All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
Student Name: __________________
ID:_______________________
Instructor: Andrew Pownuk
Exam Score: ________
1. Find the extrema for the function 𝑓(𝑥) = 2cos(𝑥) + 𝑥 on the closed interval [0,2𝜋], and
indicate which are the absolute extrema.
 
f '  x   2 sin x  1  0
f x  2 cos x  x
2 sin x  1
sin x 
x1 

6
1
2
, x2 
5
6
1.0
0.5
1
0.5
1.0
2
3
4
5
6
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
Plot[{2*Cos[x]+x,Sin[x],1/2},{x,0,2*Pi}]
5
4
3
2
1
1
2
3
4
5
1
 
f  0   2 cos  0   0  2  0  2
f x  2 cos x  x
 
  

f    2 cos     3   2.256
6
6
6 6
 5 
 5  5
5
f
 3
 0.886
  2 cos 

6
 6 
 6  6
f 2  2 cos 2  2  2  2  8.2832
 
 


    5 
gloabl _ m ax  max  f 0 , f   , f 
 , f 2   2  2  8.2832

6  6 


    5 
5

gloabl _ mi n  min  f 0 , f   , f 
 0.886
 , f 2    3 
6


6  6 

 

 
6
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
2. Consider the function 𝑓(𝑥) = 2𝑥 3 − 2𝑥 2 − 2𝑥 + 3
a. Find any critical numbers
b. Find the open intervals on which the function is increasing and/or decreasing
c. Use the First Derivative Test to find any relative extrema.
f (x )  2x 3  2x 2  2x  3
f '(x )  6x 2  4x  2
f '(x )  0
6x 2  4x  2  0
/ :2
3x 2  2x  1  0
a  3, b  2, c  1
 
b  b2  4ac  2 
x1 

2a
 
b  b2  4ac  2 
x2 

2a
 2 
2
   24  1
 4 * 3 * 1
2*3
 2 
2
6
   24 1
 4 * 3 * 1
2*3
6
 
f' x 0
Plot[-2-4 x+6 x2,{x,-1,2}]
10
5
1.0
0.5
0.5
1.0
1.5
2.0
3
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.

1
 ,  
3

x
f(x)
f’(x)
Conclusions
+

1
3
0
max
 1 
  ,1 
 3 
1
1,  
-
0
min
+
Plot[{3-2 x-2 x2+2 x3,-2-4 x+6 x2},{x,-1,2}]
10
5
1.0
0.5
0.5
1.0
1.5
2.0
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
3. Consider the function 𝑓(𝑥) = 𝑥 4 − 2𝑥 3 − 72𝑥 2 + 66𝑥 − 7
a. Find the interval on which the function is concave up or down.
b. Find any points of inflection.
f (x )  x 4  2x 3  72x 2  66x  7
f '(x )  4x 3  6x 2  144x  66
f ''(x )  12x 2  12x  144
f ''(x )  0
12x 2  12x  144  0
/ :12
x  x  12  0
2
a  1, b  1, c  12
 
 1 
b  b2  4ac
x1 

2a
 
b  b2  4ac  1 

2a
f '' x  0
x2 
 1 
2

 4 * 1 * 12
2*1
 1 
2
 4 * 1 * 12
2*1
 

  1  7  3
2
  17  4
2
12x 2  12x  144  0
x 2  x  12  0
Plot[-12-x+x2,{x,-4,5}]
5
4
2
2
4
5
10
x
 , 3
-3
(-3,4)
4
 4,  
f’’(x)
+
Concave up
0
Point of
inflection
Concave
down
0
Point of
inflection
+
Concave up
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
f(x)
5
4
2
2
4
5
10
f(x)
2000
1500
1000
500
10
5
5
500
1000
1500
10
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
4. Find the horizontal asymptotes for the function [Hint you must evaluate both lim 𝑓(𝑥)] and
𝑥→∞
lim 𝑓(𝑥)
𝑥→−∞
𝑓(𝑥) =
f (x ) 

12𝑥 − 7
√9𝑥 2 − 7𝑥 + 8
12x  7
9x 2  7x  8
12x  7
7
12 
12x  7
x
x
lim
 lim
 lim

x 
x 
x 
2
2
2
9x  7x  8
9x  7x  8
9x  7x  8
x
x2
7
1
12 
12  7
12  7  0
12
x
x
 lim
 lim


4
x 
x 
2
3
1
1
9

7

0

8

0
9x  7x  8
97 8 2
x
x
x2
y 4

lim
x 
12x  7
9x 2  7x  8
 lim 
x 
y  4
7
x
2
9x  7x
x2
12 
12x  7
7
12 
x
x
 lim
 lim

x 
x 
2
2
9x  7x  8
9x  7x  8
x
 x2
1
12  7
12  7  0
12
x
 lim 


 4
3
1
1
970 80
 8 x 
97 8 2
x
x
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
Plot[{(-7 + 12 x)/Sqrt[8 - 7 x + 9 x^2], -4, 4}, {x, -10, 10}]
4
2
10
5
5
2
4
10
All work must be shown to be awarded full credit.
Provide exact solutions to all problems, unless otherwise stated.
Only a non-graphing, non-differentiating and non-integrating calculator is allowed.
5. A large cube shaped block of ice is left out in an El Paso summer day. All edges of the
ice cube are shrinking at a rate of 2 centimeters per minute. How fast is the surface area
changing when each edge is 3 centimeters? Round your answer to three decimal places if
necessary.
𝑥
𝑥
𝑥
x  3cm
dx
cm
 2
dt
min
S  6x 2
dS
dx
 12x
dt
dt
dS
cm 2
 12 * 3 * 2  72
dt
mi n
 