MTH 106, Calculus: Fall 2005
Ch. 4 test: Applications of the Derivative
Sufficient work must be shown for the following problems that require it.
I. Use the given graph of f(x) to find the following, if they exist. State none if they do not exist.
A. Where f has an absolute maximum.________
B. Where f has an absolute minimum.________
C. Where f has a local max.________________
D. Where f has a local min.________________
E. The value(s) of x for which f has a critical point.___________
F. The value(s) of x for which f has a point of inflection._______
G. The interval(s) of x for which f is increasing.__________________________________
H. The interval(s) of x for which f ' < 0.________________________________________
I. The interval(s) of x for which f is concave up._________________________________
J. The interval(s) of x for which f ' is increasing._________________________________
K. Find all values of x that satisfy the conclusion of Rolle’s Theorem for f on the interval [-2 , 0].
_________________________________
II.
Verify that the function h(x) = cos x – sin x satisfies the hypotheses of Rolle’s Theorem on the
interval [0, 2π]. Then find all numbers c that satisfy the conclusion of the theorem.
III.
Given that the domain of a function g is that x ≠1 and that lim g(x)   ; lim g(x)  0 and
x 1
x - 
 ( x  1)
2x  4
lim g(x)  0 ;
g ( x) 
g ( x) 
and
;
g(-1) = -2.
Do NOT
3
x  
( x  1)
( x  1) 4
attempt to find g. You may leave your answers in your work, but circle and label them.
A. List the equations of all asymptotes for g.
B. Find where the graph of g is increasing and decreasing.
C. Find where the graph of g is concave up and concave down.
D. Sketch the graph of a function g (below) that satisfies the conditions above.
E. Find any local extrema and label them. Note if any are absolutes.
IV. Suppose at your new house you have $1800 to fence in two adjacent outdoor rooms as shown
below. No fence is needed on the house side. One “room” will be for the kid’s toys and one for
grilling out, etc. If the fence parallel to the house costs $12 per foot and the fence used for the sides
and interior costs $8 per foot, find the dimensions of the largest rectangular plot that can be fenced in
for the $1800. (The house serves as one wall.) Use calculus to find and show that this is a maximum.
house