Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Multiple integral wikipedia , lookup
Lebesgue integration wikipedia , lookup
Sobolev space wikipedia , lookup
Distribution (mathematics) wikipedia , lookup
Function of several real variables wikipedia , lookup
Neumann–Poincaré operator wikipedia , lookup
AP Calculus AB Lesson 3-8: Derivatives of Inverse Functions, Part 1 Name_________________________________ Date _________________________________ Learning Goal: I can calculate the numeric derivative of an inverse function I. Review: 1. Let f x 7 x 1 and g x x 1 7 a. Find f g 3 1 b. Find f g 2 c. Find f g x d. How are the functions f x and g x related? 2. Let k t 2t 5 . Find k 1 t . 3. Let h and m be inverses of each other. Find h(m( ) OVER Page 2 Let f x be some function that is one to one over its domain. Then some function g x is Theorem: the inverse of f x , written f 1 x iff f g x g f x x . II. Let f x x3 a. Find f 1 x b. Both f and f 1 are graphed to the right. Sketch the line y x . Why would we add this line to the graph? c. Find df . dx g. Find df 1 dx df dx d. e. f. df dx df dx df dx df 1 . dx h. x 1 df 1 dx x 1 x2 df 1 dx x 8 x 3 df 1 dx x 27 Conclusion: i. j. Page 3 * See geometric “proof” on SMART Board. See explanation in text book on page 165. df Theorem: If f is differentiable at every point of an interval I and is never zero on I, then f has an dx inverse and f 1 is differentiable at every point of the interval f I Example 1 Let f x x5 3x 2 , and let f 1 denote the inverse of f . Given that 1, 2 is on the graph of f, find df 1 dx x2 Example 2 If f and g are inverses, then we know that f g x x . d f g x x . Solve for g ' x . Explain how your answer makes sense with dx the conclusion from the previous page. Use the chain rule to find OVER Page 4 Practice #1 Let f x x5 2x3 x 1 (a) Find f 1 and f ' 1 . (b) Find f 1 3 and d 1 f 3 . dx Practice #2 Differentiable functions f and g have values as shown in the table. a. b. c. AP Calculus AB Lesson 3-8 , Part 1 Homework Name__________________________________ Date _________________________________ For the below problems: a) Verify that the functions have inverses b) Evaluate the derivative of the inverse at the given value 1. 2. 3. 4. f x x3 7 x 2 contains 1,10 ; Find df 1 dx f x x5 3x3 7 x 2 contains 1,13 ; Find f x x3 8x cos 3x contains 0,1 ; Find f x e 2 x x 10 df 1 dx x 13 df 1 dx x 1 df 1 9 x 4 contains 0,5 ; Find dx 3 x 5 OVER 5. Let f be the function defined by f x x3 x . If g x f 1 x and g 2 1 , what is the value of g ' 2 ? (A) 1 13 (B) 1 4 (C) 6. Let f be a function such that lim 7 4 (D) 4 f 2 h f 2 h 0 I. f is continuous at x 2 h (E) 13 5 . Which of the following must be true? II. f is differentiable at x 2 III. The derivative of f is differentiable at x 2 (A) I only (B) II only (C) I and II only (D) I and III only (E) II and III only