BC Fast
Sample Quiz
1. Evaluate the following limits. Justify your work clearly, showing all appropriate steps.
 1  cos x 
lim 
a.

x0  x  sin x 
b.

lim x 2 cot x
x0
c.
 x 
lim 

x0  ln x 
d.

lim  x  e x
x  



1
x2



Skills
BC Fast
Sample Quiz
Skills
17. Find the derivatives of each function. Do not simplify the result.
1 

a. f ( x)   x  e x2 


4
 x 2  sin x 
b. cosh 
 cos x 


c. f ( x)  x2  cos(e x )
2. An open-topped rectangular box with a square base is to be constructed so as to have a volume of 100
cubic inches. Find the dimensions of the box that minimize the amount of material used in constructing
the box.
BC Fast
3.
Sample Quiz
Skills
If V = 10 h and h is a function of t with graph shown below, use this graph to estimate the value
dV
of
when t = 2.
dt
h(t)
300
200
100
t
0
-1
0
1
2
3
4
5
6
7
BC Fast
4.
Sample Quiz
Skills
A boy (who is stationary) flies a kite at a constant height 120 feet above his hands. If the wind
carries the kite horizontally at the rate of 30 ft/min,
a. at what rate is the string being pulled out when there is 150 feet of string out?
b. at what rate is the angle of elevation of the kite changing when there is 150 feet of string out?
BC Fast
Sample Quiz
Skills
5. Use the graph of y =h(x) on [-5,6] shown below to answer the questions that follow:
Graph of h(x)
a. Do the hypotheses of the Mean Value Theorem hold for h on the interval [-4,0]?
Explain.
b. Does the conclusion of the Mean Value Theorem hold for h on [-5,6]?
If so, find an
approximate value c such that h (c) = “average rate of change of h on [-5,6]”.
c. Do the hypotheses of the Intermediate Value Theorem hold on [-4,0]?
d. Does the conclusion of the IVT hold on [-1,6] ?
Explain.
Explain.
e. Do the conclusions of the Extreme Value Theorem hold on [-5,6]?
f. Find an interval over which the conclusions of EVT hold, but the hypotheses do not hold.
BC Fast
Sample Quiz
Skills
6.
If f is continuous on the interval [0,1], the Extreme Value Theorem guarantees a maximum and a
minimum value for f on this interval. Is it possible that there is no value c in [0,1] for which
f (c)  0 ?
If yes, give a counter-example (a sketch of such a graph is sufficient). If no, explain carefully
citing appropriate theorems.
7.
Find the maximum and minimum values of f ( x)  3x 4  4 x3  6 on [2,3] .