1 Class Notes for MATH 2 Precalculus Fall 2012 Prepared by Stephanie Sorenson
2 Table of Contents 1.2 Graphs of Equations Functions Analyzing Graphs of Functions A Library of Parent Functions Transformations of Functions Combinations of Functions: Composite Functions Inverse Functions Quadratic Functions and Models Polynomial Functions of Higher Degree Polynomial and Synthetic Division Complex Numbers Zeros of Polynomial Functions Rational Functions Exponential Functions and Their Graphs Logarithmic Functions and Their Graphs Properties of Logarithms Exponential and Logarithmic Equations Exponential and Logarithmic Models Graphs of Sine and Cosine Functions Graphs of Other Trigonometric Functions Inverse Trigonometric Functions Using Fundamental Identities Verifying Trigonometric Identities Solving Trigonometric Equations Sum and Difference Formulas Multiple-Angle and Product-to-Sum Formulas Vectors in the Plane Vectors and Dot Products Trigonometric Form of a Complex Number Linear and Nonlinear Systems of Equations Two-Variable Linear Systems Multivariable Linear Systems Partial Fractions Systems of Inequalities Matrices and Systems of Equations
3 8.2 Operations with Matrices The Determinant of a Square Matrix The Inverse of a Square Matrix Applications of Matrices and Determinants Sequences and Series Arithmetic Sequences and Partial Sums Geometric Sequences and Series Mathematical Induction The Binomial Theorem Introduction to Conics: Parabolas Ellipses Hyperbolas Parametric Equations Polar Coordinates Graphs of Polar Equations
4 Section 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,
5 2 Section 1.2 Example 2 Find the - and - intercepts of the graph of the equation. (a) (b) (c)
6 Section 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.
7 4 Section 1.2 Example 3 Use the algebraic tests to check for symmetry with respect to both axes and the origin. (a) (b)
8 Section 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
9 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:
10 Section 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:
11 8 Section 1.2 Write the standard form of the equation of the circle with the given characteristics. (c) Endpoints of a diameter:
12 Section 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
13 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.
14 Section 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.
15 12 Section 1.4 Example 5 Find the domain of the function. (a) (b) (c) (d)
16 Section 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.
17 14 Section 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
18 Section 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 (a) Domain of (b) Range of f (c) (d) (e) Interval(s) for which
19 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)
20 Section 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.
21 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.
22 Section 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)
23 20 Section 1.5 Determine whether the function is even, odd, or neither. Then describe the symmetry. (b) (c)
24 Section 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.
25 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
26 Section 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.
27 24 Section 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:
28 Section 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
29 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 )
30 Section 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)
31 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.
32 Section 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.
33 30 Section 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
34 Section 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.
35 32 Section 1.8 Example 4 Given and, find and state the domain. Example 5 Find two functions and such that.
36 Section 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.
37 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
38 Section 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?
39 36 Section 1.9 Finding an Inverse Function Algebraically 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
40 Section 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 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.
41 38 Section 2.1 Example 1 Match the quadratic function with its graph. (a) (b) (c) +3 (d) (i) (ii) (iii) (iv)
42 Example 2 Sketch the graph of. Identify the vertex, axis of symmetry and - intercept(s). Section
43 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).
44 Section 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.
45 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.
46 Section 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)
47 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.
48 Section 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): 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 :
49 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
50 Section 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)
51 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.
52 Section 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.
53 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.
54 Section 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.
55 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.
56 Section 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 ( ), 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
57 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
58 Section 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
59 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.
60 Section 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 ( ). 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.
61 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.
62 Example 4 Find all the rational zeros of the function. Section
63 60 Section 2.5 Example 5 Find all solutions of the polynomial equation.
64 Section 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
65 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.
66 Section 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: Limits:
67 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)
68 Section 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)
69 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!
70 Section Example 4 Graph
71 68 Section 2.6 Example 5 Graph
72 Section Example 6 Graph
73 70 Section 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 Graph by plotting points. x f( x) x
74 Section 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 ,000 1,000,000 1 e lim 1 n n 1 1 n 1 n , , ,000,000 n 1,000, As n gets larger, the approximation for e gets better! For all intents and purposes, we can think of e as approximately x Since e 1, the graph of f ( x) e is an increasing exponential function.
75 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
76 Section 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
77 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 :
78 Example 3 Find the balance in an account after 10 years if is invested at 3% interest compounded a) monthly Section b) continuously
79 76 Section 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
80 Section 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.
81 78 Section 3.2 Properties of logarithms 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
82 Section 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.
83 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.
84 Section 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)
85 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.
86 Section 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:
87 84 Section 3.4 Example 5 Solve.
88 Section 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.
89 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?
90 Section 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.
91 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
92 Section 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.
93 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.
94 Section Example 2 Sketch the graph of. (Include two full periods.) Example 3 Sketch the graph of. (Include two full periods.)
95 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.
96 Section 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 / /6 1/ 2 3 / /6 1/ 2 3 / /4 2 / 2 2 / /3 3 / 2 1/ 2 / / 3 3 / 2 1/ / 4 2 / 2 2 / 2 5 / 6 1/ 2 3 / / 6 1/ 2 3 / 2 5 / 4 2 / 2 2 / / 3 3 / 2 1/ 2 3 / undefined Domain: Range: Period: Vertical Asymptotes: Symmetry:
97 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 / 6 1/ 2 3 / / 4 2 / 2 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 / /6 1/ 2 3 / 2 1 /4 2 / 2 2 / 2 /3 3 / 2 1/ 2 2 / / 3 3 / 2 1/ / 4 2 / 2 2 / 2 5 / 6 1/ 2 3 / undefined Domain: Range: Period: Vertical Asymptotes: Symmetry:
98 Section 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.
99 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
100 Section Example 3 Sketch the graph of. Example 4 Sketch the graph of.
101 98 Section 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
102 Section 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 =
103 100 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)
104 Section ******************************************************************************** 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
105 102 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
106 Section 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.
107 104 Section Using Fundamental Identities Fundamental Trigonometric Identities Reciprocal Identities Quotient Identities Pythagorean Identities Divide by : Divide by : Cofunction Identities Even/Odd Identities
108 Section Example 1 Given the values and, evaluate all six trig functions. Method #1) Draw triangle Method #2) Use the identities
109 106 Section 5.1 Example 2 Simplify: a) b) c) Example 3 Factor, then simplify. a) b)
110 Section 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,
111 108 Section 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:
112 Example 3 Verify the identity: Section
113 110 Section 5.2 Example 4 Verify the identity:
114 Section 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
115 112 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.
116 Section 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
117 114 Section 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)
118 Section Example 2 Find the exact value of Example 3 Find the exact value of given that and. (Both and are in Quadrant IV.)
119 116 Section 5.4 Example 4 Verify the cofunction identity
120 Section 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.
121 118 Section 5.5 Example 2 Solve in the interval. Example 3 Use the following to find.
122 Section 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.
123 120 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
124 Section 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.
125 122 Section 5.5 Example 8 Write the sum as a product. Example 9 Verify the identity.
126 Section 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.
127 124 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
128 Section 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 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.
129 126 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.
130 Section 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.
131 128 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.,,
132 Section Vectors and Dot Products Definition The dot product of and is Example 1 Find the dot product of and. Properties of the Dot Product *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.
133 130 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:
134 Section 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
135 132 Section 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.
136 Section 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.
137 134 Section 6.5 Example 2 Write the complex number in trigonometric form. a) b) Example 3 Write the complex number in standard form.
138 Section 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,.