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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

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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 )

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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

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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?

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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.

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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.

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