REVIEW
TEST 2
1
1. Find the derivative for
y = 3x 2 + 5x - 7
A. y’ = 3x + 5
C. y’ = 6x
C. y’ = 6x + 5
D. y’ = 6x + 5 - 7
E. None of the above
2
No that answer is incorrect.
You need to use the power rule on each of the
terms of the equation.
The power rule is –
If y = x n , then y’ = n x n – 1 .
Please click here to try again.
3
Yes, the answer is y’ = 6x + 5
Good work on using the power rule!
If y = x n , then y’ = n x n – 1 .
Please click here for the next question.
4
2. Find the derivative for
y = 2x – 1 – x – 3
A. y’ = - 2x 0 + 3x – 2
B. y’ = -2x – 2 – 3x – 4
C. y’ = 2x – 2 + 3x – 4
D. y’ = - 2x – 2 + 3x – 4
E. None of the above
5
No that answer is incorrect.
You need to use the power rule on each of the
terms of the equation.
The power rule is –
If y = x n , then y’ = n x n – 1 .
Watch your negative signs!
Please click here to try again.
6
Yes, the answer is y’ = - 2x – 2 + 3x – 4
Good work on using the power rule!
If y = x n , then y’ = n x n – 1 .
Please click here for the next question.
7
3. Find the derivative for
y = (2x 2 – x + 1)(x – 2)
A. y’ = (4x – 1)(1)
B. y’ = 6x 2 - 10x + 3
C. y’ = (2x 2 – x + 1)(1) + (x – 2)(4x – 1)
D. y’ = (4x – 1)(x - 1)
E. None of the above
8
Too bad that answer is incorrect.
You need to use the product rule.
The product rule is –
If
f (x) = F (x) • S (x),
Then f ’ (x) = F (x) • S’ (x) + S (x) • F ’(x)
Please click here to try again.
9
Yes, the answer is y’ = 6x 2 - 10x + 3 or
y’ = (2x 2 – x + 1)(1) + (x – 2)(4x – 1)
Good work on using the product rule
If y = f · s , then y’ = f · s’ + s · f’ .
Please click here for the next question.
10
4. Find the derivative for
2
x 3
y
5x  2
(5x  2)( 2x)  ( x 2  3)( 5)
A. y ' 
(5x  2)2
(5x  2)( 2x)  ( x 2  3)( 5)
B. y ' 
(5x  2)2
5x 2  4x  15
C. y ' 
2
( 5x  2)
D. y’ = (5x + 2)(2x) – (x 2 + 3) (5)
E. None of the above
11
Sorry, that answer is incorrect.
You need to use the quotient rule.
The quotient rule is –
If
Then
T ( x)
y
B ( x)
B ( x)  T ' ( x)  T ( x)  B ' ( x)
y' 
[ B ( x) ] 2
Please click here to try again.
12
Great, the correct answer is
5x 2  4x  15
y' 
2
(5x  2)
OR
(5x  2)( 2x)  ( x 2  3)( 5)
y' 
(5x  2)2
You used the quotient rule correctly!!
If
T ( x)
y
B ( x)
then
B ( x)  T ' ( x)  T ( x)  B ' ( x)
y' 
[ B ( x) ] 2
Please click here for the next question.
13
5. Find the derivative for
4
y
x
A. y’ = - 2 x 1/2
C. y’ = - 2
x – 3/2
B. y’ = 4 x – 3/2
D. y’ = 4 x 1/2
E. None of the above
14
Not quite, that answer is incorrect.
Change the equation to remove the radical to –
y = 4x – 1/2
and use the power rule.
Please click here to try again.
15
Hey great, the correct answer is y’ = - 2 x – 3/2
Nice work on the power rule and
negative exponents.
Please click here for the next question.
16
6. Find the derivative for
y = (x 2 – 3x + 6) 5
A. y’ = 5 (x 2 – 3x + 6) 4
B. y’ = (x 2 – 3x + 6) 5 (2x – 3)
C. y’ = (2x - 3) 5
D. y’ = 5 (x 2 – 3x + 6) 4 (2x – 3)
E. None of the above
17
Too bad, that answer is incorrect.
You need to use the chain rule with
u = x 2 – 3x + 6
d n
n  1 du
u  nu
dx
dx
Please click here to try again.
18
OK, the correct answer is
y’ = 5 (x 2 – 3x + 6) 4 (2x – 3)
Good use of the chain rule with
u = x 2 – 3x + 6 !!
d n
n  1 du
u  nu
dx
dx
Please click here for the next question.
19
7. Find the derivative for
y
x2  3
A. y’ = x (x 2 + 3) – 1/2
B. y’ = (x 2 + 3) – 1/2
C. y ' = 2x
x2 + 3
D. y’ = 1/2 (x 2 + 3) - ½ (2x)
E. None of the above
20
Too bad, that answer is incorrect.
You need to rewrite the equation without a
radical and then use the chain rule
y = (x 2 + 3) ½ and the let u = x 2 + 3
d n
n  1 du
u  nu
dx
dx
Please click here to try again.
21
Terrrrriffffic!!
Rewrite the equation without a radical and then use
the chain rule - y = (x 2 + 3) ½ and the let u = x 2 + 3
Then
y’ = x (x 2 + 3) – ½ OR
y’ = 1/2 (x 2 + 3) - ½ (2x) OR
y' 
x
x 3
2
Please click here for the next question.
22
6. If f (x) = 2x 3 - 3x 2 – 5x + 3, find the
second derivative.
A. y” = 6x – 6
B. y” = 6x 2
C. y” = 12x - 6
D. y” = 6x 2 - 6x - 5
E. None of the above
23
No that answer is incorrect.
You need to take the derivative of the derivative!
Please click here to try again.
24
Yes Yes Yes, the correct answer is y” = 12x - 6.
y’ = 6x 2 - 6x – 5 and
y” = 12x - 6
Please click here for the next question.
25
APPLICATIONS
Now, let’s try some application problems
26
9. A drug is injected into the bloodstream. The
concentration of the drug after x hours is
given by
0.20x
C( x) 
x 4
2
for 0  x  10.
A. Find the marginal concentration after 3
hours (4 decimal places).
A. C’ (x) = 0.0462 ml/cm 3
B. C’ (x) = – 0.1746 ml/cm 3
C. C’ (x) = – 0.0059 ml/cm 3
D. C’ (x) = 0.5428 ml/cm 3
E. None of the above
27
Sorry that answer is incorrect.
To find the marginal concentration graph the given
function and under the CALC menu use the dy/dx
choice at x = 3.
Please click here to try again.
28
Yes, the answer is C’ (x) = – 0.0059 ml/cm 3
from your graphing calculator.
NOTE: You will need this answer for the next
question.
Please click here for the next question.
29
10. A drug is injected into the bloodstream. The
concentration of the drug after x hours is
given by
0.20x
C( x) 
x 4
2
for 0  x  10.
B. Interpret the results of the previous
question.
Take a minute and try to write a response.
Go to a representative correct answer.
30
The drug concentration after three hours is decreasing
by about 0.0059 ml/cm 3 for the next hour.
Please click here for the next question.
31
11. The total profit from the sale of text books is
P (x) = 30 x – 0.3 x 2 - 200
0 ≤ x ≤ 100.
A. Find the marginal profit when x = 40.
A. P’ (x) = 0
B. P’ (x) = – 6
C. P’ (x) = 6
D. P’ (x) = undefined
E. None of the above
32
Oooooh too bad that answer is incorrect.
To find the marginal profit consider graphing the
function and under the CALC menu use the dy/dx
choice at x = 40.
Please click here to try again.
33
Yes, the answer is P’ (x) = 6
from your graphing calculator.
NOTE: You will need this answer for the next
question.
Please click here for the next question.
34
12. B. The total profit from the sale of text books is
P (x) = 30 x – 0.3 x 2 - 200
0 ≤ x ≤ 100.
B. Interpret P ‘ (40) = 6
Write an interpretation and then click
here to see a representative answer.
35
The profit when selling the next textbook (#41) will
increasing by about $6.
Please click here for the next question.
36
13. The total profit (in dollars) from the sale of x
calculators is P (x) = 22x – 0.2x 2 – 400
for 0  x  100.
A. Find the average profit per calculator if
50 calculators are produced and sold.
A. P (x) = $4
B. P (x) = $3
C. P (x) = $3.50
D. P (x) = $2
E. None of the above
37
Too bad that answer is incorrect.
The average profit is the profit divided by x.
Calculate the profit at 50 and divide by 50. If you
wish you can graph the average profit on your
calculator and find CALC then VALUE at x = 50.
Px  22x  0.2x 2  400
P ( x) 

x
x
Please click here to try again.
38
Yes, the answer is
P (50) ( 22)( 50)  (0.2)( 50)2  400 200
P (50) 


 $4
50
50
50
OR
Please click here for the next question.
39
14. The total profit (in dollars) from the sale of x
calculators is P (x) = 22x – 0.2x 2 – 400
for 0  x  100.
B. Find the marginal average profit at a
production level of 50 calculators.
A. P’ (x) = $ 0.04
B. P’ (x) = $ - 0.04
C. P’ (x) = $ 0.40
D. P’ (x) = $ - 0.40
E. None of the above
40
No that answer is incorrect.
There are several ways to get the marginal average
profit. Perhaps the easies is to graph the average
profit and go to CALC and dy/dx at 50.
Px  22x  0.2x 2  400
Avg Pr ofit  P ( x) 

x
x
Please click here to try again.
41
Yesaroonie, the answer is – 0.04.
Px  22x  0.2x 2  400
Graph P ( x) 

x
x
NOTE: You will need this answer for the next
question.
Please click here for the next question.
42
15. Write a brief interpretation of the results
of the previous problem.
Take a minute and write an interpretation.
Go to a representative correct answer.
43
The average profit is decreasing by about $0.04 for the
sale of the 51st calculator.
Please click here for the next question.
44
16. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
A. Find the revenue function.
A. R (x) = x (6000 - x)/30
B. R (x) = (6000 - x)/(30x)
C. R (x) = x (72000 + 60x)
D. R (x) = p (x) + C (x)
E. None of the above
45
Sorry that answer is incorrect.
To find the revenue function use R (x) = xp, where p
is the price-demand function.
Please click here to try again.
46
You have what it takes.
The correct answer is
R (x) = xp
= x (6000 - x)/30
NOTE: You will need this answer for the next
question.
Please click here for the next question.
47
17. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
A. Find the marginal revenue function.
A. R’ (x) = 60000 – x/30
B. R’ (x) = p’ (x) – C’ (x)
C. R’ (x) = 200 – x/15
D. R’ (x) = 200 – x/30
E. None of the above
48
Not the correct answer.
To find the marginal revenue function you need to
find the derivative of the revenue function from the
previous problem.
R (x) = x (6000 - x)/30
Please click here to try again.
49
Great calculus work.
The correct answer is
the derivative of R (x) = x (6000 - x)/30
R (x) = x (6000 - x)/30
= 200x – x 2/30 and
R’ (x) = 200 – 2x/30
= 200 – x/15
NOTE: You will need this answer for the next
question.
Please click here for the next question.
50
18. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
C. Find the marginal revenue at x = 1500.
A. $25
B. $100
C. $75
D. $125
E. None of the above
51
Sorry incorrect. Try the following hint.
Plug 1500 into the marginal revenue equation
(answer to the previous problem) for x and solve for
R’ (x),
Please click here to try again.
52
Great work.
The marginal revenue equation is
R’ (x) = 200 – x/15 and
R’ (1500) = 200 – 1500/15
= 200 - 100
= $100
NOTE: You will need this answer for the next
question.
Please click here for the next question.
53
19. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
D. Write a brief interpretation of the result
of the previous problem.
Go ahead, I’ll wait for you.
Click here to see a representative correct answer.
54
At a sales rate of 1500, the revenue will be about
$100 on the sale of the next gadget.
Please click here for the next question.
55
20. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
E. Find the profit function.
A. P (x) = 260x – x 2/30 - 72000
B. P (x) = C (x) – R (x)
C. P (x) = 140x – x 2/30 - 72000
D. P (x) = 140x – x/30 - 72000
E. None of the above
56
Too bad, that is incorrect.
The profit function is found by subtracting cost from
revenue, or
P (x) = R (x) – C (x)
Please click here to try again.
57
Well done!
The correct answer is P (x) = 140x – x 2/30 - 72000
P (x) = R (x) – C (x)
= (200x – x 2/30) – (72000 + 60x)
= 140x – x 2/30 - 72000
NOTE: You will need this answer for the next
question.
Please click here for the next question.
58
21. The price-demand and cost equations for
producing gadgets are given by:
p (x) = (6000 - x)/30 and C (x) = 72000 + 60x
for 0  x  6,000.
E. Find the marginal profit function.
A. P’ (x) = 140x – x 2/30 - 72000
B. P’ (x) = 140 – x/30
C. P’ (x) = 140 – x/15
D. P’ (x) = 140x – x/15
E. None of the above
59
Sorry that is incorrect.
The marginal profit is the derivative of the profit and
the profit was found in the previous problem.
Please click here to try again.
60
Well done!
The correct answer P’ (x) = 140 – x/15
Take the derivative of P in the previous problem.
P (x) = 140x – x 2/30 – 72000 and
P’ (x) = 140 – x /15
Please click here for the next question.
61
22. A company that makes CD players has
the following total cost function.
C (x) = x 2 + 2x + 2000.
C. What is the minimum average cost?
A. $ 37.54
B. $ 87.46
C. $ 44.72
D. $ 91.44
E. None of the above
62
No that is incorrect.
Remember the average cost function is the cost
divided by x or
C (x)
C ( x) 
x
Graph the average cost function on your calculator
and find the minimum. Use a window of 0 < x < 100
and 0 < y < 200.
Please click here to try again.
63
Sorry that is not the correct answer, but it is close.
You were asked for the minimum average cost. You
want the y value not the x value.
Please click here to try again.
64
Terrific!
The correct answer $91.44.
Did you use your
calculator to find that?
C (x)
C ( x) 
x
x 2  2x  2000

x
0 < x < 100 and 0 < y < 200.
Please click here for the next question.
65
23 – 25. The price-demand and cost equations for
producing gadgets are given by:
p (x) = 270 – 10x and C (x) = 100 + 70x
for 0  x  20.
23. How many gadgets should be produced to
maximize profit?.
A. 900
B. 10
C. 120
D. 12
E. None of the above.
66
Too bad that is incorrect.
This one is a lot of work. From the price
equation form the revenue equation; R = xp.
Use that revenue equation and the given cost
equation to form the profit equation. Use
your calculator to maximize the profit.
Please click here to try again.
67
That us close but incorrect.
You gave the maximum profit – you need the
x value where the maximum profit occurs.
Please click here to try again.
68
Yes indeedie calculator speedie! X = 6.
Note P(x) = R – C
= (x(270 – 10x))-(100 + 70x)
Save this info for #30 and #31.
Please click here for the next question.
69
23 – 25. The price-demand and cost equations for
producing gadgets are given by:
p (x) = 270 – 10x and C (x) = 100 + 70x
for 0  x  20.
24. What is the maximize profit?.
A. 900
B. 10
C. 120
D. 12
E. None of the above.
70
Too too too bad that is incorrect.
From the previous problem use the profit
formula and use your calculator to maximize
the profit.
Please click here to try again.
71
Yes indeedie calculator speedie! X = 6.
Save this info for #31.
Please click here for the next question.
72
23 – 25. The price-demand and cost equations for
producing gadgets are given by:
p (x) = 270 – 10x and C (x) = 100 + 70x
for 0  x  20.
25. Now for the really important question – What
price should be charged to maximize the profit?.
A. $900
B. $10
C. $170
D. $65.50
E. None of the above.
73
Well consider trying again.
The price equation is given. Use the facts
from the previous two problems to
determine what x value to substitute into
that equation.
Please click here to try again.
74
Hey, hey, hey, hey, HEY!!!!
Yes you plugged x = 10 into the price equation to
get $170 didn’t you. You cleaver devil.
Please click here for the next question.
75
That was a lot of review. I hope you found it
helpful. Good luck on the test!
76