5128_Ch03_pp098-184

Section 3.1 Derivative of a Function 105
Quick Review 3.1 (For help, go to Sections 2.1 and 2.4.)
In Exercises 1–4, evaluate the indicated limit algebraically.
1. lim 2 h 2 4
4 2. lim
h→0 h
x 3
5/2
x→2 2
y
2x 8
3. lim 1
4. lim 8
y→0 y
x→4 x 2
5. Find the slope of the line tangent to the parabola y x 2 1
at its vertex. 0
6. By considering the graph of f x x 3 3x 2 2, find the
intervals on which f is increasing. (, 0] and [2, )
In Exercises 7–10, let
x 2, x 1
f x { x 1 2 , x 1.
7. Find lim x→1
f x and lim x→1
f x. lim f(x) 0 ; lim f(x) 3
x→1 x→1
8. Find lim h→0
f 1 h. 0
9. Does lim x→1 f x exist? Explain. No, the two one-sided limits
are different.
10. Is f continuous? Explain. No, f is discontinuous at x 1 because
the limit doesn’t exist there.
Section 3.1 Exercises
In Exercises 1–4, use the definition
13.
f a lim f (a h ) f (a)
(b)
h→0 h
to find the derivative of the given function at the indicated point.
1. f (x) 1x, a 2 14 2. f (x) x 2 4, a 1 2
3. f (x) 3 x 2 , a 1 2 4. f (x) x 3 x, a 0 1
In Exercises 5–8, use the definition
14.
f (a) lim f (x (a)
) f (a)
x→a x a
to find the derivative of the given function at the indicated point.
5. f (x) 1x, a 2 14 6. f (x) x 2 4, a 1 2
7. f (x) x, 1 a 3 14 8. f (x) 2x 3, a 1 2
9. Find f x if f x 3x 12. f(x) 3
10. Find dydx if y 7x. dy/dx 7
d
11. Find x d x
2 . 2x
15.
(d)
d
12. Find f (x) if f (x) 3x 2 . 6x
dx
In Exercises 13–16, match the graph of the function with the graph of
the derivative shown here:
y'
y'
y
O
y
O
y
O
y f 1 (x)
x
y f 2 (x)
x
y f 3 (x)
x
O
x
O
x
16.
(c)
O
y
y f 4 (x)
x
(a)
(b)
O
y'
x
O
y'
x
17. If f 2 3 and f 2 5, find an equation of (a) the tangent
line, and (b) the normal line to the graph of y f x at the
point where x 2.
(a) y 5x 7
(b) y 1 5 x 1 7
5
(c)
(d)