Math 400
Exam 3
Name:_____________________________
Show all work on the problems.
1. (20pts) Two cars start moving from the same point. One travels south at 50 mph and the other
travels west at 30 mph. At what rate is the distance between the cars increasing two hours
later?
2. (10pts) Use differentials to estimate √
.
3. (20pts) Use calculus min/max techniques to find the dimensions of a rectangle with perimeter
250 ft whose area is as large as possible.
4. (10pts) Use L’Hopital’s Rule to evaluate
5. (10pts) Sketch the graph of a continuous functions that satisfies the following conditions:
( )
( )
a) ( )
( )
( )
(
)
( )
b)
(
)
( )
(
) (
)
( )
(
)
( )
( )
c)
( )
( )
d)
and
6. (20pts) A Norman window has the outline of a semicircle on top of a rectangle. Suppose
there is 8 +π feet of wood trim available for all 4 sides of the rectangle and the
semicircle. Find the dimensions of the rectangle and the radius of the semicircle that will
maximize the area of the window.
7. (20pts) Consider the function ( )
a) Find the domain of ( ) and the
of the function.
b) Calculate the first derivative of the function and use the derivative to find the intervals of
increase/decrease and any local extrema.
c) Find the second derivative and the intervals where the functions is concave up and the
intervals where the function is concave down. Also, find any inflections points.
( ) and
( )
d) Find
e) Use the information in a)-d) to draw a good sketch of ( )
. Be sure to include and
label all intercepts, local extrema and points of inflection in your graph.