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Hello AP Calculus Students, There Are Several Options for Turning in Your Hello AP Calculus students, There are several options for turning in your work: Use a google doc, or share an email, or screen shot your work on an ipad and share it with me at [email protected] . I will send out information on Remind or you can check school wires. Also check your school email from time to time. I will have video lessons from the AP program on these 2 weeks of lessons online as they are what is needed to finish up the course before taking the AP test. Book work will continue through the ipad and as always answers are available online.To have continuity of learning here is a brief overview of the next 2 weeks AP Online video is found on area here: https://www.youtube.com/watch?v=EPKKUzsnPBg&list=PLoGgviqq4844keKrijbR_EPKRNIW6hahV&inde x=12 The ap free response is found here: 2019: https://apstudents.collegeboard.org/ap/pdf/ap19-frq-calculus-ab.pdf 2018: https://apcentral.collegeboard.org/pdf/ap18-frq-calculus-ab.pdf 2017: https://apcentral.collegeboard.org/pdf/ap-calculus-ab-frq-2017.pdf 2016: https://secure-media.collegeboard.org/digitalServices/pdf/ap/ap16_frq_calculus_ab.pdf 4/16 4/17 Test on separable Derivative of an differential inverse equations Worksheets attached 4/20 4/21 4/22 4/23 4/24 Quiz on derivative Watch ap online Review area Test on area Review project of an inverse video of area between curves between curves the calculus between curves packet from conundrum and do problems before break in drop box from Study for test Worksheet video attached 4/27 4/28 4/29 4/30 2019 AP free 2018 free 2017 free 2016 free response response response response questions 1-6 questions 1-6 questions 1-6 questions 1-6 See above link See above link See above link See above link Stay Safe! Mr. Grice Finding the derivative of an inverse: Recall if f(x) and g(x) are inverses then f(g(x)) = g(f(x)) = x Also g(x) notation wise for an inverse would be f-1(x) Note: d/dx (x) = 1 and the chain rule for the derivative of f-1(x) yields d/dx f(f-1(x)) = f’(f-1(x)) •(f-1(x)’ Therefore taking the derivative of f(f-1(x)) = x yields f’(f-1(x)) •(f-1(x))’ = 1 -1 1 Dividing gives the derivative (f (x))’ = −1 f’(푓 (x)) Example: The function h is given by h(x) = x5 + 3x – 2 and h(1) =2. If h-1 is the inverse of h, what is the value of (h-1)‘(2)? Solution: h’(x) = 5x4 + 3 h(1) = 2 h-1(2) = 1 1 1 1 (h-1)‘(2) = = = ℎ′(ℎ−1(2)) ℎ′(1) 8 AP Calculus Name ________________________________ CHAPTER 7 WORKSHEET INVERSE FUNCTIONS Seat # ______ Date ____________________ Derivatives of Inverse Functions In 1-3, use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse. x 4 1. f x 2x 2 2. gx x a3 b 3. hx 2 x x3 4 4. Think About It…Find the derivative of y tan x . Notice that the subject derivative has the same sign for all values of x, so is a monotonic function. However, is not a one- to-one function. Why? In 5-6, (a) “delete” part of the graph of the function shown so that the part that remains is one-to-one. Then, (b) find the inverse of the remaining part and (c) state its domain. (Note: there is more than one correct answer for these questions!) 5. f x x 32 6. gx x 5 In 7-9, find the derivative of the inverse function at the corresponding value. 3 d 1 7. Given f (x) x 2x 1, find f (Note: you may need to use guess and check to solve an dx x 2 equation involved in this problem.) d 8. Given g(x) 2x5 x3 1, find g 1 dx x1 π π d 1 9. Given h(x) sin x on the interval , find h 2 2 dx x 1 2 SEE OTHER SIDE 10. Selected values of a strictly monotonic function (푥) and its derivative ′(푥) are shown on the table below. 푥 −3 −1 1 4 g(x) 5 1 0 −3 1 1 g’(x) −4 −2 5 6 a) Find g 1 1 b) Find g 1 3 11. Selected values of a strictly monotonic function ℎ(푥) and its derivative ℎ′(푥) are shown on the table below. 푥 −1 0 2 4 h(x) −5 −1 4 7 1 1 h’(x) 3 5 2 6 Let f x be a function such that f x h1x. a) Find f '1 b) Find f '4 True or False? In 12-15, determine whether the statement is true or false. Justify your answer. 12. If f (x) is an even function, then f 1 (x) exists. 13. If the inverse of f exists, then the y-intercept of f is an x-intercept of f -1. 14. If f (x) xn where n is odd, then exists. 15. There exists no function f such that f = f -1. AP Calculus CHAPTER 7 WORKSHEET ANSWER KEY INVERSE FUNCTIONS Derivatives of Inverse Functions 1. f 'x x3 4x xx2 1 Performing a sign analysis, f 'x 0 if x < 0, but f 'x 0 if x > 0 (except at x = 1), so this is not a strictly monotonic function and it does not have an inverse function. 2 2. g'x 3x a Performing a sign analysis, g'x 0 for all values of x, except at x = −a. So this is a strictly monotonic function and it has an inversey function. tan x 2 3. h'x 1 3x Performing a sign analysis, h'x 0 for all values of x. So this is a strictly monotonic function and it has an inverse function. 4. y' sec2 x . We have y' sec2 x 0, for all values of x included in the domain of sec x. Therefore 2 is always increasing. But the graph of has vertical tangent lines f x x 3 gx x 5 and it does not pass the horizontal line test: is not a one-to-one function. 5. 6. 1 f x x 3 1 3 g x x 5 f (x) x 2x 1 1 Domain of f x is [0, ) Domain of g 1 x is 5 3 g(x) 2x x 1 d 1 1 7. Given , f 1 dx x 2 f '1 5 d 1 8. Given , g 1 undefined dx x1 g'0 d 1 2 2 3 9. Given on the interval , h1 dx 3 3 x 1 2 h' 6 1 1 10. 푥 −3 −15 16 4 g(x) 5 1 0 −3 g’(x) −4 −2 1 a) g 1 1 5 g'1 1 1 b) g 1 3 g'4 2 1 1 11. 2 6 푥 −1 0 2 4 h(x) −5 −1 4 7 h’(x) 3 5 1 a) f '1 2 h'0 1 1 b) f '(x4) 6 f (x) h'2 12. If is an even function, then exists. FALSEf (x). Anxn even function has symmetry with respect to the y-axis and, therefore, cannot be one- to-one. 13. If the inverse of f exists, then the y-intercept of f is an x-intercept of f -1. TRUE. Switching x and y coordinates will result in switching x and y intercepts. 14. If where n is odd, then exists. TRUE. If where n is odd, its derivative is f '(x) nx n1 where n – 1 is even. So f '(x) 0 for all values of x except x = 0. Therefore is strictly monotonic. 15. There exists no function f such that f = f -1. FALSE. There are many such functions! Some examples: 푦 = 푥, 푦 = −푥, 푦 = −푥 + 푎,… π π h(x) sin x , 2 2 The Calculus Conundrum So there I was… I was talking to another calculus teacher, Mrs. AP Calca, and she relayed this story to me. Gather around and I will tell you a tale about the horrendous events of this past weekend. You may notice my bedraggled appearance accented by the dark circles under my eyes. I haven’t slept a wink in two days. Someone committed some dastardly deeds which scared me half to death. I am sorry to say that the culprit may be someone you have heard of before. Read the account which follows and see whether you can determine the identity of the perpetrator from the following list of suspects. Cross of suspects as you go. Whomever is not crossed off is the guilty suspect. Or said another way… whomever you derive to be guilty will become the prime suspect... (get it?) 1. Georg Riemann 2. Michel Rolle 3. I Sac Newton 4. Pierre de Fermat 5. Marquis de L’Hospital 6. Wilhelm Leibniz 7. Rene Descartes 8. Aria Betweencurves 9. Derry Vative 10. Int E Gruel On Friday afternoon, I left the building quite late after working on a set of calculus tests. As I walked to my car, I passed by the building and narrowly escaped being struck in the head by a falling object. Sadly, that object was the bust of Sir Isaac Newton, a cherished gift from my calculus class. As I picked up Sir Isaac’s remains, I discovered a note glued to one of the shards.
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