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Calculus: Inverse Function Derivatives

The document provides practice problems related to differentiating inverse functions in calculus, focusing on finding values of functions and their derivatives. It includes various scenarios with given values of differentiable functions f and g, as well as questions about tangent lines and the behavior of inverse functions. Additionally, it discusses the implications of certain values for the inverse of a function and includes a test prep section.

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14 views3 pages

Calculus: Inverse Function Derivatives

The document provides practice problems related to differentiating inverse functions in calculus, focusing on finding values of functions and their derivatives. It includes various scenarios with given values of differentiable functions f and g, as well as questions about tangent lines and the behavior of inverse functions. Additionally, it discusses the implications of certain values for the inverse of a function and includes a test prep section.

Uploaded by

aslihancecen007
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© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as PDF, TXT or read online on Scribd
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3.

3 Differentiating Inverse Functions


Calculus
Practice
𝟏 (𝒙)
For each problem, let 𝒇 and 𝒈 be differentiable functions where 𝒈(𝒙) = 𝒇 for all 𝒙.
1. 𝑓(3) = −2, 𝑓(−2) = 4, 2. 𝑓(1) = 5, 𝑓(2) = 4,
𝑓 (3) = 5, and 𝑓 (−2) = 1. 𝑓 (1) = −2, and 𝑓 (2) = −4.
Find 𝑔 (−2). Find 𝑔 (5).

3. 𝑓(6) = −2, 𝑓(−3) = 7, 4. 𝑓(−1) = 4, 𝑓(2) = −3,


𝑓 (6) = −1, and 𝑓 (−3) = 3. 𝑓 (−1) = −5, and 𝑓 (2) = 7.
Find 𝑔 (7). Find 𝑔 (−3).

The table below gives values of the differentiable function 𝒈 and its derivative 𝒈 at selected values of 𝒙.
Let 𝒉(𝒙) = 𝒈 𝟏 (𝒙).

𝒙 𝒈(𝒙) 𝒈 (𝒙)
−1 −2 −4
−2 −5 −2
−3 −4 −1
−4 −3 −5
−5 −1 −3

5. Find ℎ (−1) 6. ℎ (−3) 7. ℎ (−5)

Find the equation of the tangent Find the equation of the tangent Find the equation of the tangent
line to 𝑔 at 𝑥 = −1. line to 𝑔 at 𝑥 = −3. line to 𝑔 at 𝑥 = −5.
𝒇 and 𝒈 are differentiable functions. Use the table to answer the problems below. 𝒇 and 𝒈 are
NOT inverses!
𝑥 𝑓(𝑥) 𝑓 (𝑥) 𝑔(𝑥) 𝑔 (𝑥)
1 5 −5 4 5
2 1 −6 3 3
3 6 4 1 6
4 2 9 6 1
5 3 1 1 2
6 4 2 2 4
8. 𝑔 (4) 9. 𝑓 (5) 10. 𝑔 (3)

11. 𝑓 (1) 12. Find the line tangent to the graph of 𝑓 (𝑥) at
𝑥 = 2.

𝟏 (𝒙)
For each function 𝒈(𝒙), its inverse 𝒈 = 𝒇(𝒙). Evaluate the given derivative.
13. 𝑔(𝑥) = cos(𝑥) + 3𝑥 14. 𝑔(𝑥) = 2𝑥 − 𝑥 − 5𝑥
𝑔 = . Find 𝑓 𝑔(−2) = −10. Find 𝑓 (−10)

15. 𝑔(𝑥) = √8 − 2𝑥. Find 𝑓 (4)? 16. 𝑔(𝑥) = 𝑥 − 7. Find 𝑓 (20)? 17. 𝑔(𝑥) = . Find 𝑓 ?
3.3 Differentiating Inverse Functions Test Prep
18. The functions 𝑓 and 𝑔 are differentiable for all real numbers and 𝑔 is strictly increasing. The table
below gives values of the functions and their first derivatives at selected values of 𝑥. The function ℎ
is given by ℎ(𝑥) = 𝑓 𝑔(𝑥) − 6.

𝑥 𝑓(𝑥) 𝑓 (𝑥) 𝑔(𝑥) 𝑔 (𝑥)


1 6 4 2 5
2 9 2 3 1
3 10 −4 4 2
4 −1 3 6 7

(a) Explain why there must be a value 𝑟 for 1 < 𝑟 < 3 such that ℎ(𝑟) = −5.

(b) If 𝑔 is the inverse function of 𝑔, write an equation for the line tangent to the graph of
𝑦 = 𝑔 (𝑥) at 𝑥 = 2.

19. A function ℎ satisfies ℎ(3) = 5 and ℎ (3) = 7. Which of the following statements about the inverse
of ℎ must be true?

(A) (ℎ ) (5) = 3 (B) (ℎ ) (7) = 3 (C) (ℎ ) (5) = 7

(D) (ℎ ) (5) = (E) (ℎ ) (7) =

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