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Math 151 WIR, Spring 2013, Benjamin
Aurispa
Math 151 Week in Review 5
Sections 3.2 & 3.3
1. Differentiate the following functions.
√
1
+ π2
(a) f (x) = 9x + 3 x + √
53x
√
(b) f (x) = (5x5 − 7x3 + 9x + 5)(3x6 − 10x2 + e5 + cos 3)
5x2 + 7x
x3 − x
4t4 + 3t − 2
√
(d) g(t) =
t
(c) h(x) =
2. Find the equation of the tangent line to the graph of f (x) =
x
at x = 1.
x2 + 5
3. Given the graphs of f (x) and g(x) below calculate the following.
(a) h′ (1) where h(x) = g(x)[f (x) + x2 ]
f (x)
(b) 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. Find f ′ (x) for the function below. Where is f not differentiable?


4x + 11


 6 − x2
f (x) =
 −2x + 7


 2
x −8
if
if
if
if
x ≤ −2
−2<x<1
1≤x≤3
x>3
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
1
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Math 151 WIR, Spring 2013, Benjamin
Aurispa
9. From the edge of a 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?
10. An object moves in a straight line with position after t seconds given by the function s(t) = t3 −9t2 +24t
where distance is measured in ft. Determine the total distance traveled by the object in the first 6
seconds.
11. (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, determine the rate at which the
area is increasing after 2 minutes.
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