Week #6 - Taylor Series, Derivatives and Graphs Section 10.1
|
|
- Helena Preston
- 6 years ago
- Views:
Transcription
1 Week #6 - Taylor Series, Derivatives and Graphs Section 10.1 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This material is used by permission of John Wiley & Sons, Inc. QUIZ PREPARATION PROBLEMS For Exercises 1-10, find the Taylor polynomials of degree n approximating the given functions for x near 0. (Assume p is a constant.) x, n = 3,5,7 f(x) = (1 x) 1 f(0) = 1 f (x) = (1 x) ( 1) = (1 x) f (0) = 1 f (x) = (1 x) 3 ( 1) = (1 x) 3 f (0) = f (x) = 3(1 x) 4 f (0) = 6. f (n) (x) = n!(1 x) n 1 f (n) (0) = n! Using this information, P 3 (x) = 1 + (x 0) + (x 0) + 6 (x 0)3 3! = 1 + x + x + x 3 Similarly, P 5 (x) = 1 + x + x + 6 3! x3 + x4 + 5! 5! x5 = 1 + x + x + x 3 + x 4 + x 5 Following the same pattern, x, n =,3,4 P 7 (x) = 1 + x + x + x 3 + x 4 + x 5 + x 6 + x 7 1
2 f(x) = (1 + x) 1/ f(0) = 1 f (x) = 1 (1 + x) 1/ f (0) = 1 ( )( ) 1 1 f (x) = (1 + x) 3/ f (0) = 1 4 ( )( )( ) f (x) = (1 + x) 5/ f (0) = 3 8 ( )( )( )( ) f (x) = (1 + x) 7/ f (0) = Using this information, Similarly, P (x) = f(0) + f (0)(x 0) + f (0) (x 0) = x x P 3 (x) = x x = x 1 8 x x3 1 3! x3 P 4 (x) = x 1 4 x ! x = x 1 4 x x x4 x4 8. tan x, n = 3,4 Recall the useful identities that sin(0) = 0 and cos(0) = 1. f(x) = tan(x) f(0) = 0 f (x) = (cos x) f (0) = 1 f (x) = (cos x) 3 ( sin x) = (cos x) 3 (sin x) f (0) = 0 f (3) (x) = 6(cos x) 4 ( sin x)(sin x) + (cos x) 3 cos(x) = 6(cos x) 4 (sin x) + (cos x) f (3) (0) = f (4) (x) = 4(cos x) 5 ( sin x)(sin x) + 1(cos x) 4 (sin x)(cos x) 4(cos x) 3 ( sin x) = 4(cos x) 5 (sin x) (cos x) 3 (sin x) f (4) (0) = 0
3 Because the fourth derivative is zero, both P 3 (x) and P 4 (x) will be equal: P 4 (x) = x x + 1 3! x x4 = x x3 P 3 (x) = x x3 For Exercises 11-16, find the Taylor polynomial of degree n for x near the given point a. 11. e x,a = 1, n = 4 f(x) = e x f (x) = e x f (x) = e x f (x) = e x f (x) = e x f(1) = e f (1) = e f (1) = e f (1) = e f (1) = e Recall, e is just a number, with a value close to.718. We build the Taylor polynomial using the usual formula. P 4 (x) = e + e(x 1) + e (x 1) + e 3! (x 1)3 + e (x 1)4 = e + e(x 1) + e (x 1) + e 6 (x 1)3 + e (x 1) x, a = 1, n = 3 f(x) = (1 + x) 1/ f(1) = ( ) 1 f (x) = (1 + x) 1/ f (1) = 1 ( )( ) 1 1 f (x) = (1 + x) 3/ f (1) = 1 1 = 4 3/ 8 ( )( ) ( ) f (x) = (1 + x) 5/ f 3 (1) = 8 5/ = 3 3 P 3 (x) = + 1 (x 1) 1 8! (x 1) (x 1)3 3! 3
4 13. sin x, a = π/, n = 4 Recall from the graphs of sin and cos that sin(π/) = 1 and cos(π/) = 0. f(x) = sin x f(π/) = 1 f (x) = cos x f (π/) = 0 f (x) = sin x f (π/) = 1 f (x) = cos x f (π/) = 0 f (x) = sin x f (π/) = 1 P 4 (x) = 1 + 0(x π/) 1! (x π/) + 0 3! (x π/)3 + 1 (x π/)3 = 1 1! (x π/) + 1 (x π/)4 14. cos x, a = π/4, n = 3 π/4 radians is 45 o. From the standard triangles, this gives sin(π/4) = cos(π/4) = 1. f(x) = cos x f(π/4) = 1 f (x) = sin x f (π/4) = 1 f (x) = cos x f (π/4) = 1 f (x) = sin x f (π/4) = 1 P 3 (x) = 1 1 (x π/4) 1! (x π/4) + 1 3! (x π/4) ln(x ), a = 1, n = 4 f(x) = ln(x ) f(1) = 0 f (x) = 1 x x = x 1 f (1) = f (x) = x f (1) = f (x) = 4x 3 f (1) = 4 f (x) = 1x 4 f (1) = 1 4
5 P 3 (x) = 0 + (x 1)! (x 1) + 4 3! (x 1)3 1 (x 1)4 3! = (x 1) (x 1) + 3 (x 1)3 1 (x 1)4 16. sin(x), a = π/4, n = 4 Recall from the graphs of sin and cos that sin(π/) = 1 and cos(π/) = 0. f(x) = sin(x) f(π/4) = sin(π/) = 1 f (x) = cos(x) f (π/4) = 0 f (x) = 4sin(x) f (π/4) = 4 f (x) = 8cos(x) f (π/4) = 0 f (x) = 16sin(x) f (π/4) = 16 P 4 (x) = 1 + 0(x π/4) 4! (x π/4) + 0 3! (x π/4) (x π/4)4 = 1 (x π/4) + (x π/4)4 3 For Problems 17-0, suppose P (x) = a+bx+cx is the second degree Taylor polynomial for the function f about x = 0. What can you say about the signs of a, b, c if f has the graph given below? 18. Since P (x) is the second degree Taylor polynomial for f(x) about x = 0, we know from the definition that P(0) = a = f(0) y-intercept P (0) = b = f (0) slope at x = 0 P (0) = c = f (0) concavity at x = 0 As with the previous problem, a > 0, b < 0 and c < As with the original problem, a < 0, b < 0 and c > 0. 5
6 1. The function f(x) is approximated near x = 0 by the second degree Taylor polynomial P (x) = 5 7x + 8x Give the value of (a) f(0) (b) f (0) (c) f (0) Using the fact that f(x) P (x) = f(0) + f (0)x + f (0) x and identifying coefficients with those given for P (x), we obtain the following: (a) f(0) = constant term = 5, (b) f (0) = coefficient of x = -7, (c) f (0) = coefficient of x = 8, so f (0) = Find the second-degree Taylor polynomial for f(x) = 4x 7x + about x = 0. What do you notice? f(x) = 4x 7x + f(0) = f (x) = 8x 7 f (0) = 7 f (x) = 8 f (0) = 8, so P (x) = + ( 7)x + 8 x = 4x 7x +. We notice that f(x) = P (x) in this case. 6. Find the third-degree Taylor polynomial for f(x) = x 3 + 7x 5x + 1 about x = 0. What do you notice? f (x) = 3x + 14x 5, f (x) = 6x + 14, f (x) = 6. Thus, about a = 0, P 3 (x) = ! x + 14! x + 6 3! x3 = 1 5x + 7x + x 3 = f(x). Once again, we notice that the Taylor polynomial of a function which already is a polynomial gives us the same polynomial back. 7. (a) Based on your observations in Problems 5-6, make a conjecture about Taylor approximations in the case when f is itself a polynomial. (b) Show that your conjecture is true. 6
7 f (0) = (a 1 + a x + + na n x n 1 ) x=0 = a 1 ; f (0) = (a + 3 a 3 x + + n(n 1)a n x n ) x=0 =! a. (a) We ll make the following conjecture: If f(x) is a polynomial of degree n, i.e. f(x) = a 0 + a 1 x + a x + + a n 1 x n 1 + a n x n ; then P n (x), the n th degree Taylor polynomial for f(x) about x = 0, is f(x) itself. (b) All we need to do is to calculate P n (x), the n th degree Taylor polynomial for f about x = 0 and see if it is the same as f(x). f(0) = a 0 ; If we continue doing this, we ll see in general Therefore, f (k) (0) = k! a k, k = 1,,3,...,n. P n (x) = f(0) + f (0) x + f (0) x + + f(n) (0) 1!! n! = a 0 + a 1 x + a x + + a n x n = f(x). 33. (a) Find the Taylor polynomial approximation of degree 4 about x = 0 for the function f(x) = e x. (b) Compare this result to the Taylor polynomial approximation of degree for the function f(x) = e x about x = 0. What do you notice? (c) Use your observation in part (b) to write out the Taylor polynomial approximation of degree 0 for the function in part (a). (d) What is the Taylor polynomial approximation of degree 5 for the function f(x) = e x? (a) f(x) = e x. f (x) = xe x,f (x) = (1 + x )e x,f (x) = 4(3x + x 3 )e x, f (4) = 4(3 + 6x )e x + 4(3x + x 3 )xe x. The Taylor polynomial about x = 0 is x n P 4 (x) = ! x +! x + 0 3! x3 + 1 x4 = 1 + x + 1 x4. 7
8 (b) f(x) = e z. The Taylor polynomial of degree is Q (x) = 1 + x 1! + x! = 1 + x + 1 x. If we substitute x for x in the Taylor polynomial for e x of degree, we will get P 4 (x), the Taylor polynomial for e x of degree 4: Q (x ) = 1 + x + 1 (x ) = 1 + x + 1 x4 = P 4 (x). (c) Let Q 10 (x) = 1 + x 1! + x! + + x10 10! be the Taylor polynomial of degree 10 for e x about x = 0. Then P 0 (x) = Q 10 (x ) (d) Let e x Q 5 (x) = 1 + x 1! + + x5 5!. Then = 1 + x 1! + (x ) + + (x ) 10! 10! = 1 + x 1! + x4! + + x0 10!. e x Q 5 ( x) = 1 + x 1! + ( x)! + ( x)3 3! + ( x)4 = 1 x + x 4 3 x3 + 3 x x5. + ( x)5 5! 8
Week #16 - Differential Equations (Euler s Method) Section 11.3
Week #16 - Differential Equations (Euler s Method) Section 11.3 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used
More informationWeek #1 The Exponential and Logarithm Functions Section 1.2
Week #1 The Exponential and Logarithm Functions Section 1.2 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More information8.7 Taylor s Inequality Math 2300 Section 005 Calculus II. f(x) = ln(1 + x) f(0) = 0
8.7 Taylor s Inequality Math 00 Section 005 Calculus II Name: ANSWER KEY Taylor s Inequality: If f (n+) is continuous and f (n+) < M between the center a and some point x, then f(x) T n (x) M x a n+ (n
More informationWeek #1 The Exponential and Logarithm Functions Section 1.3
Week #1 The Exponential and Logarithm Functions Section 1.3 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More informationMath 152 Take Home Test 1
Math 5 Take Home Test Due Monday 5 th October (5 points) The following test will be done at home in order to ensure that it is a fair and representative reflection of your own ability in mathematics I
More informationWeek #1 The Exponential and Logarithm Functions Section 1.4
Week #1 The Exponential and Logarithm Functions Section 1.4 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by
More informationMTH 122: Section 204. Plane Trigonometry. Test 1
MTH 122: Section 204. Plane Trigonometry. Test 1 Section A: No use of calculator is allowed. Show your work and clearly identify your answer. 1. a). Complete the following table. α 0 π/6 π/4 π/3 π/2 π
More information1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. Ans: x = 4, x = 3, x = 2,
1. The graph of a function f is given above. Answer the question: a. Find the value(s) of x where f is not differentiable. x = 4, x = 3, x = 2, x = 1, x = 1, x = 2, x = 3, x = 4, x = 5 b. Find the value(s)
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 CHAPTER
More informationUnit #24 - Lagrange Multipliers Section 15.3
Unit #24 - Lagrange Multipliers Section 1.3 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 200 by John Wiley & Sons, Inc. This material is
More informationThe Area bounded by Two Functions
The Area bounded by Two Functions The graph below shows 2 functions f(x) and g(x) that are continuous between x = a and x = b and f(x) g(x). The area shaded in green is the area between the 2 curves. We
More informationMath 226 Calculus Spring 2016 Practice Exam 1. (1) (10 Points) Let the differentiable function y = f(x) have inverse function x = f 1 (y).
Math 6 Calculus Spring 016 Practice Exam 1 1) 10 Points) Let the differentiable function y = fx) have inverse function x = f 1 y). a) Write down the formula relating the derivatives f x) and f 1 ) y).
More informationWeek #13 - Integration by Parts & Numerical Integration Section 7.2
Week #3 - Inegraion by Pars & Numerical Inegraion Secion 7. From Calculus, Single Variable by Hughes-Halle, Gleason, McCallum e. al. Copyrigh 5 by John Wiley & Sons, Inc. This maerial is used by permission
More informationChapter 5 Notes. 5.1 Using Fundamental Identities
Chapter 5 Notes 5.1 Using Fundamental Identities 1. Simplify each expression to its lowest terms. Write the answer to part as the product of factors. (a) sin x csc x cot x ( 1+ sinσ + cosσ ) (c) 1 tanx
More informationMA1021 Calculus I B Term, Sign:
MA1021 Calculus I B Term, 2014 Final Exam Print Name: Sign: Write up your solutions neatly and show all your work. 1. (28 pts) Compute each of the following derivatives: You do not have to simplify your
More informationSection 4.8 Anti Derivative and Indefinite Integrals 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 4.8 Anti Derivative and Indefinite Integrals 2 Lectures College of Science MATHS 101: Calculus I (University of Bahrain) 1 / 28 Indefinite Integral Given a function f, if F is a function such that
More informationMath 113 (Calculus 2) Exam 4
Math 3 (Calculus ) Exam 4 November 0 November, 009 Sections 0, 3 7 Name Student ID Section Instructor In some cases a series may be seen to converge or diverge for more than one reason. For such problems
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationMath 180, Final Exam, Fall 2012 Problem 1 Solution
Math 80, Final Exam, Fall 0 Problem Solution. Find the derivatives of the following functions: (a) ln(ln(x)) (b) x 6 + sin(x) e x (c) tan(x ) + cot(x ) (a) We evaluate the derivative using the Chain Rule.
More informationTangent Lines Sec. 2.1, 2.7, & 2.8 (continued)
Tangent Lines Sec. 2.1, 2.7, & 2.8 (continued) Prove this Result How Can a Derivative Not Exist? Remember that the derivative at a point (or slope of a tangent line) is a LIMIT, so it doesn t exist whenever
More informationTaylor and Maclaurin Series. Approximating functions using Polynomials.
Taylor and Maclaurin Series Approximating functions using Polynomials. Approximating f x = e x near x = 0 In order to approximate the function f x = e x near x = 0, we can use the tangent line (The Linear
More informationSection 5.5 More Integration Formula (The Substitution Method) 2 Lectures. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 5.5 More Integration Formula (The Substitution Method) 2 Lectures College of Science MATHS : Calculus I (University of Bahrain) Integrals / 7 The Substitution Method Idea: To replace a relatively
More informationThis practice exam is intended to help you prepare for the final exam for MTH 142 Calculus II.
MTH 142 Practice Exam Chapters 9-11 Calculus II With Analytic Geometry Fall 2011 - University of Rhode Island This practice exam is intended to help you prepare for the final exam for MTH 142 Calculus
More informationMath 181, Exam 2, Study Guide 2 Problem 1 Solution. 1 + dx. 1 + (cos x)2 dx. 1 + cos2 xdx. = π ( 1 + cos π 2
Math 8, Exam, Study Guide Problem Solution. Use the trapezoid rule with n to estimate the arc-length of the curve y sin x between x and x π. Solution: The arclength is: L b a π π + ( ) dy + (cos x) + cos
More informationWeek #5 - Related Rates, Linear Approximation, and Taylor Polynomials Section 4.6
Week #5 - Related Rates, Linear Approximation, and Taylor Polynomials Section 4.6 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 005 by John Wiley & Sons, Inc. This
More informationUniversity Calculus I. Worksheet # 8 Mar b. sin tan e. sin 2 sin 1 5. b. tan. c. sec sin 1 ( x )) cos 1 ( x )) f. csc. c.
MATH 6 WINTER 06 University Calculus I Worksheet # 8 Mar. 06-0 The topic covered by this worksheet is: Derivative of Inverse Functions and the Inverse Trigonometric functions. SamplesolutionstoallproblemswillbeavailableonDL,
More informationMath 112, Precalculus Mathematics Sample for the Final Exam.
Math 11, Precalculus Mathematics Sample for the Final Exam. Phone use is not allowed on this exam. You may use a standard two sided sheet of note paper and a calculator. The actual final exam consists
More informationSANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET
SANDERSON HIGH SCHOOL AP CALCULUS AB/BC SUMMER REVIEW PACKET 017-018 Name: 1. This packet is to be handed in on Monday August 8, 017.. All work must be shown on separate paper attached to the packet. 3.
More informationMath 112, Precalculus Mathematics Solutions to Sample for the Final Exam.
Math 11, Precalculus Mathematics Solutions to Sample for the Final Exam. Phone and calculator use is not allowed on this exam. You may use a standard one sided sheet of note paper. The actual final exam
More informationIntroduction Derivation General formula List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 21 Adrian Jannetta Recap: Binomial Series Recall that some functions can be rewritten as a power series
More informationJUST THE MATHS UNIT NUMBER DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) A.J.Hobson
JUST THE MATHS UNIT NUMBER.5 DIFFERENTIATION APPLICATIONS 5 (Maclaurin s and Taylor s series) by A.J.Hobson.5. Maclaurin s series.5. Standard series.5.3 Taylor s series.5.4 Exercises.5.5 Answers to exercises
More informationMATH 163 HOMEWORK Week 13, due Monday April 26 TOPICS. c n (x a) n then c n = f(n) (a) n!
MATH 63 HOMEWORK Week 3, due Monday April 6 TOPICS 4. Taylor series Reading:.0, pages 770-77 Taylor series. If a function f(x) has a power series representation f(x) = c n (x a) n then c n = f(n) (a) ()
More informationdx dx [x2 + y 2 ] = y d [tan x] + tan x = 2x + 2y = y sec 2 x + tan x dy dy = tan x dy dy = [tan x 2y] dy dx = 2x y sec2 x [1 + sin y] = sin(xy)
Math 7 Activit: Implicit & Logarithmic Differentiation (Solutions) Implicit Differentiation. For each of the following equations, etermine x. a. tan x = x 2 + 2 tan x] = x x x2 + 2 ] = tan x] + tan x =
More informationMath 106: Review for Exam II - SOLUTIONS
Math 6: Review for Exam II - SOLUTIONS INTEGRATION TIPS Substitution: usually let u a function that s inside another function, especially if du (possibly off by a multiplying constant) is also present
More informationIntegration by Parts
Calculus 2 Lia Vas Integration by Parts Using integration by parts one transforms an integral of a product of two functions into a simpler integral. Divide the initial function into two parts called u
More informationSection 8.7. Taylor and MacLaurin Series. (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications.
Section 8.7 Taylor and MacLaurin Series (1) Definitions, (2) Common Maclaurin Series, (3) Taylor Polynomials, (4) Applications. MATH 126 (Section 8.7) Taylor and MacLaurin Series The University of Kansas
More informationMath 112, Precalculus Mathematics Sample for the Final Exam.
Math 11, Precalculus Mathematics Sample for the Final Exam. Solutions. There is no promise of infallibility. If you get a different solution, do not be discouraged, but do contact me. (1) If the graph
More informationSection 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series
Section 9.7 and 9.10: Taylor Polynomials and Approximations/Taylor and Maclaurin Series Power Series for Functions We can create a Power Series (or polynomial series) that can approximate a function around
More informationHello Future Calculus Level One Student,
Hello Future Calculus Level One Student, This assignment must be completed and handed in on the first day of class. This assignment will serve as the main review for a test on this material. The test will
More informationChapter 8: Taylor s theorem and L Hospital s rule
Chapter 8: Taylor s theorem and L Hospital s rule Theorem: [Inverse Mapping Theorem] Suppose that a < b and f : [a, b] R. Given that f (x) > 0 for all x (a, b) then f 1 is differentiable on (f(a), f(b))
More informationDifferential Equations: Homework 2
Differential Equations: Homework Alvin Lin January 08 - May 08 Section.3 Exercise The direction field for provided x 0. dx = 4x y is shown. Verify that the straight lines y = ±x are solution curves, y
More informationMth Review Problems for Test 2 Stewart 8e Chapter 3. For Test #2 study these problems, the examples in your notes, and the homework.
For Test # study these problems, the examples in your notes, and the homework. Derivative Rules D [u n ] = nu n 1 du D [ln u] = du u D [log b u] = du u ln b D [e u ] = e u du D [a u ] = a u ln a du D [sin
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 23 Motivation Goal: We want to derive rules to find the derivative
More informationMath 12 Final Exam Review 1
Math 12 Final Exam Review 1 Part One Calculators are NOT PERMITTED for this part of the exam. 1. a) The sine of angle θ is 1 What are the 2 possible values of θ in the domain 0 θ 2π? 2 b) Draw these angles
More informationMath 250 Skills Assessment Test
Math 5 Skills Assessment Test Page Math 5 Skills Assessment Test The purpose of this test is purely diagnostic (before beginning your review, it will be helpful to assess both strengths and weaknesses).
More informationa x a y = a x+y a x a = y ax y (a x ) r = a rx and log a (xy) = log a (x) + log a (y) log a ( x y ) = log a(x) log a (y) log a (x r ) = r log a (x).
You should prepare the following topics for our final exam. () Pre-calculus. (2) Inverses. (3) Algebra of Limits. (4) Derivative Formulas and Rules. (5) Graphing Techniques. (6) Optimization (Maxima and
More informationMath 1552: Integral Calculus Final Exam Study Guide, Spring 2018
Math 55: Integral Calculus Final Exam Study Guide, Spring 08 PART : Concept Review (Note: concepts may be tested on the exam in the form of true/false or short-answer questions.). Complete each statement
More informationUnit #3 - Differentiability, Computing Derivatives, Trig Review
Unit #3 - Differentiability, Computing Derivatives, Trig Review Some problems and solutions selected or adapted from Hughes-Hallett Calculus. Derivative Interpretation and Existence 1. The cost, C (in
More informationMATH 1004E Get out of your seat and mingle until you find someone with a name that either has:
Monday 11:35-1:5 CORE CONCEPTS: Pre-calc review (key terms, exponents, factoring, rationalizing) Basic functions (absolute, trigonometric) Intro to limits (one-sided) OPENER (10min): Mingling Name Game
More informationWeek #6 - Taylor Series, Derivatives and Graphs Section 4.1
Week #6 - Talor Series, Derivatives and Graphs Section 4.1 From Calculus, Single Variable b Hughes-Hallett, Gleason, McCallum et. al. Copright 2005 b John Wile & Sons, Inc. This material is used b permission
More informationWeek #8 - Optimization and Newton s Method Section 4.5
Week #8 - Optimization and Newton s Method Section 4.5 From Calculus, Single Variable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material is used by permission
More informationSection 3.6 The chain rule 1 Lecture. Dr. Abdulla Eid. College of Science. MATHS 101: Calculus I
Section 3.6 The chain rule 1 Lecture College of Science MATHS 101: Calculus I (University of Bahrain) Logarithmic Differentiation 1 / 1 Motivation Goal: We want to derive rules to find the derivative of
More informationMultiple Choice Answers. MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March Question
MA 113 Calculus I Spring 2018 Exam 2 Tuesday, 6 March 2018 Name: Section: Last 4 digits of student ID #: This exam has 12 multiple choice questions (five points each) and 4 free response questions (ten
More informationMath 229 Mock Final Exam Solution
Name: Math 229 Mock Final Exam Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and that it
More informationCalculus I Review Solutions
Calculus I Review Solutions. Compare and contrast the three Value Theorems of the course. When you would typically use each. The three value theorems are the Intermediate, Mean and Extreme value theorems.
More informationPower Series. x n. Using the ratio test. n n + 1. x n+1 n 3. = lim x. lim n + 1. = 1 < x < 1. Then r = 1 and I = ( 1, 1) ( 1) n 1 x n.
.8 Power Series. n x n x n n Using the ratio test. lim x n+ n n + lim x n n + so r and I (, ). By the ratio test. n Then r and I (, ). n x < ( ) n x n < x < n lim x n+ n (n + ) x n lim xn n (n + ) x
More informationAP CALCULUS SUMMER WORKSHEET
AP CALCULUS SUMMER WORKSHEET DUE: First Day of School, 2011 Complete this assignment at your leisure during the summer. I strongly recommend you complete a little each week. It is designed to help you
More informationM152: Calculus II Midterm Exam Review
M52: Calculus II Midterm Exam Review Chapter 4. 4.2 : Mean Value Theorem. - Know the statement and idea of Mean Value Theorem. - Know how to find values of c making the theorem true. - Realize the importance
More information1 Arithmetic calculations (calculator is not allowed)
1 ARITHMETIC CALCULATIONS (CALCULATOR IS NOT ALLOWED) 1 Arithmetic calculations (calculator is not allowed) 1.1 Check the result Problem 1.1. Problem 1.2. Problem 1.3. Problem 1.4. 78 5 6 + 24 3 4 99 1
More informationMAT137 Calculus! Lecture 9
MAT137 Calculus! Lecture 9 Today we will study: Limits at infinity. L Hôpital s Rule. Mean Value Theorem. (11.5,11.6, 4.1) PS3 is due this Friday June 16. Next class: Applications of the Mean Value Theorem.
More informationNAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018
NAME: DATE: CLASS: AP CALCULUS AB SUMMER MATH 2018 A] Refer to your pre-calculus notebook, the internet, or the sheets/links provided for assistance. B] Do not wait until the last minute to complete this
More informationCalculus I Exam 1 Review Fall 2016
Problem 1: Decide whether the following statements are true or false: (a) If f, g are differentiable, then d d x (f g) = f g. (b) If a function is continuous, then it is differentiable. (c) If a function
More informationMTH30 Review Sheet. y = g(x) BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE
BRONX COMMUNITY COLLEGE of the City University of New York DEPARTMENT OF MATHEMATICS & COMPUTER SCIENCE MTH0 Review Sheet. Given the functions f and g described by the graphs below: y = f(x) y = g(x) (a)
More informationReview for Final. The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study:
Review for Final The final will be about 20% from chapter 2, 30% from chapter 3, and 50% from chapter 4. Below are the topics to study: Chapter 2 Find the exact answer to a limit question by using the
More informationWeek 1: need to know. November 14, / 20
Week 1: need to know How to find domains and ranges, operations on functions (addition, subtraction, multiplication, division, composition), behaviors of functions (even/odd/ increasing/decreasing), library
More informationMath 230 Mock Final Exam Detailed Solution
Name: Math 30 Mock Final Exam Detailed Solution Disclaimer: This mock exam is for practice purposes only. No graphing calulators TI-89 is allowed on this test. Be sure that all of your work is shown and
More informationMath 131 Final Exam Spring 2016
Math 3 Final Exam Spring 06 Name: ID: multiple choice questions worth 5 points each. Exam is only out of 00 (so there is the possibility of getting more than 00%) Exam covers sections. through 5.4 No graphing
More informationFamilies of Functions, Taylor Polynomials, l Hopital s
Unit #6 : Rule Families of Functions, Taylor Polynomials, l Hopital s Goals: To use first and second derivative information to describe functions. To be able to find general properties of families of functions.
More informationIntroduction Derivation General formula Example 1 List of series Convergence Applications Test SERIES 4 INU0114/514 (MATHS 1)
MACLAURIN SERIES SERIES 4 INU0114/514 (MATHS 1) Dr Adrian Jannetta MIMA CMath FRAS Maclaurin Series 1/ 19 Adrian Jannetta Background In this presentation you will be introduced to the concept of a power
More information14 Increasing and decreasing functions
14 Increasing and decreasing functions 14.1 Sketching derivatives READING Read Section 3.2 of Rogawski Reading Recall, f (a) is the gradient of the tangent line of f(x) at x = a. We can use this fact to
More informationAim: How do we prepare for AP Problems on limits, continuity and differentiability? f (x)
Name AP Calculus Date Supplemental Review 1 Aim: How do we prepare for AP Problems on limits, continuity and differentiability? Do Now: Use the graph of f(x) to evaluate each of the following: 1. lim x
More informationAbsolute and Local Extrema
Extrema of Functions We can use the tools of calculus to help us understand and describe the shapes of curves. Here is some of the data that derivatives f (x) and f (x) can provide about the shape of the
More informationUnit #17 - Differential Equations Section 11.7
Unit #17 - Differential Equations Section 11.7 Some material from Calculus, Single and MultiVariable by Hughes-Hallett, Gleason, McCallum et. al. Copyright 2005 by John Wiley & Sons, Inc. This material
More informationMath 261 Calculus I. Test 1 Study Guide. Name. Decide whether the limit exists. If it exists, find its value. 1) lim x 1. f(x) 2) lim x -1/2 f(x)
Math 261 Calculus I Test 1 Study Guide Name Decide whether the it exists. If it exists, find its value. 1) x 1 f(x) 2) x -1/2 f(x) Complete the table and use the result to find the indicated it. 3) If
More informationFinal Exam Review Quesitons
Final Exam Review Quesitons. Compute the following integrals. (a) x x 4 (x ) (x + 4) dx. The appropriate partial fraction form is which simplifies to x x 4 (x ) (x + 4) = A x + B (x ) + C x + 4 + Dx x
More informationMATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series.
MATH 1231 MATHEMATICS 1B CALCULUS. Section 5: - Power Series and Taylor Series. The objective of this section is to become familiar with the theory and application of power series and Taylor series. By
More informationA special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule.
The Chain Rule A special rule, the chain rule, exists for differentiating a function of another function. This unit illustrates this rule. In order to master the techniques explained here it is vital that
More informationMATH 408N PRACTICE FINAL
05/05/2012 Bormashenko MATH 408N PRACTICE FINAL Name: TA session: Show your work for all the problems. Good luck! (1) Calculate the following limits, using whatever tools are appropriate. State which results
More informationSESSION CLASS-XI SUBJECT : MATHEMATICS FIRST TERM
TERMWISE SYLLABUS SESSION-2018-19 CLASS-XI SUBJECT : MATHEMATICS MONTH July, 2018 to September 2018 CONTENTS FIRST TERM Unit-1: Sets and Functions 1. Sets Sets and their representations. Empty set. Finite
More informationSummer 2017 Review For Students Entering AP Calculus AB/BC
Summer 2017 Review For Students Entering AP Calculus AB/BC Holy Name High School AP Calculus Summer Homework 1 A.M.D.G. AP Calculus AB Summer Review Packet Holy Name High School Welcome to AP Calculus
More informationName: Answer Key No calculators. Show your work! 1. (21 points) All answers should either be,, a (finite) real number, or DNE ( does not exist ).
Mat - Final Exam August 3 rd, Name: Answer Key No calculators. Sow your work!. points) All answers sould eiter be,, a finite) real number, or DNE does not exist ). a) Use te grap of te function to evaluate
More information(c) Find the equation of the degree 3 polynomial that has the same y-value, slope, curvature, and third derivative as ln(x + 1) at x = 0.
Chapter 7 Challenge problems Example. (a) Find the equation of the tangent line for ln(x + ) at x = 0. (b) Find the equation of the parabola that is tangent to ln(x + ) at x = 0 (i.e. the parabola has
More informationNext, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations.
Section 6.3 - Solving Trigonometric Equations Next, we ll use all of the tools we ve covered in our study of trigonometry to solve some equations. These are equations from algebra: Linear Equation: Solve:
More information1.7 Sums of series Example 1.7.1: 2. Real functions of one variable Example 1.7.2: 2.1 General definitions Example 2.1.3: Example 2.1.
7 Sums of series We often want to sum a series of terms, for example when we look at polynomials As we already saw, we abbreviate a sum of the form For example and u + u + + u r by r u i i= a n x n + a
More informationMATH 115 Precalculus Spring, 2015, V1.2
MULTIPLE CHOICE 1. Solve, and express the answer in interval notation: 6 5x 4. 1. A. (, 2] [2/5, ) B. [2/5, 2] C. (, 2/5] D. (, 2/5] [2, ) 2. Which of the following polynomials has a graph which exhibits
More informationMATH 150 TOPIC 16 TRIGONOMETRIC EQUATIONS
Math 150 T16-Trigonometric Equations Page 1 MATH 150 TOPIC 16 TRIGONOMETRIC EQUATIONS In calculus, you will often have to find the zeros or x-intercepts of a function. That is, you will have to determine
More informationFinal practice, Math 31A - Lec 1, Fall 2013 Name and student ID: Question Points Score Total: 90
Final practice, Math 31A - Lec 1, Fall 13 Name and student ID: Question Points Score 1 1 1 3 1 4 1 5 1 6 1 7 1 8 1 9 1 Total: 9 1. a) 4 points) Find all points x at which the function fx) x 4x + 3 + x
More informationMath 106 Fall 2014 Exam 2.1 October 31, ln(x) x 3 dx = 1. 2 x 2 ln(x) + = 1 2 x 2 ln(x) + 1. = 1 2 x 2 ln(x) 1 4 x 2 + C
Math 6 Fall 4 Exam. October 3, 4. The following questions have to do with the integral (a) Evaluate dx. Use integration by parts (x 3 dx = ) ( dx = ) x3 x dx = x x () dx = x + x x dx = x + x 3 dx dx =
More informationFall 2016, MA 252, Calculus II, Final Exam Preview Solutions
Fall 6, MA 5, Calculus II, Final Exam Preview Solutions I will put the following formulas on the front of the final exam, to speed up certain problems. You do not need to put them on your index card, and
More informationAs f and g are differentiable functions such that. f (x) = 20e 2x, g (x) = 4e 2x + 4xe 2x,
srinivasan (rs7) Sample Midterm srinivasan (690) This print-out should have 0 questions. Multiple-choice questions may continue on the next column or page find all choices before answering. Determine if
More informationFunctions Modeling Change A Preparation for Calculus Third Edition
Powerpoint slides copied from or based upon: Functions Modeling Change A Preparation for Calculus Third Edition Connally, Hughes-Hallett, Gleason, Et Al. Copyright 2007 John Wiley & Sons, Inc. 1 Section
More informationMath 147 Exam II Practice Problems
Math 147 Exam II Practice Problems This review should not be used as your sole source for preparation for the exam. You should also re-work all examples given in lecture, all homework problems, all lab
More informationMATH 32 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS
MATH 2 FALL 2012 FINAL EXAM - PRACTICE EXAM SOLUTIONS (1) ( points) Solve the equation x 1 =. Solution: Since x 1 =, x 1 = or x 1 =. Solving for x, x = 4 or x = 2. (2) In the triangle below, let a = 4,
More informationAP Calculus AB One Last Mega Review Packet of Stuff. Take the derivative of the following. 1.) 3.) 5.) 7.) Determine the limit of the following.
AP Calculus AB One Last Mega Review Packet of Stuff Name: Date: Block: Take the erivative of the following. 1.) x (sin (5x)).) x (etan(x) ) 3.) x (sin 1 ( x3 )) 4.) x (x3 5x) 4 5.) x ( ex sin(x) ) 6.)
More informationSolutions to Second Midterm(pineapple)
Math 125 Solutions to Second Midterm(pineapple) 1. Compute each of the derivatives below as indicated. 4 points (a) f(x) = 3x 8 5x 4 + 4x e 3. Solution: f (x) = 24x 7 20x + 4. Don t forget that e 3 is
More informationIF you participate fully in this boot camp, you will get full credit for the summer packet.
18_19 AP Calculus BC Summer Packet NOTE - Please mark July on your calendars. We will have a boot camp in my room from 8am 11am on this day. We will work together on the summer packet. Time permitting,
More informationSET 1. (1) Solve for x: (a) e 2x = 5 3x
() Solve for x: (a) e x = 5 3x SET We take natural log on both sides: ln(e x ) = ln(5 3x ) x = 3 x ln(5) Now we take log base on both sides: log ( x ) = log (3 x ln 5) x = log (3 x ) + log (ln(5)) x x
More informationMATH 151 Engineering Mathematics I
MATH 151 Engineering Mathematics I Spring 2019, WEEK 10 JoungDong Kim Week 10 Section 4.2, 4.3, 4.4 Mean Value Theorem, How Derivatives Affect the Shape of a Graph, Indeterminate Forms and L Hospital s
More informationMA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, True/False 1 T F 2 T F 3 T F 4 T F 5 T F. Name: Section:
MA 113 Calculus I Fall 2016 Exam Final Wednesday, December 14, 2016 Name: Section: Last 4 digits of student ID #: This exam has five true/false questions (two points each), ten multiple choice questions
More informationMA Practice Exam #2 Solutions
MA 123 - Practice Exam #2 Solutions Name: Instructions: For some of the questions, you must show all your work as indicated. No calculators, books or notes of any form are allowed. Note that the questions
More information