c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
Math 151 Week in Review 5
Section 3.2-3.3
1. Differentiate the following functions.
√
1
+ π2
(a) f (x) = 9x + 3 x + √
53x
√
(b) f (x) = (5x5 − 7x3 + 5)(3x6 − 10x2 + e5 + cos 3)
√
√
10
7
− 4x x
(c) g(x) = x3 +
x
5s2 + 7s
(d) h(s) = 3
s −s
4t4 + 3t − 2
√
(e) g(t) =
t
2. Find the equation of the tangent line to the graph of f (x) =
x
at x = 1.
x2 + 5
3. Consider the graphs of f (x) and g(x) below.
(a) Find f (1), f (3), f ′ (1), f ′ (3).
(b) Find g(1), g(3), g ′ (1), g′ (3).
(c) Calculate h′ (1) where h(x) = f (x)g(x)
f (x)
(d) Calculate u′ (3) where u(x) =
xg(x)
4. Consider the function f (x) = 2x(x2 − 1).
(a) Find the values of x for which the tangent line to the graph of f is horizontal.
(b) Find the values of x for which the tangent line to the graph of f is parallel to the line 8x − 2y = 9.
5. For what values of a and b is the line y = 3x + b tangent to the graph of f (x) = ax2 when x = −2.
6. At what points on the graph of f (x) = −x2 + 4 does the tangent line also pass through the point
(1, 7)?
7. For f (x) defined below, determine where f not differentiable?
f (x) =


4x + 11



2




6−x
−2x + 7
x2 − 8
if
if
if
if
x ≤ −2
−2<x<1
1≤x≤3
x>3
1
c
Math 151 WIR, Spring 2014, Benjamin
Aurispa
8. Given f (x) below, find the values of a and b that make f differentiable everywhere.
f (x) =
(
ax + b
x2 − x
if x ≤ 3
if x > 3
9. The graph below represents the position after t seconds of an object moving in a straight line. When
is the object moving forward? backward? at rest?
10. From the edge of an 80 ft building, a ball is thrown straight up into the air with a speed of 64 ft/s.
After t seconds, its height from the ground is given by the function s(t) = −16t2 + 64t + 80.
(a) What is the maximum height the ball reaches?
(b) What is the ball’s velocity when it hits the ground?
11. An object moves in a straight line with position after t seconds, t ≥ 0, given by the function
s(t) = t3 − 9t2 + 24t where distance is measured in ft.
(a) When is the object at rest? moving forward? moving backward?
(b) Determine the total distance traveled by the object in the first 6 seconds.
12. (a) Determine the rate at which the area of a circle is changing with respect to the radius when r = 2
inches.
(b) If the radius of the circle is changing at a rate of 3 inches/min (so that r = 3t), determine the
rate at which the area is increasing after 2 minutes.
2