MTH 106, Calculus, Fall 2005
Chapter 3, Derivatives Test
Sufficient work must be shown for problems that require it.
I.
The graph of f is shown to the right.
1. State, with reasons, the values of x at which f is not differentiable.
Where? Why?
2. (a) f ' (-1) = _______ .
(b) Which is larger: f ' (-4) or f ' (3)? ________
(c) How do you know?
II.
The graph of g is shown below; graph g ' on the coordinate plane to the right.
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III.
State the derivative of each of the following. DO NOT SIMPLIFY.
5
A. y  7 x 3  8 x  5  csc 10 x  tan x  6
x
1 

B. f ( x)  (sin 2 x) x  2 
x 

C. g ( x) 
( x  1) 3
2x  x 4
D. h(x) = cot(sec(nx)) , where n is a real number.
IV.
Find the slope of the tangent line for 3y2 – 4x3 = 5x2y -2 at the point (1,2).
V.
Given s(t) = t 4 – 2t 2 + 5, t ≥ 0 is the position of a particle, where s is in feet and t is in
minutes. Find the following:
A. Find the velocity function for the particle.
B. Find the acceleration function for the particle.
C. At what time, t, is the particle stopped?
VI.
Use the definition of the derivative, either form, to find F'(3) if F(x) = 5x – x2.
VII.
Suppose that a volunteer holds a small cluster of black and gold balloons at Project
Pumpkin while standing on level ground. Suddenly the volunteer sees a guest start to trip
and fall about 30 feet away. The volunteer lets go of the balloons which go straight up in
the air at a rate of 6 feet per second, and she runs toward the child at a rate of 10 feet per
second. How fast is the distance from the volunteer to the balloons increasing 2 seconds
after the balloons were released?