CPC
NAME _________________________
Practice Test on Chapter 5:
Applications of Differentiation (100 pts)
CALCULATORS ARE ALLOWED. ( 10 problems at 10 points each)
1. Locate the absolute and relative extrema of the following function on the given closed
interval.
f x  x  3x  7
 0, 3 
2. Determine if Rolle’s Theorem can be applied to f on the closed interval given. If
Rolle’s Theorem can be applied, find all the values of c in the open interval such that
f c  0.
f x  
1
x2
  2 , 2
3. Find the point(s) of g guaranteed by the Mean Value Theorem for the given closed
interval:
g x   4  x 2
1, 2 
4. Find the critical numbers of the following function, the open intervals on which it is
increasing and the open intervals on which it is decreasing:
hx   x  2 x  3
2
Critical numbers: _______________
Increasing: ____________________
Decreasing: ___________________
5. Use the First Derivative Test to find any relative extrema of the following function.
f x   2 x 3  3x 2  12 x  4
Relative maximum at ( _____, ______ )
Relative minimum at ( _____, ______ )
6. Find the inflection point of the function in problem #5. Determine the concavity on
either side of the inflection point.
Point of inflection: ( _______ , _______ )
For what values of x is the function concave up? _________________________
For what values of x is the function concave down? _________________________
7. Find the horizontal and vertical asymptotes for the following function. Make a sketch
and label the asymptotes on your sketch.
f x  
x2  x  6
x2 1
5
-5
5
-5
What are the x-intercepts? __________________
What is the y-intercept? ____________________
Give the equation of the horizontal asymptote: ________________________________
Give the equations of the vertical asymptotes: ________________________________
8. Analyze the following function using derivatives and a sketch. Show your work.
You may check your work with the calculator.
f x   x  1 x  4
2
SKETCH:
20
15
10
5
-10
10
-5
-10
List all the x-intercepts _____________________
Give the y-intercept _______________________
Relative maximum ( ________ , ________ )
Relative minimum ( ________ , ________ )
Inflection Point ( __________ , __________ )
9. Analyze the following function using derivatives and a sketch. Show your work. You
may check your work with the calculator.
f x  
x  12
x2
SKETCH:
10
5
-10
10
-5
-10
List all the x-intercepts _____________________
Give the y-intercept _______________________
Relative maximum ( ________ , ________ )
Relative minimum ( ________ , ________ )
Vertical asymptote:
x = ____________
BONUS: Give the equation of the slant asymptote: __________________________
10. Find the area of the largest rectangle that has its upper corners at points ( x , y ) and
( -x , y ) which is on the curve y  9  x 2 and its other corners at ( x , 0 ) and ( -x , 0 ).
fx = 9-x2
10
(- x ,y )
(x ,y )
5
( -x , 0)
-2
(x , 0)
2
First, recall that the area of a rectangle is the base times the height to write an equation
for the area of the rectangle. Get the equation in terms of just x.
Find the derivative of the area as a function of x.
When is this derivative = 0 ?
What is the maximum area of the rectangle?