Page 79 - 2024-calc4e-SE proofs-4e.indd

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

Chapter 2 • Review Exercises 229

2.4 1 Differentiate the product of two functions (p. 203) 1, 2 27, 28, 36, 46, 47–50, 53–56, 60 6, 7, 9
2 Differentiate the quotient of two functions (p. 206) 3–6 29–35, 37–43, 57–59, 68, 73, 74 3, 10
3 Find higher-order derivatives (p. 208) 7, 8 61–66, 71, 72 8, 12
4 Find the acceleration of an object moving on a 9 71, 72 8, 12
line (p. 210)
2.5 1 Differentiate trigonometric functions (p. 218) 1–6 49–60, 70 1, 9, 11
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REVIEW EXERCISES

In Problems 1 and 2, use a definition of the derivative to find the rate 17. Use the information in the graph of y = f (x) to sketch the graph
of change of f at the indicated numbers. of y = f (x).
′
√
1. f (x) = x at (a) c = 1 (b) c = 4 y
(c) c any positive real number (2, 4)
4
2 (0, 3)
2. f (x) = at (a) c = 0 (b) c = 2 (4, 2) y f(x)
x − 1
(c) c any real number, c 6= 1 1
(6, 0) (2, 1) (5, 0)
In Problems 3–8, use a definition of the derivative to find the derivative
6 4 2 2 4 6 x
of each function at the given number.
2
3. F(x) = 3x + 6 at −2 4. f (x) = 8x + 1 at −1 f!(6) f!(4) f!(2) f!(0) f!(2) f!(5)
3 2 0 0 1 0 3
2
5. f (x) = 3x + 5x at 0 6. f (x) = at 1
x
√ x + 1 18. Match the graph of y = f (x) with the graph of its derivative.
7. f (x) = 4x + 1 at 0 8. f (x) = at 1
2x − 3 y
8 y f(x)
In Problems 9–12, use a definition of the derivative to find the 6 (5, 8)
derivative of each function. Graph f and f on the same set of axes. (3, 6)
′
2 (0, 3)
9. f (x) = x − 6 10. f (x) = 7 − 3x 2 x
1 5 3 1 1 3 5
11. f (x) = 12. f (x) = π 4
2x 3
In Problems 13 and 14, determine whether the function f has a
derivative at c. If it does, find the derivative. If it does not, explain why. y y
Graph each function.
3 3
3
13. f (x) = |x − 1| at c = 1 (3, 3)
2 (5, 1)
4 − 3x if x ≤ −1 1 1
14. f (x) = 3 at c = −1
−x if x > −1
5 3 1 1 3 5 x 5 3 1 3 5 x
1 1
In Problems 15 and 16, determine whether the graphs represent a
′
function f and its derivative f . If they do, indicate which is the graph
of f and which is the graph of f . 3 3
′
15. 16.
(A) (B)
y
y 6
6 y y
4
5 (5, 5) 5
4
2
3 3
2
(5, 1)
1 1 x 1 1 1
3
2 1 1 2 x 2
5 3 1 (0, 0) 3 5 x 5 (0, 0) 3 5 x
2 (3, 1)
4
3 3
5 5
(C) (D)
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