BC1
Sample Quiz
Sections 3.10-4.2
1. An open topped box is to be made by cutting out squares with sides of x inches from each corner a
rectangular (10 inch x 16 inch) sheet of cardboard, then turning the sides up. Find the value of x
which will maximize the volume of the box.
10
16
x
x
BC1
Sample Quiz
2. Find the maximum and minimum values of f ( x)  x 
Sections 3.10-4.2
1
1 
on  ,9  .
2x
4 
3. Suppose ABC is a right triangle with right angle at C, and that vertex A is at the origin, vertex
1
B is on the curve y  4e  x ,  x  4 , and vertex C is along the positive x-axis. Find the
2
minimum area of such a triangle.
BC1
Sample Quiz
Sections 3.10-4.2
4. Given the graph of y  f ( x) shown below, sketch the graph of y  f ( x) .
y
x
5.
Find values for a and b such that h  x   x3  2ax  b has a local minimum at (2, 8).
BC1
6.
Sample Quiz
Sections 3.10-4.2
x
.
x 1
a. Do the hypotheses of the EVT hold for g on [2, 2] ? Explain.
Let g ( x) 
b. Do the conclusions of the EVT hold on [2, 2] ?
maximum values of g on [2, 2] . If not, explain.
If so, find the minimum and
c. Do the hypotheses of the MVT hold for g on [0, 2]? Explain.
d. Do the conclusions of the MVT hold for g on [0, 2] ?
c  (0, 2) guaranteed by the conclusion of the MVT.
7.
If so, find all values
On Saturday morning, I got in my car at 6:28 am and the temperature (in my relatively warm
garage) was 52°F. By 6:36 am, the car thermometer showed 28°F.
What assumptions must be made to apply the MVT? Be specific to this situation.
Assuming the MVT does apply, what does the theorem allow you to conclude? Be specific
to this situation.
BC1
8.
Sample Quiz
Sections 3.10-4.2
For each of the following conjectures, decide if it is true or false. For the true statements, give
some coherent justification for the statement using complete sentences. For the false
statement, draw or otherwise describe a function f which provides a counterexample, and
explain why it does. Always assume that [a, b] implies that a  b .
a. Let a function f on [a, b] be given which is differentiable on (a, b) and continuous on
f (b)  f (a )
[a,b]. Then there exists a number c  (a, b) such that f (c) 
.
ba
b. Let f be a continuous function with domain [0, ) , then for each real number a > 0, if L is
between f (0) and f (a ) there is number c  [0, a ] such that f (c)  L .
c. Let f be a continuous function with domain (a, b) , then f has a maximum and a minimum
value on (a,b).
d. Suppose that f is continuous on [a, b] . If f is differentiable at c  (a, b) , then
f (b)  f (a )
f (c) 
.
ba
BC1
Sample Quiz
Sections 3.10-4.2
9. When you cough, your windpipe contracts. The speed v(r) , with which you expel air depends
on the radius r , of your windpipe. If a is the normal (rest) radius of your windpipe, then for
0 £ r £ a the speed is given by
v(r )  k (a  r ) r
where k is a positive constant.
What value of r maximizes the speed? For what value is the speed minimized?