EXAM 2 – REVIEW SHEET
1. Differentiate the following functions:
a. y  x 5 e 3 x
j. y  (5x 2  3x  4)10
b. y  2 x  4
k. y  35 x  5 x 3  ln( 5 x)
c. y  ln( 8 x  1)
x4
l. y 
ln( x)
d. y 
ln( x)
x
m. y  e ( 2 x5)
2
7x  5
8x  3
e. y  log( x 2  1)
n. y 
f. y  83 x
o. y  12(8x 2  5x  2) 2
g. y  x e  e x
p. y 
h. y  x 4 ln( x)
q. y  4 x ( x 2  1) 3
i. y  log 2 (6x  3)
r. Use logarithmic differentiation
3x
( x  7) 2
y
(8 x  3) 5 (2 x  1) 3
( x  4) 3
2. Find
dy
by implicit differentiation:
dx
a. x 3 y 5  e y ln( x)  2  x 2
b. ( x 2  y 2 ) 2  2( x 2  y 2 )
c. What is the equation of the tangent line to the curve in part a at the point (1, 1).
3. Consider the following functions:
I. f ( x)  1  4 x  x 3 II. g ( x)  2 x 3  8 x  5
III. h( x)  x 3  12 x 2  45x  6
Answer the following questions for each of the functions above.
a. Use interval notation to find the interval on which the function is increasing.
b. Use interval notation to find the interval on which the function is decreasing.
c. Find the coordinates of the relative minimum.
d. Find the coordinates of the relative maximum.
4. The cost of producing x DVDs is given by C ( x)  700  110 x  110 ln( x).
C ( x)
a. Find the average cost C ( x) 
.
x
b. Find the number of DVDs that minimize the average cost.
c. What is the minimum average cost of producing the DVDs?
Find the absolute max/min for each
5. f ( x)  3x 2  12 x  5 on [0, 3]
6. f ( x)  x 3  6 x 2  9 x  2 on [–1, 4]
7. A local doughnut shop sell 150 chocolate doughnuts a day for 85 cents apiece. For
every 2 cents reduction in price 10 more doughnuts will be sold. At what price
should you sell the doughnuts to maximize revenue?
8. A 300 room hotel in Las Vegas is filled to capacity every night at $80. For each $1
increase in rent, 3 fewer rooms are rented. If each rented room costs $10 to service
per day, how much should management charge each room to maximize profit? What
is the maximum profit?
9. A rancher has 200 feet of fence to enclose two adjacent corrals. What dimensions
should be used so that enclosed area will be a maximum? What is the maximum
enclosed area?
10. A candy box is to be made out of cardboard that measures 8 by 12 inches. Squares of
equal size will be cut from each corner and then the ends will be folded up to form a
rectangular box. What size square should be cut from each corner to maximize
volume? What is the maximum volume?
11. A company manufactures and sells x digital cameras per week. The price function is given as
p ( x)  400  0.4 x and the cost function is C ( x)  2000  160 x.
a.
b.
c.
d.
e.
f.
g.
h.
What is the revenue function, R (x) ?
How many cameras should be produced to maximize revenue?
What is the maximum revenue?
What price should the company charge for the cameras to maximize revenue?
What is the profit function, P (x) ?
How many cameras should be produced to maximize profit?
What is the maximum profit?
What price should the company charge for the cameras to maximize profit?
12. The total profit (in dollars) from the sale of charcoal grills is
P( x)  30 x  0.1x 2  295.
(A) Find the average profit per grill if 40 grills are produced.
(B) Find the marginal average profit at a production level of 40 grills.
(C) Use the results from parts (A) and (B) to estimate the average profit per grill if 41 grills are
produced.
13. The total cost (in dollars) of producing x coffee machines is
C ( x)  2400  50 x  0.9 x 2 .
(A) Find the exact cost of producing the 21st machine.
(B) Use marginal cost to approximate the cost of producing the 21st machine.
ANSWERS
1. a. y'  5x 4 e 3 x  3x 5 e 3 x
d. y ' 
1
b. y ' 
1  ln( x)
x2
e. y ' 
g. y '  ex e1  e x
c. y ' 
2x  4
2x
( x  1) ln( 10)
8
8x  1
f. y'  3 ln( 8)(83x )
2
h. y'  4 x 3 ln( x)  x 3
i. y ' 
6
(6 x  3) ln( 2)
j. y'  10(5 x 2  3x  4) 9 (10 x  3) k. y '  5 ln( 3)(35 x )  15 x 2  1x
l. y ' 
4 x 3 ln( x)  x 3
(ln( x)) 2
m. y'  4(2 x  5)e ( 2 x5)
21  3 x
p. y ' 
( x  7) 3
3
o. y'  24(8 x  5 x  2) (16 x  5)
2
r. y ' 
2
n. y ' 
61
(8 x  3) 2

1
2
q. y'  2 x ( x 2  1) 3  24 x x ( x 2  1) 2
(8 x  3) 5 (2 x  1) 3  40
6
3 




3
( x  4)
 8x  3 2 x  1 x  4 
2 5
dy  2 x  ex  3x y

2. a.
dx 5 x 3 y 4  e y ln( x)
y
3. I. Critical values are 
b.
2
dy x(1  x 2  y 2 )

dx y (1  x 2  y 2 )
5e
c. y  1  
( x  1)
 5 
inflection point at 0.
3
 2 3 3  16 
 2 3 3  16 
 2 2 


2   2
 d. 

,
a.  
,
 b.   ,
  
,   c.  

 3, 3 3 
3
3
3
3
3
3
3



 





II critical values are at 
2
3
inflection point at 0.
 2 15 3  32 
 2 15 3  32 


 2 2 
2   2
 d.  

,
,
  
,   b.  
,
 c. 
a.   ,



3
3
3
3
3
3
3  3 
3 3






III critical values are at 3 and 5 inflection point at 4.
a. (–∞, 3) U (5, ∞)
4. C ( x) 
b. (3, 5) c. (5, 44) d. (3, 60)
700
ln( x)
 110  110
x
x
1578 DVDs avg cost is $110.
5. abs max is 5, abs. min. –7
6. abs. max 6, abs. min –14
7. R( x)  (150  10 x)(85  2 x) , $0.575
8. R( x)  (300  3x)(80  x) , C ( x)  10(300  3x) ,
P( x)  R( x)  C ( x)  (300  3x)(70  x)
P' ( x)  6 x  90  0 x = 15 rent $95, rooms 255, max profit $21,675
2 x  3 y  200
9.
A  xy
A( y )  100 y  1.5 y 2
A( y )  100  3 y  0
10. V ( x)  x(8  2 x)(12  2 x)
x = 50 ft, y = 100/3 ft, A = 5000/3 sq ft
x = 1.57 inches
V = 67.6 cubic inches
11. a. R( x)  xp  400 x  0.4 x 2
b. x = 500
c. $100,000
d. p = $200
e. P( x)  R( x)  C ( x)  0.4 x 2  240 x  2000
f. x = 300
g. $66,400
h. p = $280
30 x  0.1x 2  295
P( x) 
x
12. A.
30(4)  0.1(40)2  295
P(40) 
 $18.625
40
M P  P' ( x)  (30  0.1x  295 x 1 )'  0.1  295 x 2
B.
P'(40)  $0.084375
C. $18.625 + $0.084375 = $18.709375
13. 21st machine C(21) – C(20) = $13.10
C’(x) ≈ C(x+1) – C(x)
C’(x) = 50 – 1.8x, thus C’(20) = $14