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Unit 2 Review - Differentiation: Definition & Fundamental Properties

This document provides a review of key concepts related to differentiation including: 1) Finding the average rate of change of functions over given intervals using appropriate units 2) Finding the derivative of various functions using the definition of the derivative 3) Interpreting the meaning of function values in terms of the context 4) Finding derivatives, tangent lines, and normal lines of various functions

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Azra Ozen
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293 views3 pages

Unit 2 Review - Differentiation: Definition & Fundamental Properties

This document provides a review of key concepts related to differentiation including: 1) Finding the average rate of change of functions over given intervals using appropriate units 2) Finding the derivative of various functions using the definition of the derivative 3) Interpreting the meaning of function values in terms of the context 4) Finding derivatives, tangent lines, and normal lines of various functions

Uploaded by

Azra Ozen
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Name: _______________________________ Date:_______________ Period: _____ Review

Unit 2 Review – Differentiation: Definition & Fundamental


Properties
Reviews do NOT cover all material from the lessons but will hopefully remind you of key points. To be prepared,
you should review all packets from Unit 2.
Find the average rate of change of each function on the given interval. Use appropriate units if necessary.
1. 𝑤(𝑥) = ln 𝑥 ; 1 ≤ 𝑥 ≤ 7 2. 𝑠(𝑡) = −𝑡 − 𝑡 + 4; [1, 5]
𝑡 represents seconds
𝑠 represents feet

( ) ( )
3. Find the derivative of 𝑦 = 2𝑥 + 3𝑥 − 1 by using the definition of the derivative. lim
→

4. For the function ℎ(𝑡), ℎ is the temperature of the oven in Fahrenheit, and 𝑡 is the time measured in minutes.
a. Explain the meaning of the equation ℎ(15) = 420.

b. Explain the meaning of the equation ℎ (43) = −11.

Find the derivative of each function.


5. 𝑓(𝑥) = 4 − 6. 𝑔(𝑥) = 3√𝑥 − + 5𝜋 7. ℎ(𝑥) = 4𝑒 − 2 cos 𝑥

8. 𝑠(𝑡) = 𝑡 sin(𝑡) 9. 𝑑(𝑡) = 3√𝑡 ln 𝑡


10. 𝑦 = − sec 𝑥 11. ℎ(𝑥) =

Find the equation of the tangent line of the function at the given x-value.
12. 𝑓(𝑥) = −2𝑥 + 3𝑥 at 𝑥 = −1. 13. 𝑓(𝑥) = 4 sin 𝑥 − 2 at 𝑥 = 𝜋

14. Find the equation for the normal line of 𝑦 = 𝑥 + 𝑥 − 4 at 𝑥 = −3

15. If 𝑓(𝑥) = 3 sin 𝑥 − 2𝑒 find 𝑓 (0). No calculator!

A calculator is allowed on the following problems.


16. If 𝑓(𝑥) = 𝑥 sin(3𝑥 − 2); find 𝑓 (7). 17. If 𝑓(𝑥) = csc(3𝑥) at 𝑥 = 2.

18. Use the table below to estimate the value of 𝑑 (120). Indicate units of measures.

𝑡
2 13 60 180 500
seconds
𝑑(𝑡)
10 81 412 808 2,105
feet
19. Is the function differentiable at 𝑥 = 2?
3𝑥 − 3𝑥 − 5, 𝑥 < 2
𝑓(𝑥) =
7 − 9𝑥, 𝑥≥2

20. What values of 𝑎 and 𝑏 would make the function differentiable at 𝑥 = 4?


𝑎√𝑥 + 𝑏𝑥 − 1, 𝑥 < 4
𝑓(𝑥) = 16
+ 𝑏𝑥, 𝑥≥4
𝑥

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