Page 37 - 2024-calc4e-SE proofs-4e.indd

Sullivan 04 apcalc4e 45342 ch02 166 233 5pp August 7, 2023 12:54

Section 2.2 • The Derivative as a Function; Differentiability 187

2.2 Assess Your Understanding
Concepts and Vocabulary
1. True or False The domain of a function f and the domain of its In Problems 23–26, for each figure determine if the graphs represent a
′
′
derivative function f are always equal. function f and its derivative f . If they do, indicate which is the graph
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2. True or False If a function is continuous at a number c, then it of f and which is the graph of f .
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is differentiable at c.
23. 24.
3. Multiple Choice If f is continuous at a number c and
y y
f (x) − f (c) 4 4
if lim is infinite, then the graph of f has
x→c x − c
2 2
[(a) a horizontal (b) a vertical (c) no]
tangent line at c. 4 2 2 4 x 2 2 4 x
4. The instruction, “Differentiate f ,” means to find the of f. 2 2
4 4
Skill Building
In Problems 5–10, find the derivative of each function f at any real
25. 26.
number c. Use Form (1) on page 179.
y y
4
5. f (x) = 10 10
2
6. f (x) = −4
7. f (x) = 5x + 8 5 5 10 x 4 2 2 4 x
2
8. f (x) = 6x − 4 10
9. f (x) = 2 − x 2 4
2
10. f (x) = 2x + 4
In Problems 27–30, use the graph of f to obtain the graph of f .
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In Problems 11–16, differentiate each function f and determine the
domain of f . Use Form (2) on page 179.
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27. 28.
11. f (x) = 5 y y
4 4
12. f (x) = −2 y  f(x)
2 2
2
13. f (x) = 4x + x + 6 y f(x)
2
14. f (x) = 3x − x − 8 4 2 2 4 x 4 2 2 4 x
PAGE √ 2 2
180 15. f (x) = 5 x − 1
√ 4 4
16. f (x) = 4 x + 3
PAGE
In Problems 17–22, differentiate each function f . Graph y = f (x) 182 29. 30.
and y = f (x) on the same set of coordinate axes. y y
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8
1
17. f (x) = x + 1 2
3 y  f(x) 6
2
18. f (x) = −4x − 5 y  f(x)
4 2 4 x 4
PAGE 2
181 19. f (x) = 2x − 5x 4
2
2
20. f (x) = −3x + 2 6
3
21. f (x) = x − 8x 4 2 2 4 x
3
22. f (x) = −x − 8

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