 2 2 / 2 1 2.5 3 4 / 3 3 / 2 1/ 2 3 / 2 1 0 1 0 undefined Domain: Range: Period: Vertical Asymptotes: Symmetry:
94 Section 4.6 Let s graph y cot x: x sin x cos x 0 1 cos x y cot x sin x undefined 1 0 5 / 6 1/ 2 3 / 2 3 1.7 3 3 / 4 2 / 2 2 / 2 1 2 / 3 3 / 2 1/ 2 2 /2 1 0 /3 3 / 2 1/ 2 1 /4 2 / 2 2 / 2 /6 1/ 2 3 / 2 0 0 1 5 3 2 6 4 3 2 3 4 6 6 2 3 5 4 3 2 3 4 6 /6 1/ 2 3 / 2 1 /4 2 / 2 2 / 2 /3 3 / 2 1/ 2 2 /2 1 0 2 / 3 3 / 2 1/ 2 3 3 / 4 2 / 2 2 / 2 5 / 6 1/ 2 3 / 2 0 1 1 0 undefined Domain: Range: Period: Vertical Asymptotes: Symmetry:
Section 4.6 95 To sketch the graph of * 4. Solve the 3-part inequality to find an interval of one period and establish two consecutive vertical asymptotes. 5. At the midpoint between two consecutive vertical asymptotes is an -intercept of the graph. Plot this point as well as two other points. 6. Sketch at least one additional cycle to the left or right. To sketch the graph of * 1. Solve the 3-part inequality to find an interval of one period and establish two consecutive vertical asymptotes. 2. At the midpoint between two consecutive vertical asymptotes is an -intercept of the graph. Plot this point as well as two other points. 3. Sketch at least one additional cycle to the left or right. *Note that Steps 2 and 3 are the same for both graphs. Example 1 Sketch the graph of.
96 Section 4.6 Example 2 Sketch the graph of. Graphs of the Reciprocal Functions To get the graphs of the reciprocal functions and, we first sketch and then take the reciprocals of the - coordinates. *Note that is undefined whenever and is undefined whenever. 2 3 2 2 0 3 2 2 3 0 2 2 2 2 3 2 2 2
Section 4.6 97 Example 3 Sketch the graph of. Example 4 Sketch the graph of.
98 Section 4.7 4.7 Inverse Trigonometric Functions Sketch a quick graph of. Does the function have an inverse function? However, if we restrict the domain to the interval, then the restricted domain and thus we can define the inverse function! is 1-1 on Recall, in general, if and only if. Inverse Sine Function Restriction: Domain of = Range of = The graph of the inverse sine function is obtained by reflecting the restricted sine function about the line. Sketch the graph of
Section 4.7 99 Example 1 If possible, find the exact value. a) b) c) ******************************************************************************** Sketch a quick graph of. Does the function have an inverse function? However, if we restrict the domain to the interval, then is 1-1 on the restricted domain and thus we can define the inverse function! Inverse Cosine Function Restriction: Domain of = Range of =
100 Section 4.7 The graph of the inverse cosine function is obtained by reflecting the restricted cosine function about the line. Sketch the graph of Example 2 If possible, find the exact value. a) b) c)
Section 4.7 101 ******************************************************************************** Sketch a quick graph of. Does the function have an inverse function? However, if we restrict the domain to the interval, then is 1-1 on the restricted domain and thus we can define the inverse function! Inverse Tangent Function Restriction: Domain of = Range of = The graph of the inverse tangent function is obtained by reflecting the restricted tangent function about the line. Sketch the graph of
102 Section 4.7 Example 3 If possible, find the exact value. a) b) c) Example 4 Use a calculator to approximate the value (if possible). a) b) Example 5 Write as a function of. 5
Section 4.7 103 Example 6 Evaluate the expression. a) b) c) Example 7 Find the exact value of the expression. a) b) c) Example 8 Write an algebraic expression that is equivalent to the expression.
104 Section 5.1 5.1 Using Fundamental Identities Fundamental Trigonometric Identities Reciprocal Identities Quotient Identities Pythagorean Identities Divide by : Divide by : Cofunction Identities Even/Odd Identities
Section 5.1 105 Example 1 Given the values and, evaluate all six trig functions. Method #1) Draw triangle Method #2) Use the identities
106 Section 5.1 Example 2 Simplify: a) b) c) Example 3 Factor, then simplify. a) b)
Section 5.1 107 Example 4 Perform the indicated operation, then simplify. a) b) Example 5 Use the trig. substitution to write the algebraic expression as a trig function of, where,
108 Section 5.2 5.2 Verifying Trigonometric Identities Guidelines for Verifying Trig Identities: 1. Work with one side of the equation at a time. Start with the more complicated side. Keep in mind the goal is to simplify this side to look like the other side of the equation! 2. Look for opportunities to factor an expression, find a common denominator to add fractions, or create a monomial denominator by multiplying numerator and denominator by the same quantity. 3. Use the reciprocal/quotient identities to convert all terms to sines and cosines if this helps simplify the expression. 4. If both sides are complicated, work each side separately to obtain a common 3 rd expression. Example 1 Verify the identity: Example 2 Verify the identity:
Example 3 Verify the identity: Section 5.2 109
110 Section 5.2 Example 4 Verify the identity:
Section 5.3 111 5.3 Solving Trigonometric Equations When solving a trigonometric equation, your primary goal is to isolate the trigonometric function involved in the equation. It may be necessary to do one or more of the following: 1) arrange all terms on one side of the equation and factor 2) rewrite the equation with a single trig function using the identities 3) square both sides and remember to check your answer for extraneous solutions ******************************************************************************** Example 1 Solve the equation Example 2 Solve the equation
112 Section 5.3 Example 3 Find all solutions of the equation in the interval. Example 4 Find all solutions of the equation in the interval.
Section 5.3 113 To solve equations involving trigonometric functions of multiple angles of the forms, first solve the equation for. Then isolate. and Example 5 Solve the equation Example 6 Solve the equation
114 Section 5.4 5.4 Sum and Difference Formulas Sum and Difference Formulas*: *Proofs are on page 424 in the book. Example 1 Find the exact value of a) b)
Section 5.4 115 Example 2 Find the exact value of Example 3 Find the exact value of given that and. (Both and are in Quadrant IV.)
116 Section 5.4 Example 4 Verify the cofunction identity
Section 5.5 117 5.5 Multiple-Angle and Product-to-Sum Formulas Double-Angle Formulas: Derivation of Double-Angle Formula for Sine: (The others are derived similarly.) Example 1 Find using the figure.
118 Section 5.5 Example 2 Solve in the interval. Example 3 Use the following to find.
Section 5.5 119 Power-Reducing Formulas: Derivation of Power-Reducing Formula for Sine: (The others are derived similarly.) Example 4 Rewrite in terms of the first power of cosine.
120 Section 5.5 Half-Angle Formulas*: The sign depends on the quadrant in which lies. *To derive, replace with in the power reducing formulas. Example 5 ind Example 6 ind given
Section 5.5 121 Product-to-Sum Formulas*: *may be verified using the sum and difference formulas in the preceding section Example 7 Write the product as a sum or difference. Sum-to-Product Formulas*: *Proofs are on page 426 in the book. Slightly complicated substitution is required.
122 Section 5.5 Example 8 Write the sum as a product. Example 9 Verify the identity.
Section 6.3 123 6.3 Vectors in the Plane Quantities such as force and velocity involve both magnitude and direction. We represent such quantities by a vector: Terminal point (Tip) Initial point (Tail) Its magnitude (or length) is denoted by and can be found using the distance formula. Vectors have no fixed orientation in the plane. They may be translated up down left or right. All that defines a vector is its direction (angle) and magnitude (length). Equal Not Equal Not Equal Same Magnitude Different Magnitudes Different Directions Same Direction When we position a vector in the plane such that its initial point (or tail) is at the origin, we say that the vector is in standard position. (Many books call such a vector a position vector). The component form of a vector is: where and is the horizontal component is the vertical component. Note: A vector with initial point and terminal point has component form and magnitude If, then is called a unit vector. If, then is called the zero vector and denoted.
124 Section 6.3 Example 1 Find the component form and the magnitude of the vector with initial point and terminal point. The negative (or opposite) of is the vector which has the same magnitude, but opposite direction. The scalar multiple of times is the vector which has magnitude times the magnitude of. The sum of and is the vector, often referred to as the resultant vector, which is formed by placing the initial point of at the terminal point of ( tip to tail ). This is represented using the parallelogram law: Note: The difference of and is the vector
Section 6.3 125 Example 2 Let and. Find and sketch the following: (a) (b) (c) Properties of Vector Addition and Scalar Multiplication Let, and be vectors and let and be scalars. Then the following properties are true. 1. 2. 3. 4. 5. 6. 7. 8. and 9. If we divide a vector by its magnitude, the resulting vector has a magnitude of 1 and the same direction as. The vector is called a unit vector in the direction of. Example 3 Find a unit vector in the direction of.
126 Section 6.3 The unit vectors and are called the standard unit vectors and are denoted by and Sketch and : In fact, any vector may be represented as a linear combination of the standard unit vectors as follows: Example 4 Sketch the vector. Then write the component form of. Example 5 Let and. Find.
Section 6.3 127 If is a unit vector in standard position, then it s tip lies on the unit circle. Label the diagram: The angle is the direction angle of the vector. If is any vector that makes an angle with the positive -axis, then is the scalar multiple of times, where is the unit vector with direction angle. Note: If has direction angle, then.
128 Section 6.3 Example 6 Find the magnitude and direction angle of the vector: a) b) Example 7 Find the component form of given its magnitude and the angle it makes with the positive -axis. and Example 8 Find the component form of the sum of and with direction angles and.,,
Section 6.4 129 6.4 Vectors and Dot Products Definition The dot product of and is Example 1 Find the dot product of and. Properties of the Dot Product 1. 2. 3. *There are other properties listed in the book, but we will only study these ones. Proof of Property 1: Proof of Property 2: Proof of Property 3: Example 2 Use the dot product to find the magnitude of.
130 Section 6.4 The Angle Between Two Vectors If is the angle ( ) between two nonzero vectors and, then Or alternatively, *The proof is on page 492 in the book and uses Law of Cosines. Example 3 Find the angle between and. Example 4 Use vectors to find the interior angles of the triangle with the given vertices:
Section 6.4 131 Note, if the angle between and is a right angle (or ), then Definition The vectors and are orthogonal if Example 5 Determine whether and are orthogonal, parallel, or neither. *Note: Two vectors are parallel if one vector can be written as a scalar multiple of the other. a) and b) and
132 Section 6.5 6.5 Trigonometric Form of a Complex Number The Complex Plane We may represent any complex number as the point in the complex plane as follows: Plot: (a) (b) Definition The absolute value of the complex number is Example 1 Find the absolute value of.
Section 6.5 133 Trigonometric Form of a Complex Number Find expressions for and in terms of and. The trigonometric (or polar) form of the complex number is The number is the of, and is called an of. Note: The trigonometric form is not unique since there are infinitely many choices for. However normally is given in radians and restricted to the interval.
134 Section 6.5 Example 2 Write the complex number in trigonometric form. a) b) Example 3 Write the complex number in standard form.
Section 6.5 135 Suppose we are given two complex numbers: and Find the product. Product and Quotient of Two Complex Numbers Let and be complex numbers. In words, to find the product,. In words, to find the quotient,.
136 Section 6.5 Example 4 Perform the operation and leave the result in trigonometric form. Powers of Complex Numbers Let s apply the product rule to find powers of.
Section 6.5 137 DeMoivre s Theorem If is a complex number and is a positive integer, then Example 5 Use DeMoivre s Theorem to find. Write the result in standard form. The complex number is an th root of the complex number if Example 6 Find all the fourth roots of 1.
138 Section 6.5 In general th Roots of Unity The distinct th roots of 1 are called the th roots of unity. The possible values for here are the th roots of unity. ******************************************************************************** To find the th roots of unity: Example 7 Find all the cube roots of unity.
Section 7.1 139 7.1 Linear and Nonlinear Systems of Equations We can use substitution to solve both linear and nonlinear systems. Graphically, the solutions are the points of intersection of the two graphs. Example 1 Solve by substitution. Sometimes it is convenient to solve a system graphically when it is difficult to solve by substitution. Example 2 Solve by graphing.
140 Section 7.1 Break-Even Analysis The total cost of producing units is given by the equation The total revenue from selling units is given by the equation Both of these equations describe a line. The break-even point occurs when, that is revenue cost, which corresponds to the point of intersection of the two graphs. *Note: Profit Revenue Cost Example 3 A restaurant invests $5000 to produce a new food item that will sell for $3.49. Each item can be produced for $2.16. (a) How many items must be sold to break even? (b) How many items must be sold to make a profit of $8500?
Section 7.2 141 7.2 Two-Variable Linear Systems A system of two linear equations containing two variables represents a pair of lines. The points of intersection are the solutions of the system. Intersect at exactly one point Parallel They are the same line One solution No solution Infinite number of solutions Consistent system Independent system Inconsistent system Consistent system Dependent system We can use elimination to solve linear systems fast and efficiently. Example 1 Solve Example 2 Solve
142 Section 7.2 Example 3 Solve Applications: Wind Problems: Speed of plane with the wind Plane airspeed Wind speed Speed of plane against the wind Plane airspeed Wind speed Example 4 An airplane flying into a headwind travels 250 miles in 2 hours and 25 minutes. The return flight takes only 2 hours. Find the airspeed of the plane and the speed of the wind.
Section 7.2 143 In a free market, the demand for a product is related to the price of the product. As price decreases, demand increases and thus the amount producers are able or willing to supply decreases. The equilibrium point is the price and number of units that satisfy both the demand and supply equations. Price per unit, Equilibrium Point Supply Demand Example 5 Demand equation: Supply equation: Find the equilibrium point. Number of units,
144 Section 7.3 7.3 Multivariable Linear Systems The following is an example of a multivariable linear system: Refer to Sections 8.1, 8.3 and 8.5 for matrix methods used to solve multivariable linear systems.
Section 7.4 145 7.4 Partial Fractions We call the sum on the RHS a partial fraction decomposition of the rational expression on the LHS. How to Decompose a Rational Expression into Partial Fractions: Step 1 If the rational expression is improper (ie. the degree of numerator the degree of denominator) first divide using long division. Then proceed to decompose the proper rational expression. Step 2 Completely factor the denominator into linear and irreducible (non-factorable) quadratic factors. Step 3 For each linear factor of the form, include in the decomposition Step 4 For each irreducible quadratic factor of the form, include in the decomposition Step 5 Proceed to find the coefficients in the numerators as follows: If the basic equation contains only linear factors 1. Substitute the zeros of the distinct linear factors into the basic equation. 2. For repeated linear factors, use the coefficients determined in the previous step to rewrite the basic equation. Then substitute other convenient values of and solve for the remaining coefficients. If the basic equation contains any quadratic factors 1. Expand the basic equation. 2. Collect terms according to power of. 3. Equate the coefficients of like terms to obtain equations involving and so on. 4. Use a system of linear equations to solve for
146 Section 7.4 In the following example, we will practice Steps 2 through 4 only of the decomposition process. Once we have mastered setting up partial fraction decomposition problems, we will be ready to move on to Examples 2-5 where we will complete the entire process. Example 1 Write the form of the partial fraction decomposition of the rational expression. Do not solve for the constants.
Section 7.4 147 Example 2 (Distinct Linear Factors) Write the partial fraction decomposition of the rational expression.
148 Section 7.4 Example 3 (Repeated Linear Factors) Write the partial fraction decomposition of the improper rational expression.
Section 7.4 149 Example 4 (Distinct Linear & Quadratic Factors) Write the partial fraction decomposition of the rational expression.
150 Section 7.4 Example 5 (Repeated Quadratic Factors) Write the partial fraction decomposition of the rational expression.
Section 7.5 151 7.5 Systems of Inequalities How to Sketch the Graph of an Inequality: 1) Graph the boundary line by replacing the inequality with an equal sign. Use a dashed or solid line for the boundary as follows: 2) Shade the correct region. (One method for shading: Test a point not on the line in the original inequality. If a true statement results, shade the region containing the test point. If a false statement results, shade the OTHER region.) Example 1 Sketch the graph of the inequality
152 Section 7.5 How to Sketch the Graph of a System of Inequalities: 1) Sketch the graph of each inequality separately. 2) The solution region is the region that is common to every graph in the system. (Note: If there is no common region, the system of inequalities has no solution) 3) Find and label the vertices of the solution region and indicate each with either an open or closed dot as appropriate. WARNING: There is a mistake in many of the answers to HW problems in the back of the book. The author of the answer key forgot to include open dots when a vertex should not be part of the solution. (Hint: To find the vertices of the solution region, solve the systems of corresponding equations obtained by taking pairs of equations representing the boundary of the solution region.) Example 2 Sketch the graph (and label the vertices) of the solution set of the system.
Example 3 Sketch the graph (and label the vertices) of the solution set of the system. Section 7.5 153
154 Section 8.1 8.1 Matrices and Systems of Equations If and are positive integers, an matrix is a rectangular array in which each entry,, of the matrix is a number. A matrix having rows and columns is said to be of order. Matrices are usually denoted by capital letters. The plural of matrix is matrices. If a matrix has the same number of rows as columns, then the matrix is square of order. For a square matrix, the entries are the main diagonal entries. Circle the main diagonal entries on the square matrices given below: square matrix: square matrix: A matrix that has only one row is called a row matrix, and a matrix that has only one column is called a column matrix. Example 1 Determine the order of each matrix.
Section 8.1 155 Consider the following system of equations: System: A matrix derived from a system of linear equations is the augmented matrix of the system. Augmented Matrix: The matrix derived from the coefficients of the system (but not including the constant terms) is the coefficient matrix of the system. Coefficient Matrix: Example 2 Write the system of equations corresponding to the augmented matrix.
156 Section 8.1 Row-Echelon Form and Back Substitution Consider the following two systems of linear equations. Write the corresponding augmented matrix for each system. Augmented Matrices: The second system is said to be in row-echelon form, which means that the coefficient matrix has a stair-step pattern with leading coefficients of 1. A matrix is in row-echelon form when the following conditions are met: 1. The first nonzero entry in each row (called the lead entry) is 1. 2. Lead entries appear farther to the right as we move down the rows of the matrix. 3. Any rows containing only 0 s are at the bottom of the matrix Examples: It may be verified that both systems above have the same solution,, and, which can be written as the ordered triple. Two systems of equations are equivalent if they have the same solution set. Hence, the two systems given above are equivalent. After comparing the two systems, it should be clear that it is easier to solve the system given in row-echelon form! To solve a system that is not in row-echelon form, first convert it to an equivalent system that is in row-echelon form by using the following operations. Elementary Row Operations Each of the following row operations performed on the augmented matrix of a system of linear equations produces an equivalent system of linear equations. 1. Interchange two rows. 2. Multiply a row by a nonzero constant. 3. Add a multiple of a row to another row.
Section 8.1 157 Consider the system of equations: 1. Interchange two rows: This corresponds to the following move involving the augmented matrix: 3 / 2 7 / 2 1 1 2 1 r r 1 2 2. Multiply a row by a nonzero constant. This corresponds to the following move involving the augmented matrix: 1 2 1 3 / 2 7 / 2 1 R 2r 2 2 3. Add a multiple of a row to another row. This corresponds to the following move involving the augmented matrix: 1 2 1 3 7 2 R 3r r 2 1 2 After performing the above three row operations, we get an equivalent system in row-echelon form. Now solve by back-substitution:
158 Section 8.1 To solve a system of linear equations using matrices, we perform elementary row operations on the augmented matrix to obtain a matrix that is in row-echelon form. This method of solving a system of equations is called Gaussian elimination with back-substitution: Step 1: Step 2: Step 3: Write the augmented matrix of the system of linear equations. Use elementary row operations to rewrite the augmented matrix in rowechelon form. Write the system of linear equations corresponding to the matrix in rowechelon form, and use back-substitution to find the solution. Example 3 Use matrices and the method of Gaussian elimination to solve the system of equations. 3x 3y 3 8 4x 2y 3
Section 8.1 159 How to Solve a System of Linear Equations using Gaussian Elimination: Step 1 Write the augmented matrix that represents the system. Step 2 Perform row operations to get a leading 1 in the first row, first column entry. 1 a b i c d e j f g h k Step 3 Perform row operations to get a column of 0 s below the first leading 1, while leaving the first leading 1 unchanged. 1 a b i 0 j k n 0 l m o Step 4 Perform row operations to get a leading 1 in the second row, second column entry. (If this entry is already 0, then get a leading 1 in the third column of this row.) 1 a b i 0 1 p q 0 l m o Step 5 Perform row operations to get a column of 0 s below the second leading 1, while leaving the previous 1 s and 0 s unchanged. 1 a b i 0 1 p q 0 0 r s Step 6 Now repeat step 4, placing a 1 in the next row, but one column to the right. Continue until the bottom row or the vertical bar is reached. 1 a b i 0 1 p q Row Echelon Form 0 0 1 t Step 7 Write the system of equations and solve.
160 Section 8.1 Example 4 Use matrices and the method of Gaussian elimination to solve the system of equations. 2x y 4 2 y 4z 0 3x 2z 11
Example 5 Use matrices and the method of Gaussian elimination to solve the system of equations. Section 8.1 161 Example 6 Use matrices and the method of Gaussian elimination to solve the system of equations. 2x 3y z 0 x 2y z 5 3x 4 y z 1
162 Section 8.1 A modification of Gaussian elimination, called Gauss-Jordan elimination, uses row operations to produce a matrix in reduced row-echelon form. With this method, we can obtain the solution of the system from that matrix directly, and back substitution is not needed. A matrix is in reduced row-echelon form when the following conditions are met: 1. It is in row echelon form. 2. Entries above each lead entry are also zero. Examples: To get a matrix in reduced row echelon form requires the extra step of getting 0 s above and not just below leading 1 s. Example 7 Use matrices and the method of Gauss-Jordan elimination to solve the system of equations.
Example 8 Use matrices and the method of Gauss-Jordan elimination to solve the system of equations. Section 8.1 163
164 Section 8.2 8.2 Operations with Matrices In Section 8.1, we used matrices to solve systems of linear equations. There is a rich mathematical theory of matrices, and its applications are numerous. This section and the sections that follow introduce some fundamentals of matrix theory. Matrix Representation A matrix can be denoted by: 1. an uppercase letter such as. 2. a representative element enclosed in brackets, such as. 3. a rectangular array of numbers. Two matrices and are equal if they have the same order and their corresponding entries are equal, that is. Example 1 Find and Definition of Matrix Addition If and are matrices of order, their sum is the matrix given by. In other words, add two matrices by adding their corresponding entries. The sum of two matrices of different orders is undefined. Example 2 (a)
Section 8.2 165 (b) In operations with matrices, numbers are usually referred to as scalars. Definition of Scalar Multiplication If is an matrix and is a scalar, the scalar multiple of by is the matrix given by In other words, multiply a matrix by a scalar by multiplying each entry in by. Note: The symbol represents the scalar product. Thus, Example 3 Suppose that and. Find the following: (a) (b)
166 Section 8.2 Properties of Matrix Addition and Scalar Multiplication Let and be matrices and let and be scalars. 1. 2. 3. 4. 5. 6. If is an matrix and is the zero matrix consisting entirely of zeros, then. zero matrix: and zero matrix: Example 4 Solve for in the matrix equation, where and
Section 8.2 167 Definition of Matrix Multiplication If is an matrix and is an matrix, the product is an matrix where. In other words, the entry in the th row and th column of the product is obtained by multiplying the entries in the th row of by the corresponding entries in the th column of and then adding the results. We can think of this as a dot product of rows of by columns of : Example 5 and If possible, find each of the following products. (a) (b)
168 Section 8.2 Example 6,,, and If possible, find each of the following products and state the order of the result. (a) (b) (c)
Section 8.2 169 Properties of Matrix Multiplication Let and be matrices and let be a scalar. 1. 2. 3. 4. Note: In general,. Definition of Identity Matrix The square matrix that consists of 1 s on its main diagonal and 0 s elsewhere is called the identity matrix of order and is denoted by. When the order is understood from the context, you can denote the identity matrix simply by. ; Example 7 If, find the following: (a) (b) The identity matrix has the property that.
170 Section 8.4 8.4 The Determinant of a Square Matrix Determinant of a Matrix The determinant of the matrix is given by *Note: The determinant of a square matrix is a real number. Example 1 Find the determinant of each matrix. (a) (b) Example 2 Solve for.
Section 8.4 171 To define the determinant of a square matrix of order the concepts of minors and cofactors. or higher, it is convenient to introduce Minors and Cofactors of a Square Matrix If is a square matrix, the minor of the entry is the determinant of the matrix obtained by deleting the th row and th column of. The cofactor of the entry is equal to the minor with the correct sign attached. Sign pattern for cofactors (in case): Example 2 Find all the minors and cofactors of
172 Section 8.4 Determinant of a Square Matrix If is a square matrix (of order or greater), the determinant of is the sum of the entries in any row (or column) of multiplied by their respective cofactors. For instance, expanding along the first row yields Applying this definition to find a determinant is called expanding by cofactors. Example 3 Find the determinant of the matrix. Expand using the indicated row or column. (a) Row 1 (b) Column 2
Example 4 Find the determinant of the matrix. Section 8.4 173
174 Section 8.3 8.3 The Inverse of a Square Matrix Definition of the Inverse of a Square Matrix Let be an matrix and let be the identity matrix. If there exists a matrix such that Then is called the inverse of, and is denoted with the symbol. *********************************************************************************** Note: Only square matrices can have an inverse. However, not all square matrices have an inverse. If a matrix has an inverse, it is called invertible (or ). Otherwise, if a matrix does not have an inverse, it is called. Example 1 Show that is the inverse of A.,
Section 8.3 175 Finding an Inverse Matrix If is, the elementary row operations performed on the augmented matrix will transform it into. If it is not possible to row reduce to, then is not invertible and thus does not have an inverse. Refer to Example 2 (pg. 600) in the book for the rationale of this method. Example 2 Find the inverse of the matrix (if it exists).
176 Section 8.3 Example 3 Find the inverse of the matrix (if it exists). Finding the Inverse of a (Using the Formula) Matrix If is a matrix given by the inverse can be found using the formula Note: If, then does not have an inverse. What is the determinant of the matrix in Example 3?
Section 8.3 177 Example 4 Find the inverse of the matrix (if it exists). Use the formula. Example 5 Find the inverse of the matrix (if it exists). Use the formula.
178 Section 8.3 Consider the system of linear equations: The system can be represented by the matrix equation, where is the coefficient matrix of the system, and and are column matrices. The column matrix is called a constant matrix. Verify that this matrix equation represents the system of linear equations. Thus, solving the matrix equation for will yield the solution to the system of equations. Solve for in the matrix equation: Solving a System of Equations Using an Inverse Matrix If is an invertible matrix, the system of linear equations represented by has a unique solution given by
Example 6 Use the inverse matrix found in Example 2 to solve the system of linear equations. Section 8.3 179
180 Section 8.5 8.5 Applications of Matrices and Determinants So far, we have studied 2 methods for solving a system of linear equations using matrices: 1) Gaussian (or Gauss-Jordan) Elimination performed on the augmented matrix 2) Using an Inverse Matrix to solve the corresponding matrix equation In this section, we will study one more method, Cramer s Rule, named after Gabriel Cramer (1704-1752). Consider the system of 2 equations We define the following determinants: *********************************************************************************** Now, for a system of 3 equations we have the following:
Section 8.5 181 Cramer s Rule If a system of linear equations in variables has a coefficient matrix with a nonzero determinant, the solution of the system is If the determinant of the coefficient matrix is zero (ie. ), the system has either no solution or infinitely many solutions. In such a case, Cramer s Rule does not apply, and another technique should be used to solve the system. Example 1 Use Cramer s Rule to solve the system of linear equations.
182 Section 8.5 Example 2 Use Cramer s Rule to solve the system of linear equations. Hint: Use the following information to save time on this problem.
Section 8.5 183 Another application of determinants is finding the area of a triangle. Area of a Triangle The area of a triangle with vertices,, and is where the appropriate sign should be chosen to yield a positive area. Example 3 Find the area of the triangle.
184 Section 9.1 9.1 Sequences and Series Definition of Sequence An infinite sequence is a function whose domain is the set of positive integers. The function values are the terms of the sequence. If the domain of the function consists of the first only, the sequence is a finite sequence. positive integers Examples: (finite sequence) * (infinite sequence) *This last sequence is known as the Fibonacci sequence. Note: On occasion it is convenient to begin subscripting a sequence with 0 instead of 1 so that the terms of the sequence become When this is the case, the domain includes 0. Example 1 Write the first four terms of the sequence. (Assume that begins with 1.) (a) (b)
Example 2 Write an expression for the apparent th term of the sequence. (Assume that begins with 1) (a) Section 9.1 185 (b) (c)
186 Section 9.1 A sequence can be defined recursively by giving one or more of the first few terms and a rule showing how to obtain the next term from the previous term(s). Example 3 Find the first five terms of the sequence defined recursively. Definition of Factorial If is a positive integer, factorial is defined as As a special case, zero factorial is defined as
Section 9.1 187 Example 4 Write the first five terms of the sequence. (Assume that begins with 0.) Note, based on the definition of factorial: Example 5 Simplify the factorial expression.
188 Section 9.1 The sum of the first Summation Notation (or Sigma Notation) terms of a sequence is represented by where is called the index of summation, the lower limit of summation. is the upper limit of summation, and in this case 1 is Examples: Note that the lower limit of a summation does not have to be 1. Also note that the index of summation does not have to be the letter. Example 6 Find each sum.
Section 9.1 189 Example 7 Use sigma notation to write the sum Note: The representation of a sum using sigma notation is not unique. Variations in the upper and lower limits of summation can produce quite different-looking summation notations for the same sum. Example 8 Find two representations for the following sum (a) Use 1 as the lower limit of summation. (b) Use 0 as the lower limit of summation.
190 Section 9.1 Definition of Series Consider the infinite sequence 1. The sum of the first terms of the sequence is called a finite series (or the nth partial sum of the sequence) and is denoted by 2. The sum of all the terms of the infinite sequence is called an infinite series and is denoted by Example 9 Use a calculator to find the sum of the following finite series. Based on your answer in Example 7, can you guess the following sum of the infinite series?
Section 9.2 191 9.2 Arithmetic Sequences and Partial Sums Consider the sequence: 2, 5, 8, 11, 14, Compute the differences of consecutive terms: Each term (except the first term) is found by adding the constant to the preceding term. An arithmetic sequence is a sequence of the form where there is a common difference between consecutive terms. The th term of an Arithmetic Sequence Example 1 Write the first five terms of the arithmetic sequence with and. Then find a formula for the th term of the sequence.
192 Section 9.2 Example 2 Find the 100 th term of an arithmetic sequence with and. Example 3 Write the first five terms of the arithmetic sequence defined recursively: Then find the common difference and write the th term of the sequence as a function of.
Section 9.2 193 The Sum of a Finite Arithmetic Sequence (nth partial sum) Let represent the sum of the first terms of an arithmetic sequence. Then, Note: This formula only works for arithmetic sequences! Historical Note: An elementary school teacher of a young Carl Friedrich Gauss (1777-1855) asked him to add all the integers from 1 to 100. When Gauss returned with the correct answer after only a few moments, the teacher could only look at him in astounded silence. This is what it is speculated that Gauss did:
194 Section 9.2 Example 4 Find the 10 th partial sum of the arithmetic sequence Example 5 Find the partial sum.
Section 9.2 195 Example 6 Determine the seating capacity of a movie theater with 36 rows of seats if there are 15 seats in the first row, 18 seats in the second row, 21 seats in the third row, and so on.
196 Section 9.3 9.3 Geometric Sequences and Series Consider the sequence: 2,,,,, Compute the ratios of consecutive terms: Each term (except the first term) is found by multiplying the preceding term by the constant. A geometric sequence is a sequence of the form where there is a common ratio between consecutive terms. The th term of a Geometric Sequence
Section 9.3 197 Example 1 Write the first five terms of a geometric sequence with and. Then find a formula for the th term of the sequence. Example 2 Write the first five terms of the geometric sequence. Determine the common ratio and write the th term of the sequence as a function of.
198 Section 9.3 Example 3 Find the 6 th term of a geometric sequence if and. Assume that the terms of the sequence are positive. Method #1 (Brute Force) Method #2 (Algebraically)
Section 9.3 199 The Sum of a Finite Geometric Sequence ( th Partial Sum) Let represent the sum of the first terms of a geometric sequence. Then for, Note: This formula only works for finite geometric sequences! Example 4 Find the sum of each finite geometric sequence.
200 Section 9.3 Let sequence. The Sum of an Infinite Geometric Series represent the sum of all the terms of an infinite geometric Then for, *Note: If, the series does not have a sum. Note: This formula only works for infinite geometric sequences! Example 5 Find the sum of each infinite geometric series.
Section 9.3 201 Recall, the balance in an account after years due to a principal invested at an annual interest rate (in decimal form) compounded times per year is: If a deposit is compounded monthly, then, so the balance in the account after months ( years) is given by Example 6 A deposit of $50 is made at the beginning of each month* in an account that pays 8%, compounded monthly. The balance in the account at the end of 4 years (or 48 months) is Find. *This type of savings plan is called an increasing annuity.
202 Section 9.4 9.4 Mathematical Induction The Principle of Mathematical Induction If a statement involving the positive integer has the following two properties: 1. The statement is true for, and 2. If the statement is true for, then it is true for then the statement is true for all positive integers. How to Prove a Statement using Mathematical Induction 1. Base Step: Show the statement is true for. 2. Induction Step: Assume the statement is true for some integer. Show that this implies the statement is true for the next integer. Example 1 Use induction the prove that the following formula is true for every positive integer. Base Step: Induction Step:
Section 9.4 203 Example 2 Use induction the prove that the following formula is true for every positive integer. Base Step: Induction Step:
204 Section 9.4 Example 3 Use induction the prove that the following formula is true for every positive integer. Base Step: Induction Step:
Section 9.5 205 9.5 The Binomial Theorem In this section, we learn how to expand binomials of the form. Observe: Notes: 1) In each expansion there are terms. 2) In each expansion, and have symmetrical roles. The powers of by in successive terms, whereas the powers of 3) The sum of the powers of each terms is. by. 4) The coefficients increase and then decrease in a symmetrical pattern as represented by illustrated below: *Each entry (other than the 1 s) is the sum of the closest pair of numbers in the line above it. If is any positive number, then The Binomial Theorem where the coefficients, called binomial coefficients, are defined by and may either be found using the nth row of Pascal s Triangle or computed on a scientific calculator using the feature.
206 Section 9.5 Note: The Binomial Theorem may be expressed using sigma notation. Also, the symbol place of to denote binomial coefficients. is often used in Hence, the following is an alternative form of the Binomial Theorem: The Binomial Theorem (using Sigma Notation) Example 1 Use the Binomial Theorem to expand and simplify the expression (a) (b)
Example 2 Use the Binomial Theorem to expand the complex number. Simplify your result. Section 9.5 207
208 Section 10.2 10.2 Introduction to Conics: Parabolas A conic section (or conic) is the intersection of a plane and a double-napped cone. There are 4 basic types: Geometric properties of conics were discovered during the classical Greek period (600 to 300 B.C.) When the plane passes through the vertex, the resulting figure is a degenerate conic. The result is either a point, line or two intersecting lines. General Equation of Conics For example, a circle is defined as the collection (locus) of all points fixed point. that are equidistant from a which is equivalent to
Section 10.2 209 We have already discussed quadratic functions whose graphs are parabolas that open up or down. They can be written in the form: where is the vertex. We now discuss parabolas in a more general sense, where they may open up, down, left or right. Definition of Parabola A parabola is the set (locus) of all points in a plane equidistant from a fixed line (called the directrix) and a fixed point (called the focus) that is not on line. The vertex and axis (of symmetry) are defined as before. Sketch of a parabola using the definition:
210 Section 10.2 Activity Let s derive the equation of a parabola (using the definition)! For simplicity, we will derive the equation of a parabola with: Vertex, Focus, Directrix, Directrix 1. The distance from to, denoted, is found using the distance formula: 2. The distance from to the directrix is. 3. By definition of a parabola, the distances found in steps (1) and (2) are equal. Thus, 4. Now solve for (in terms of and ). Simplify your answer by FOILing.
Section 10.2 211 Standard Equation of a Parabola with Vertex : Opens Up or Down : Opens Right or Left : Focus: Focus: Directrix: Directrix: Axis of Symmetry: Vertical, -axis Axis of Symmetry: Horizontal, -axis distance from vertex to focus distance from vertex to directrix. Example 1 Find the standard equation of the parabola with vertex at the origin and focus. Example 2 Find the vertex, focus, and directrix of the parabola and sketch its graph.
212 Section 10.2 Translations can be applied to parabolas with vertex to obtain parabolas with vertex : Standard Equation of a Parabola with Vertex : Opens Up or Down : Opens Right or Left : Focus: Directrix: Axis of Symmetry: Vertical, distance from vertex to focus Focus: Directrix: Axis of Symmetry: Horizontal, distance from vertex to directrix. Example 3 Find the standard equation of the parabola with vertex at and directrix.
Example 4 Find the vertex, focus, and directrix of the parabola and sketch its graph. Section 10.2 213
214 Section 10.3 10.3 Ellipses An ellipse is the set of all points in a plane such that the sum of the distances from to two distinct fixed points and (called the foci) is constant. where: 1/2 the length of the major axis distance from the center to a vertex 1/2 the length of the minor axis distance from the center to a focus Label,, and on the following ellipse: Interesting Fact! Any light or sound that starts at one focus of an ellipse will be reflected through the other. Some examples: Whispering galleries (Statuary Hall in Washington D.C. originally chamber of House of Reps) Lithotripsy kidney stone at one focus is pulverized by shock waves originating at the other focus.
Section 10.3 215 Standard Equation of an Ellipse with Center Center:, Foci:, Vertices: Center:, Foci:, Vertices: Major axis: Horizontal (along -axis) Major axis: Vertical (along -axis) Example 1 Find the equation of the ellipse with center at the origin, major axis of length 10 along the -axis, and minor axis of length 8.
216 Section 10.3 Example 2 Find the equation of the ellipse with center at the origin, focus at, and vertex at. Standard Equation of an Ellipse with Center Center:, Foci:, Vertices: Center:, Foci:, Vertices: Major axis: Horizontal (parallel to -axis) Major axis: Vertical (parallel to -axis)
Section 10.3 217 Example 3 Find the standard form of the equation of the ellipse with focus at and vertices at and. Example 4 Find the center, vertices and foci of the ellipse, and sketch its graph.
218 Section 10.3 Example 5 Find the center, vertices and foci of the ellipse, and sketch its graph.
Section 10.4 219 10.4 Hyperbolas A hyperbola is the set of all points in a plane such that the difference of the distances from to two distinct fixed points and (called the foci) is a positive constant. where: distance from the center to a vertex 1/2 the length of the conjugate axis distance from the center to a focus Label,, and on the following hyperbola:
220 Section 10.4 Standard Equation of a Hyperbola with Center : Center:, Foci:, Vertices: Center:, Foci:, Vertices: Transverse axis: Horizontal (along -axis) Transverse axis: Vertical (along -axis) Asymptotes: Asymptotes: Example 1 Find the standard equation of the hyperbola with center at the origin, vertices and and foci and.
Section 10.4 221 Standard Equation of a Hyperbola with Center Center:, Foci:, Vertices: Center:, Foci:, Vertices: Transverse axis: Horizontal (parallel to -axis) Transverse axis: Vertical (parallel to -axis) Asymptotes: Asymptotes: Example 2 Find the standard equation of the hyperbola with vertices and and foci and.
222 Section 10.4 Example 3 Find the center, vertices, foci, and the equations of the asymptotes of the hyperbola, and sketch its graph. Example 4 Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. a. b. c. d.
Example 5 Find the center, vertices and foci of the hyperbola, and sketch its graph. Section 10.4 223
224 Section 10.6 10.6 Parametric Equations Consider the following path followed by a little ant on the ground who discovers a piece of cake and then proceeds to bring a piece back to the ant hill: Ant Hill Piece of Cake The ant follows the parabolic path, (Rectangular Equation) However, this equation does not tell the whole story. Although it does tell you where the ant has been, it doesn t tell you when the ant was at a given point on the path. To determine this time, we can introduce a third variable, called a parameter. It is possible to write both and as functions of to obtain the parametric equations ill in the table to find the ant s position at time (in minutes). 4 6 How long does it take the ant to reach the piece of cake from his starting point? How long does it take the ant to travel from the piece of cake to the ant hill?
Section 10.6 225 If and are continuous functions of on an interval, the set of ordered pairs is a plane curve. The equations and are parametric equations for, and is the parameter. Plotting the points in the order of increasing values of traces the curve in a specific direction. This is called the orientation of the curve. Example 1 Consider the parametric equations a) Sketch the curve represented by the parametric equations (indicate the orientation of the curve) b) Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Hint: Solve for in one equation. Then substitute in other equation & simplify. c) Adjust the domain of the resulting rectangular equation, if necessary.
226 Section 10.6 The next example illustrates one of the most useful applications of parametric equations. If we let the parameter be the angle, we may describe and graph circular and elliptical curves with ease. To eliminate the parameter in equations involving trigonometric functions, try using the identity Example 2 Consider the parametric equations Fill in the table of values. Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve. Sketch the curve represented by the parametric equations (indicate the orientation of the curve).
Section 10.6 227 Example 3 Consider the parametric equations Sketch the curve represented by the parametric equations (indicate the orientation of the curve). Eliminate the parameter and write the corresponding rectangular equation whose graph represents the curve.
228 Section 10.7 10.7 Polar Coordinates Rectangular Coordinate System -axis Polar Coordinate System -axis Pole Polar axis Each point in the plane can be assigned polar coordinates as follows: directed distance from to, counterclockwise from polar axis to segment directed angle Polar axis Example 1 Plot the point given in polar coordinates. a. b. c.
Section 10.7 229 Example 2 Find three additional polar representations of the point using. Because lies on a circle of radius, it follows that:,, Coordinate Conversion The polar coordinates are related to the rectangular coordinates as follows. Polar-to-Rectangular Rectangular-to-Polar Example 3 Convert the point given in polar coordinates to rectangular coordinates.
230 Section 10.7 Example 4 Convert the point given in rectangular coordinates to polar coordinates. a. b. To convert a rectangular equation to polar form, simply replace by and by. Example 5 Convert the rectangular equation to polar form.
Section 10.7 231 To convert a polar equation to rectangular form requires a little more ingenuity. Example 6 Convert the polar equation to rectangular form. a. b. c. d.
232 Section 10.8 10.8 Graphs of Polar Equations Polar Equation Form: Example 1 Sketch the graph of the polar equation A table of values has been provided to save time. 4 3.5 2 0
Section 10.8 233 Polar Equation Form: Example 2 Sketch the graph of the polar equation Restriction on A table of values has been provided to save time.
234 Section 10.8 Polar Equation Form: Example 3 Sketch the graph of the polar equation A table of values has been provided to save time. 1 0.5 0 0 1
Section 10.8 235 Polar Equation Form: Example 4 Sketch the graph of the polar equation A table of values has been provided to save time. 4 3.7 3 2 1 0.3 0
236 Section 10.8 Polar Equation Form: Example 5 Sketch the graph of the polar equation A table of values has been provided to save time. 0 3.5 5 3.5 0