Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = e x. y = exp(x) if and only if x = ln(y)


 Lenard Grant
 1 years ago
 Views:
Transcription
1 Lecture 3 : The Natural Exponential Function: f(x) = exp(x) = Last day, we saw that the function f(x) = ln x is onetoone, with domain (, ) and range (, ). We can conclude that f(x) has an inverse function f (x) = exp(x) which we call the natural exponential function. The definition of inverse functions gives us the following: The cancellation laws give us: y = f (x) if and only if x = f(y) y = exp(x) if and only if x = ln(y) f (f(x)) = x and f(f (x)) = x exp(ln x) = x and ln(exp(x)) = x. We can draw the graph of y = exp(x) by reflecting the graph of y = ln(x) in the line y = x. 5 y = exphxl = H, e L 5 H, el y = lnhxl H7, e 7 L H, L He, L He, L 5 H, L 55 He 7, 7L  We have that the graph y = exp(x) is onetoone and continuous with domain (, ) and range (, ). Note that exp(x) > for all values of x. We see that In fact for any rational number r, we have exp() = since ln = exp() = e since ln e =, exp() = e since ln(e ) =, exp( 7) = e 7 since ln(e 7 ) = 7. exp(r) = e r since ln(e r ) = r ln e = r,
2 by the laws of Logarithms. When x is rational or irrational, we define to be exp(x). = exp(x) Note: This agrees with definitions of given elsewhere, since the definition is the same when x is a rational number and the exponential function is continuous. Restating the above properties given above in light of this new interpretation of the exponential function, we get: = y if and only if We can use these formulas to solve equations. Example Solve for x if ln(x + ) = 5 ln y = x e ln x = x and ln = x Solving Equations Example Solve for x if = From the graph we see that x ex =, Limits =. x Example Find the it x.
3 Rules of Exponents The following rules of exponents follow from the rules of logarithms: +y = e y, y = ex e y, (ex ) y = y. Proof We have ln(+y ) = x + y = ln( ) + ln(e y ) = ln( e y ). Therefore +y = e y. The other rules can be proven similarly. Example Simplify ex e x+ ( ). d dx ex = Derivatives d dx eg(x) = g (x)e g(x) Proof We use logarithmic differentiation. If y =, we have ln y = x and differentiating, we get dy dy = or = y = y dx dx ex. The derivative on the right follows from the chain rule. Example Find d dx esin x and d dx sin ( ) dx = + C Integrals g (x)e g(x) dx = e g(x) + C Example Find x + dx. 3
4 Old Exam Questions Old Exam Question The function f(x) = x 3 + 3x + e x is onetoone. Compute f( ) (). Old Exam Question Compute the it x e x e x e x. Old Exam Question Compute the Integral. ln + dx
5 Extra Examples (please attempt these before looking at the solutions) Example Find the domain of the function g(x) = 5. Example Solve for x if ln(ln(x )) = Example Let f(x) = e x+3, Show that f is a onetoone function and find the formula for f (x). Example Evaluate the integral 3e ( 3e x ln x 3 ) 3 dx. Example Find the it x and x. Example Find π (cos x)esin x dx. Example Find the first and second derivatives of h(x) = ex. Sketch the graph of h(x) with horizontal, and vertical asymptotes, showing where the function is increasing and decreasing and showing intervals of concavity and convexity. 5
6 Extra Examples: Solutions Example Find the domain of the function g(x) = 5. The domain of g is {x 5 }. 5 if and only if 5 if and only if ln 5 ln( ) = x or x ln 5 since ln(x) is an increasing function. Example Solve for x if ln(ln(x )) = We apply the exponential function to both sides to get e ln(ln(x )) = e or ln(x ) = e. Applying the exponential function to both sides again, we get e ln(x) = e e or x = e e. Taking the square root of both sides, we get x = e e. Example Let f(x) = e x+3, Show that f is a onetoone function and find the formula for f (x). We have the domain of f is all real numbers. To find a formula for f, we use the method given in lecture. y = e x+3 is the same as x = f (y). we solve for x in the equation on the left, first we apply the logarithm function to both sides ln(y) = ln(e x+3 ) = x + 3 x = ln(y) 3 x = ln(y) 3 Now we switch th and y to get y = ln(x) 3 = f (x). = f (y). Example Evaluate the integral 3e ( 3e x ln x 3 ) 3 dx. We try the substitution u = ln x 3. du = 3 x 3 dx = x dx, u(3e ) =, u(3e ) =. 3e ( 3e x ln x 3 ) 3 dx = u du = u3 6 = u
7 = Example Find the it x x x ( )(6) ( )() = 8 3 = 3 3 and x = =. x x ( ) = =. x x ( ) = = 9. Example Find π (cos x)esin x dx. We use substitution. Let u = sin x, then du = cos x dx, u() = and u(π/) =. π (cos x)e sin x dx = e u du = e u = e e = e. Example Find the first and second derivatives of h(x) = ex. Sketch the graph of h(x) with horizontal, and vertical asymptotes, showing where the function is increasing and decreasing and showing intervals of concavity and convexity. yint: h() = 9 xint: h(x) = if and only if =, this is impossible, so there is no x intercept. e H.A. : In class, we saw x x = and above, we saw x ex =. So the H.A. s are y = and y =. V.A. : The graph has a vertical asymptote at x if =, that is if = or x = ln( ). Inc/Dec (h (x)) To determine where the graph is increasing or decreasing, we calculate the derivative using the quotient rule h (x) = (ex ) ( ) ( ) = ex ( ) ( ) = ( ). Since h (x) is always negative, the graph of y = h(x) is always decreasing. Concave/Convex To determine intervals of concavity and convexity, we calculate the second derivative. h (x) = d dx h (x) = d dx ( ) = d dx ( ). I m going to use logarithmic differentiation here y = differentiating both sides, we get Multiplying across by y = ( ) ln(y) = ln( ) ln( ) = x ln( ) dy y dx = ex = ex. ( ), we get dy dx = ( ) ( ) 7.
8 = ex ( ) ( ) = ex ( ) = ex ( + ) ( ) 3 ( ) 3 ( ) 3 h (x) = dy dx = ex ( + ) ( ) 3 We see that the numerator is always positive here. From our calculations above, we have < if x < ln(/) and > if x > ln(/). Therefore h (x) < if x < ln(/) and h (x) > if x > ln(/) and The graph of y = h(x) is concave down if x < ln(/) and concave up if x > ln(/). Putting all of this together, you should get a graph that looks like: Check it and other functions out in Mthematica 8
9 Answers to Old Exam Questions Old Exam Question The function f(x) = x 3 + 3x + e x is onetoone. Compute f( ) (). We use the formula (f ) () = f (f ()) b = f () same as f(b) = b 3 + 3b + e b =. Solving for b is very difficult, but we can work by trail and error. If we try b =, we see that it works, since e =. Therefore f () =. We also need to calculate f (x), we get f (x) = 3x e x. (f ) () = Old Exam Question Compute the it f (f ()) = f () = 3 + = 5. x e x e x e x. We divide both numerator and denominator by the highest power of in the denominator which is e x in this case. e x ( e x )/e x e x e 3x = = = x e x e x x (e x e x )/ex x e x =. Old Exam Question Compute the Integral. ln We make the substitution u = +. We have + dx du = dx, u() =, u(ln ) = + e ln = 3. We get ln 3 + e dx = x u du = ln u = ln 3 ln = ln(3/). 3 9
Review for Calculus Rational Functions, Logarithms & Exponentials
Definition and Domain of Rational Functions A rational function is defined as the quotient of two polynomial functions. F(x) = P(x) / Q(x) The domain of F is the set of all real numbers except those for
More informationFINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA
FINAL EXAM SECTIONS AND OBJECTIVES FOR COLLEGE ALGEBRA 1.1 Solve linear equations and equations that lead to linear equations. a) Solve the equation: 1 (x + 5) 4 = 1 (2x 1) 2 3 b) Solve the equation: 3x
More informationDerivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x):
Derivatives: rules and applications (Stewart Ch. 3/4) The derivative f (x) of the function f(x): f f(x + h) f(x) (x) = lim h 0 h (for all x for which f is differentiable/ the limit exists) Property:if
More informationCalculus 1: Sample Questions, Final Exam, Solutions
Calculus : Sample Questions, Final Exam, Solutions. Short answer. Put your answer in the blank. NO PARTIAL CREDIT! (a) (b) (c) (d) (e) e 3 e Evaluate dx. Your answer should be in the x form of an integer.
More informationApr 23, 2015. Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23, 2015 sketching 1 / and 19pa
Calculus with Algebra and Trigonometry II Lecture 23 Final Review: Curve sketching and parametric equations Apr 23, 2015 Calculus with Algebra and Trigonometry II Lecture 23Final Review: Apr Curve 23,
More information3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS. Copyright Cengage Learning. All rights reserved.
3.2 LOGARITHMIC FUNCTIONS AND THEIR GRAPHS Copyright Cengage Learning. All rights reserved. What You Should Learn Recognize and evaluate logarithmic functions with base a. Graph logarithmic functions.
More information6.4 Logarithmic Equations and Inequalities
6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.
More informationHomework # 3 Solutions
Homework # 3 Solutions February, 200 Solution (2.3.5). Noting that and ( + 3 x) x 8 = + 3 x) by Equation (2.3.) x 8 x 8 = + 3 8 by Equations (2.3.7) and (2.3.0) =3 x 8 6x2 + x 3 ) = 2 + 6x 2 + x 3 x 8
More informationAPPLICATIONS OF DIFFERENTIATION
4 APPLICATIONS OF DIFFERENTIATION APPLICATIONS OF DIFFERENTIATION So far, we have been concerned with some particular aspects of curve sketching: Domain, range, and symmetry (Chapter 1) Limits, continuity,
More informationMath 120 Final Exam Practice Problems, Form: A
Math 120 Final Exam Practice Problems, Form: A Name: While every attempt was made to be complete in the types of problems given below, we make no guarantees about the completeness of the problems. Specifically,
More information100. In general, we can define this as if b x = a then x = log b
Exponents and Logarithms Review 1. Solving exponential equations: Solve : a)8 x = 4! x! 3 b)3 x+1 + 9 x = 18 c)3x 3 = 1 3. Recall: Terminology of Logarithms If 10 x = 100 then of course, x =. However,
More informationList the elements of the given set that are natural numbers, integers, rational numbers, and irrational numbers. (Enter your answers as commaseparated
MATH 142 Review #1 (4717995) Question 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Description This is the review for Exam #1. Please work as many problems as possible
More informationDefinition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left.
Vertical and Horizontal Asymptotes Definition 2.1 The line x = a is a vertical asymptote of the function y = f(x) if y approaches ± as x approaches a from the right or left. This graph has a vertical asymptote
More information55x 3 + 23, f(x) = x2 3. x x 2x + 3 = lim (1 x 4 )/x x (2x + 3)/x = lim
Slant Asymptotes If lim x [f(x) (ax + b)] = 0 or lim x [f(x) (ax + b)] = 0, then the line y = ax + b is a slant asymptote to the graph y = f(x). If lim x f(x) (ax + b) = 0, this means that the graph of
More informationDefinition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: f (x) =
Vertical Asymptotes Definition of Vertical Asymptote The line x = a is called a vertical asymptote of f (x) if at least one of the following is true: lim f (x) = x a lim f (x) = lim x a lim f (x) = x a
More informationcorrectchoice plot f(x) and draw an approximate tangent line at x = a and use geometry to estimate its slope comment The choices were:
Topic 1 2.1 mode MultipleSelection text How can we approximate the slope of the tangent line to f(x) at a point x = a? This is a Multiple selection question, so you need to check all of the answers that
More informationCurve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x and yintercepts.
Curve Sketching GUIDELINES FOR SKETCHING A CURVE: A. Domain. B. Intercepts: x and yintercepts. C. Symmetry: even (f( x) = f(x)) or odd (f( x) = f(x)) function or neither, periodic function. ( ) ( ) D.
More informationAsymptotes. Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if
Section 2.1: Vertical and Horizontal Asymptotes Definition. The vertical line x = a is called a vertical asymptote of the graph of y = f(x) if, lim x a f(x) =, lim x a x a x a f(x) =, or. + + Definition.
More informationLogarithmic Functions
Logarithmic Functions The function e x is unique exponential function whose tangent at (0, ) has slope The number e = Math e 2740 and hence Lecture 2 < e < 3 2 graph e x Write in form Eir (method : easy
More informationSimplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression.
MAC 1105 Final Review Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression. 1) 8x 249x + 6 x  6 A) 1, x 6 B) 8x  1, x 6 x 
More informationD f = (2, ) (x + 1)(x 3) (b) g(x) = x 1 solution: We need the thing inside the root to be greater than or equal to 0. So we set up a sign table.
. Find the domains of the following functions: (a) f(x) = ln(x ) We need x > 0, or x >. Thus D f = (, ) (x + )(x 3) (b) g(x) = x We need the thing inside the root to be greater than or equal to 0. So we
More information5.1 Derivatives and Graphs
5.1 Derivatives and Graphs What does f say about f? If f (x) > 0 on an interval, then f is INCREASING on that interval. If f (x) < 0 on an interval, then f is DECREASING on that interval. A function has
More informationMATHS WORKSHOPS. Functions. Business School
MATHS WORKSHOPS Functions Business School Outline Overview of Functions Quadratic Functions Exponential and Logarithmic Functions Summary and Conclusion Outline Overview of Functions Quadratic Functions
More informationLearning Objectives for Math 165
Learning Objectives for Math 165 Chapter 2 Limits Section 2.1: Average Rate of Change. State the definition of average rate of change Describe what the rate of change does and does not tell us in a given
More informationSolving DEs by Separation of Variables.
Solving DEs by Separation of Variables. Introduction and procedure Separation of variables allows us to solve differential equations of the form The steps to solving such DEs are as follows: dx = gx).
More informationMULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.
Math 110 Review for Final Examination 2012 Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Match the equation to the correct graph. 1) y = 
More informationPRACTICE FINAL. Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 10cm.
PRACTICE FINAL Problem 1. Find the dimensions of the isosceles triangle with largest area that can be inscribed in a circle of radius 1cm. Solution. Let x be the distance between the center of the circle
More informationMath 21A Brian Osserman Practice Exam 1 Solutions
Math 2A Brian Osserman Practice Exam Solutions These solutions are intended to indicate roughly how much you would be expected to write. Comments in [square brackets] are additional and would not be required.
More informationItems related to expected use of graphing technology appear in bold italics.
 1  Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing
More information1 Lecture: Integration of rational functions by decomposition
Lecture: Integration of rational functions by decomposition into partial fractions Recognize and integrate basic rational functions, except when the denominator is a power of an irreducible quadratic.
More informationInverse Functions and Logarithms
Section 3. Inverse Functions and Logarithms 1 Kiryl Tsishchanka Inverse Functions and Logarithms DEFINITION: A function f is called a onetoone function if it never takes on the same value twice; that
More informationCHAPTER FIVE. Solutions for Section 5.1. Skill Refresher. Exercises
CHAPTER FIVE 5.1 SOLUTIONS 265 Solutions for Section 5.1 Skill Refresher S1. Since 1,000,000 = 10 6, we have x = 6. S2. Since 0.01 = 10 2, we have t = 2. S3. Since e 3 = ( e 3) 1/2 = e 3/2, we have z =
More informationx 2 + y 2 = 1 y 1 = x 2 + 2x y = x 2 + 2x + 1
Implicit Functions Defining Implicit Functions Up until now in this course, we have only talked about functions, which assign to every real number x in their domain exactly one real number f(x). The graphs
More informationMath 181 Spring 2007 HW 1 Corrected
Math 181 Spring 2007 HW 1 Corrected February 1, 2007 Sec. 1.1 # 2 The graphs of f and g are given (see the graph in the book). (a) State the values of f( 4) and g(3). Find 4 on the xaxis (horizontal axis)
More informationAlgebra 2: Themes for the Big Final Exam
Algebra : Themes for the Big Final Exam Final will cover the whole year, focusing on the big main ideas. Graphing: Overall: x and y intercepts, fct vs relation, fct vs inverse, x, y and origin symmetries,
More informationSolutions to Study Guide for Test 3. Part 1 No Study Guide, No Calculator
Solutions to Study Guide for Test 3 Part 1 No Study Guide, No Calculator 1. State the definition of the derivative of a function. Solution: The derivative of a function f with respect to x is the function
More informationTOPIC 4: DERIVATIVES
TOPIC 4: DERIVATIVES 1. The derivative of a function. Differentiation rules 1.1. The slope of a curve. The slope of a curve at a point P is a measure of the steepness of the curve. If Q is a point on the
More informationDerivatives and Graphs. Review of basic rules: We have already discussed the Power Rule.
Derivatives and Graphs Review of basic rules: We have already discussed the Power Rule. Product Rule: If y = f (x)g(x) dy dx = Proof by first principles: Quotient Rule: If y = f (x) g(x) dy dx = Proof,
More informationRational Polynomial Functions
Rational Polynomial Functions Rational Polynomial Functions and Their Domains Today we discuss rational polynomial functions. A function f(x) is a rational polynomial function if it is the quotient of
More information3.4 Limits at Infinity  Asymptotes
3.4 Limits at Infinity  Asymptotes Definition 3.3. If f is a function defined on some interval (a, ), then f(x) = L means that values of f(x) are very close to L (keep getting closer to L) as x. The line
More informationExponential and Logarithmic Functions
Chapter 6 Eponential and Logarithmic Functions Section summaries Section 6.1 Composite Functions Some functions are constructed in several steps, where each of the individual steps is a function. For eample,
More informationChapter 11  Curve Sketching. Lecture 17. MATH10070  Introduction to Calculus. maths.ucd.ie/modules/math10070. Kevin Hutchinson.
Lecture 17 MATH10070  Introduction to Calculus maths.ucd.ie/modules/math10070 Kevin Hutchinson 28th October 2010 Z Chain Rule (I): If y = f (u) and u = g(x) dy dx = dy du du dx Z Chain rule (II): d dx
More informationSept 20, 2011 MATH 140: Calculus I Tutorial 2. ln(x 2 1) = 3 x 2 1 = e 3 x = e 3 + 1
Sept, MATH 4: Calculus I Tutorial Solving Quadratics, Dividing Polynomials Problem Solve for x: ln(x ) =. ln(x ) = x = e x = e + Problem Solve for x: e x e x + =. Let y = e x. Then we have a quadratic
More informationPractice with Proofs
Practice with Proofs October 6, 2014 Recall the following Definition 0.1. A function f is increasing if for every x, y in the domain of f, x < y = f(x) < f(y) 1. Prove that h(x) = x 3 is increasing, using
More informationChapter Usual types of questions Tips What can go ugly 1 Algebraic Fractions
C3 Cheat Sheet Last Updated: 9 th Nov 2015 Chapter Usual types of questions Tips What can go ugly 1 Algebraic Fractions Almost always adding or subtracting fractions. Factorise everything in each fraction
More informationMicroeconomic Theory: Basic Math Concepts
Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts
More informationAlgebra Graphing an Exponential Function
Graphing an Exponential Function Graphing an exponential or logarithmic function is a process best described by example. Each step of the process is described and illustrated in the examples over the next
More informationArea Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x b is given by
MATH 42, Fall 29 Examples from Section, Tue, 27 Oct 29 1 The First Hour Area Between Curves. The idea: the area between curves y = f(x) and y = g(x) (if the graph of f(x) is above that of g(x)) for a x
More informationLecture 3: Derivatives and extremes of functions
Lecture 3: Derivatives and extremes of functions Lejla Batina Institute for Computing and Information Sciences Digital Security Version: spring 2011 Lejla Batina Version: spring 2011 Wiskunde 1 1 / 16
More informationExponential Functions and Their Graphs
Exponential Functions and Their Graphs Definition of Exponential Function f ( x) a x where a > 0, a 1, and x is any real number Parent function for transformations: f(x) = b a x h + k The graph of f moves
More informationTOPIC 3: CONTINUITY OF FUNCTIONS
TOPIC 3: CONTINUITY OF FUNCTIONS. Absolute value We work in the field of real numbers, R. For the study of the properties of functions we need the concept of absolute value of a number. Definition.. Let
More informationPreCalculus Review Lesson 1 Polynomials and Rational Functions
If a and b are real numbers and a < b, then PreCalculus Review Lesson 1 Polynomials and Rational Functions For any real number c, a + c < b + c. For any real numbers c and d, if c < d, then a + c < b
More informationExponential and Logarithmic Functions
Exponential and Logarithmic Functions Exponential Functions Overview of Objectives, students should be able to: 1. Evaluate exponential functions. Main Overarching Questions: 1. How do you graph exponential
More informationMidChapter Quiz: Lessons 31 through 33
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 1. f (x) = 5 x Evaluate the function
More informationPRECALCULUS GRADE 12
PRECALCULUS GRADE 12 [C] Communication Trigonometry General Outcome: Develop trigonometric reasoning. A1. Demonstrate an understanding of angles in standard position, expressed in degrees and radians.
More information2 Integrating Both Sides
2 Integrating Both Sides So far, the only general method we have for solving differential equations involves equations of the form y = f(x), where f(x) is any function of x. The solution to such an equation
More informationChapter 7  Roots, Radicals, and Complex Numbers
Math 233  Spring 2009 Chapter 7  Roots, Radicals, and Complex Numbers 7.1 Roots and Radicals 7.1.1 Notation and Terminology In the expression x the is called the radical sign. The expression under the
More information1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some
Section 3.1: First Derivative Test Definition. Let f be a function with domain D. 1. Then f has a relative maximum at x = c if f(c) f(x) for all values of x in some open interval containing c. The number
More informationApplications of Integration Day 1
Applications of Integration Day 1 Area Under Curves and Between Curves Example 1 Find the area under the curve y = x2 from x = 1 to x = 5. (What does it mean to take a slice?) Example 2 Find the area under
More informationMcMurry University Pretest Practice Exam. 1. Simplify each expression, and eliminate any negative exponent(s).
1. Simplify each expression, and eliminate any negative exponent(s). a. b. c. 2. Simplify the expression. Assume that a and b denote any real numbers. (Assume that a denotes a positive number.) 3. Find
More informationThe Derivative. Philippe B. Laval Kennesaw State University
The Derivative Philippe B. Laval Kennesaw State University Abstract This handout is a summary of the material students should know regarding the definition and computation of the derivative 1 Definition
More information31 Exponential Functions
Sketch and analyze the graph of each function. Describe its domain, range, intercepts, asymptotes, end behavior, and where the function is increasing or decreasing. 1. f (x) = 2 x 4 16 2 4 1 2 0 1 1 0.5
More information3. Exponential and Logarithmic functions
3. ial and s ial and ic... 3.1. Here are a few examples to remind the reader of the definitions and laws for expressions involving exponents: 2 3 = 2 2 2 = 8, 2 0 = 1, 2 1 = 1 2, 2 3 = 1 2 3 = 1 8, 9 1/2
More informationMain page. Given f ( x, y) = c we differentiate with respect to x so that
Further Calculus Implicit differentiation Parametric differentiation Related rates of change Small variations and linear approximations Stationary points Curve sketching  asymptotes Curve sketching the
More informationCore Maths C3. Revision Notes
Core Maths C Revision Notes October 0 Core Maths C Algebraic fractions... Cancelling common factors... Multipling and dividing fractions... Adding and subtracting fractions... Equations... 4 Functions...
More informationLIMITS AND CONTINUITY
LIMITS AND CONTINUITY 1 The concept of it Eample 11 Let f() = 2 4 Eamine the behavior of f() as approaches 2 2 Solution Let us compute some values of f() for close to 2, as in the tables below We see from
More information6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms
AAU  Business Mathematics I Lecture #6, March 16, 2009 6 Rational Inequalities, (In)equalities with Absolute value; Exponents and Logarithms 6.1 Rational Inequalities: x + 1 x 3 > 1, x + 1 x 2 3x + 5
More informationMA4001 Engineering Mathematics 1 Lecture 10 Limits and Continuity
MA4001 Engineering Mathematics 1 Lecture 10 Limits and Dr. Sarah Mitchell Autumn 2014 Infinite limits If f(x) grows arbitrarily large as x a we say that f(x) has an infinite limit. Example: f(x) = 1 x
More informationOfficial Math 112 Catalog Description
Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A
More informationCollege Algebra. Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381
College Algebra Course Text Barnett, Raymond A., Michael R. Ziegler, and Karl E. Byleen. College Algebra, 8th edition, McGrawHill, 2008, ISBN: 9780072867381 Course Description This course provides
More information5 Indefinite integral
5 Indefinite integral The most of the mathematical operations have inverse operations: the inverse operation of addition is subtraction, the inverse operation of multiplication is division, the inverse
More informationSituation: Dividing Linear Expressions
Situation: Dividing Linear Expressions Date last revised: June 4, 203 Michael Ferra, Nicolina Scarpelli, Mary Ellen Graves, and Sydney Roberts Prompt: An Algebra II class has been examining the product
More informationEquations. #110 Solve for the variable. Inequalities. 1. Solve the inequality: 2 5 7. 2. Solve the inequality: 4 0
College Algebra Review Problems for Final Exam Equations #110 Solve for the variable 1. 2 1 4 = 0 6. 2 8 7 2. 2 5 3 7. = 3. 3 9 4 21 8. 3 6 9 18 4. 6 27 0 9. 1 + log 3 4 5. 10. 19 0 Inequalities 1. Solve
More informationSection 4.4 Rational Functions and Their Graphs
Section 4.4 Rational Functions and Their Graphs p( ) A rational function can be epressed as where p() and q() are q( ) 3 polynomial functions and q() is not equal to 0. For eample, is a 16 rational function.
More informationLarson, R. and Boswell, L. (2016). Big Ideas Math, Algebra 2. Erie, PA: Big Ideas Learning, LLC. ISBN
ALG B Algebra II, Second Semester #PR0, BK04 (v.4.0) To the Student: After your registration is complete and your proctor has been approved, you may take the Credit by Examination for ALG B. WHAT TO
More informationThe data is more curved than linear, but not so much that linear regression is not an option. The line of best fit for linear regression is
CHAPTER 5 Nonlinear Models 1. Quadratic Function Models Consider the following table of the number of Americans that are over 100 in various years. Year Number (in 1000 s) 1994 50 1996 56 1998 65 2000
More informationMath 113 HW #9 Solutions
Math 3 HW #9 Solutions 4. 50. Find the absolute maximum and absolute minimum values of on the interval [, 4]. f(x) = x 3 6x 2 + 9x + 2 Answer: First, we find the critical points of f. To do so, take the
More informationThis is a square root. The number under the radical is 9. (An asterisk * means multiply.)
Page of Review of Radical Expressions and Equations Skills involving radicals can be divided into the following groups: Evaluate square roots or higher order roots. Simplify radical expressions. Rationalize
More informationThe Mean Value Theorem
The Mean Value Theorem THEOREM (The Extreme Value Theorem): If f is continuous on a closed interval [a, b], then f attains an absolute maximum value f(c) and an absolute minimum value f(d) at some numbers
More informationLecture 5 : Continuous Functions Definition 1 We say the function f is continuous at a number a if
Lecture 5 : Continuous Functions Definition We say the function f is continuous at a number a if f(x) = f(a). (i.e. we can make the value of f(x) as close as we like to f(a) by taking x sufficiently close
More informationChapter 7 Outline Math 236 Spring 2001
Chapter 7 Outline Math 236 Spring 2001 Note 1: Be sure to read the Disclaimer on Chapter Outlines! I cannot be responsible for misfortunes that may happen to you if you do not. Note 2: Section 7.9 will
More information1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N.
CHAPTER 3: EXPONENTS AND POWER FUNCTIONS 1. The algebra of exponents 1.1. Natural Number Powers. It is easy to say what is meant by a n a (raised to) to the (power) n if n N. For example: In general, if
More informationa. all of the above b. none of the above c. B, C, D, and F d. C, D, F e. C only f. C and F
FINAL REVIEW WORKSHEET COLLEGE ALGEBRA Chapter 1. 1. Given the following equations, which are functions? (A) y 2 = 1 x 2 (B) y = 9 (C) y = x 3 5x (D) 5x + 2y = 10 (E) y = ± 1 2x (F) y = 3 x + 5 a. all
More informationZero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
More informationMATH SOLUTIONS TO PRACTICE FINAL EXAM. (x 2)(x + 2) (x 2)(x 3) = x + 2. x 2 x 2 5x + 6 = = 4.
MATH 55 SOLUTIONS TO PRACTICE FINAL EXAM x 2 4.Compute x 2 x 2 5x + 6. When x 2, So x 2 4 x 2 5x + 6 = (x 2)(x + 2) (x 2)(x 3) = x + 2 x 3. x 2 4 x 2 x 2 5x + 6 = 2 + 2 2 3 = 4. x 2 9 2. Compute x + sin
More information3.5 Summary of Curve Sketching
3.5 Summary of Curve Sketching Follow these steps to sketch the curve. 1. Domain of f() 2. and y intercepts (a) intercepts occur when f() = 0 (b) yintercept occurs when = 0 3. Symmetry: Is it even or
More information1.5 ANALYZING GRAPHS OF FUNCTIONS. Copyright Cengage Learning. All rights reserved.
1.5 ANALYZING GRAPHS OF FUNCTIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use the Vertical Line Test for functions. Find the zeros of functions. Determine intervals on which
More informationCCSS: F.IF.7.e., F.IF.8.b MATHEMATICAL PRACTICES: 3 Construct viable arguments and critique the reasoning of others
7.1 Graphing Exponential Functions 1) State characteristics of exponential functions 2) Identify transformations of exponential functions 3) Distinguish exponential growth from exponential decay CCSS:
More information4.4 Concavity and Curve Sketching
Concavity and Curve Sketching Section Notes Page We can use the second derivative to tell us if a graph is concave up or concave down To see if something is concave down or concave up we need to look at
More informationDraft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then
CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function,
More informationPre Cal 2 1 Lesson with notes 1st.notebook. January 22, Operations with Complex Numbers
0 2 Operations with Complex Numbers Objectives: To perform operations with pure imaginary numbers and complex numbers To use complex conjugates to write quotients of complex numbers in standard form Complex
More informationSection 2.7 OnetoOne Functions and Their Inverses
Section. OnetoOne Functions and Their Inverses OnetoOne Functions HORIZONTAL LINE TEST: A function is onetoone if and only if no horizontal line intersects its graph more than once. EXAMPLES: 1.
More information1 Introduction to Differential Equations
1 Introduction to Differential Equations A differential equation is an equation that involves the derivative of some unknown function. For example, consider the equation f (x) = 4x 3. (1) This equation
More informationBlue Pelican Calculus First Semester
Blue Pelican Calculus First Semester Teacher Version 1.01 Copyright 20112013 by Charles E. Cook; Refugio, Tx Edited by Jacob Cobb (All rights reserved) Calculus AP Syllabus (First Semester) Unit 1: Function
More information1. [2.3] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions. Indeterminate FormEliminate the Common Factor
Review for the BST MTHSC 8 Name : [] Techniques for Computing Limits Limits of Polynomials/Rational Functions/Continuous Functions Evaluate cos 6 Indeterminate FormEliminate the Common Factor Find the
More informationTHE DERIVATIVE. Summary
THE DERIVATIVE Summary 1. Derivative of usual functions... 3 1.1. Constant function... 3 1.2. Identity function... 3 1.3. A function at the form... 3 1.4. Exponential function (of the form with 0):...
More informationALGEBRA REVIEW LEARNING SKILLS CENTER. Exponents & Radicals
ALGEBRA REVIEW LEARNING SKILLS CENTER The "Review Series in Algebra" is taught at the beginning of each quarter by the staff of the Learning Skills Center at UC Davis. This workshop is intended to be an
More informationMA107 Precalculus Algebra Exam 2 Review Solutions
MA107 Precalculus Algebra Exam 2 Review Solutions February 24, 2008 1. The following demand equation models the number of units sold, x, of a product as a function of price, p. x = 4p + 200 a. Please write
More informationNotes on Curve Sketching. B. Intercepts: Find the yintercept (f(0)) and any xintercepts. Skip finding xintercepts if f(x) is very complicated.
Notes on Curve Sketching The following checklist is a guide to sketching the curve y = f(). A. Domain: Find the domain of f. B. Intercepts: Find the yintercept (f(0)) and any intercepts. Skip finding
More informationALGEBRA I A PLUS COURSE OUTLINE
ALGEBRA I A PLUS COURSE OUTLINE OVERVIEW: 1. Operations with Real Numbers 2. Equation Solving 3. Word Problems 4. Inequalities 5. Graphs of Functions 6. Linear Functions 7. Scatterplots and Lines of Best
More information