Math 151 WIR, Spring 2014, c Benjamin Aurispa
Section 3.2-3.3
1. Differentiate the following functions.
(a) f ( x ) = 9 x + 3
√ x +
5
1
3 x
+ π
2
(b) f ( x ) = (5 x 5
− 7 x 3 +
√
5)(3 x 6
− 10 x 2 + e 5 + cos 3)
(c) g ( x ) =
√ x 3 +
10 x
− 4 x
√ x
(d) h ( s ) =
5 s 2 + 7 s s 3
− s
(e) g ( t ) =
4 t
4 + 3
√ t − 2 t
1

Math 151 WIR, Spring 2014, c Benjamin Aurispa
2. Find the equation of the tangent line to the graph of f ( x ) = x x 2 + 5 at x = 1.
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 )
(d) Calculate u ′ (3) where u ( x ) = f ( x ) xg ( x )
2

Math 151 WIR, Spring 2014, c Benjamin Aurispa
4. Consider the function f ( x ) = 2 x ( x 2
− 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 8 x − 2 y = 9.
5. For what values of a and b is the line y = 3 x + b tangent to the graph of f ( x ) = ax
2 when x = − 2.
6. At what points on the graph of f ( x ) = − x 2
(1 , 7)?
+ 4 does the tangent line also pass through the point
3

Math 151 WIR, Spring 2014, c Benjamin Aurispa
7. For f ( x ) defined below, determine where f not differentiable?
 f ( x ) =








4 x + 11 if x ≤ − 2
6 − x 2 if − 2 < x < 1
− 2 x + 7 if 1 ≤ x ≤ 3 x 2
− 8 if x > 3
8. Given f ( x ) below, find the values of a and b that make f differentiable everywhere.
f ( x ) =
( ax + b if x ≤ 3 x
2
− x 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?
4

Math 151 WIR, Spring 2014, c Benjamin Aurispa
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 ) = − 16 t
2 + 64 t + 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 ) = t 3
− 9 t 2 + 24 t 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.
5

Math 151 WIR, Spring 2014, c Benjamin Aurispa
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 = 3 t ), determine the rate at which the area is increasing after 2 minutes.
6