5128_Ch03_pp098-184

114 Chapter 3 Derivatives
Quick Review 3.2 (For help, go to Sections 1.2 and 2.1.)
In Exercises 1–5, tell whether the limit could be used to define f a 6. Find the domain of the function y x 43 . All reals
(assuming that f is differentiable at a).
7. Find the domain of the function y x 34 . [0, )
1. lim f a h f a
Yes 2. lim f a h f h
No
8. Find the range of the function y x 2 3. [3, )
h→0 h
h→0 h
3. lim f x f a
Yes 4. lim f a 9. Find the slope of the line y 5 3.2x p. 3.2
f x
Yes
x→a x a
x→a a x
10. If f x 5x, find
f a h f a h
f 3 0.001 f 3 0.001
5. lim No
. 5
h→0 h 0.002
Section 3.2 Exercises
In Exercises 1–4, compare the right-hand and left-hand derivatives
to show that the function is not differentiable at the point P. Find all
points where f is not differentiable.
1. y
2. y
y f (x)
y f(x)
y x 2
y 2x
P(0, 0)
3. y
4.
y f(x)
1
0
P(1, 1)
y 1 x
–
1
y x
y 2x 1
x
x
y 2
y x
y f(x)
(a) All points in [3, 3] except x 0 (b) None (c) x 0
7. y
8. y
y f (x)
y f (x)
D: –3 ≤ x ≤3
D: –2 ≤ x ≤3
3
1
2
x
–3 –2 –1 0 1 2 3
1
–1
x
–2
–2 –1 0 1 2 3
(a) All points in [2, 3] except x 1, 0, 2
(b) x 1 (c) x 0, x 2
2
1
0
1
y
P(1, 2)
1 2
P(1, 1)
y 1 – x
1
x
x
(a) All points in [1, 2] except x 0
(b) x 0 (c) None
9. y
10.
y f(x)
D: –1 ≤ x ≤2
x
x
–1 0 1 2
–3 –2 –1 0 1 2 3
(a) All points in [3, 3] except x 2, 2
(b) x 2, x 2 (c) None
In Exercises 11–16, the function fails to be differentiable at x 0.
Tell whether the problem is a corner, a cusp, a vertical tangent, or a
discontinuity. Discontinuity
tan
11.
1 x, x 0
y { 12. y x 45 Cusp
1, x 0
13. y x x 2 2 Corner 14. y 3 3 x Vertical tangent
15. y 3x 2x 1 Corner 16. y 3 x Cusp
y y f(x)
D: –3 ≤ x ≤3
4
In Exercises 17–26, find the numerical derivative of the given function
at the indicated point. Use h 0.001. Is the function differentiable
at the indicated point?
17. f (x) 4x x
In Exercises 5–10, the graph of a function over a closed interval D is
2 , x 0 4, yes 18. f (x) 4x x 2 , x 3 2, yes
given. At what domain points does the function appear to be
19. f (x) 4x x 2 , x 1 2, yes 20. f (x) x 3 4x, x 0
(a) differentiable? (b) continuous but not differentiable? 21. f (x) x 3 4x, x 2 22. f (x) x 3 3.999999, yes
4x, x 2
(c) neither continuous nor differentiable?
23. f (x) x 23 8.000001, yes
8.000001, yes
, x 0 24. f (x) x 3, x 3
0, no
0, no
(a) All points in [3, 2] (b) None (c) None (a) All points in [2, 3] (b) None (c) None
5. y f (x) y
6. y
25. f (x) x 2/5 , x 0 26. f (x) x 4/5 , x 0
y f (x)
0, no
0, no
D: –3 ≤ x ≤2
D: – 2 ≤x ≤ 3
2
2
Group Activity In Exercises 27–30, use NDER to graph the derivative
of the function. If possible, identify the derivative function by
1
1
looking at the graph.
x
x
–3 –2 –1 0 1 2
–2 –1 0 1 2 3
27. y cos x 28. y 0.25x 4
–1
–1
xx
–2
–2
29. y 30. y ln cos x
2
1
In Exercises 31–36, find all values of x for which the function is
differentiable.
x
31. f x 3 8
x 2 32. hx 3 3x 6 5
4x
5
All reals except x 2
33. Px sin x 1 34. Qx 3 cos x All reals
x 1 2 , x 0
35. gx 2x 1, 0 x 3 All reals except x 3
{
4 x 2 , x 3
31. All reals except x 1, 5
33. All reals except x 0
2