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REVIEW TEST 4. 1. Find the following integral. A. x 8 + c. B. x 8 /8 + c. D. x 6 /6 + c. C. 7 x 6 + c. E. None of the above. Sorry, that answer is incorrect. Remember the power rule for integration –. Please click here to try again. - PowerPoint PPT Presentation
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2
1. Find the following integral
A. x 8 + c B. x 8 /8 + c
C. 7 x 6 + c D. x 6 /6 + c
E. None of the above
dxx7
3
Sorry, that answer is incorrect.
Remember the power rule for integration –
Please click here to try again.
c1n
xdxx
1nn
4
Great, the correct answer is x 8 /8 + c, by substituting the power 7 into the power rule.
Please click here for the next question.
c1n
xdxx
1nn
5
2. Find the following integral
A. x –3 /- 3 + c B. x –1 /- 3 + c
C. - x – 1 + c D. - 2x –3 + c
E. None of the above
dxx
12
6
Not quite, that answer is incorrect.
Remember the power rule for integration –
Please click here to try again.
1n
xdxx
1nn
7
Hey great, the correct answer is - x – 1 + c, by substituting the power - 2 into the power rule.
Please click here for the next question.
c1n
xdxx
1nn
8
3. Find the following integral
A. B.
C.
E. None of the above
dxx
3x3
33
c2
x3
4
x3 24
c4
x3
4
x3 44
c2
x3x9
22
c4
x3x9
42
D.
9
Too bad, that answer is incorrect.
Remember the power rule for integration –
Please click here to try again.
1n
xdxx
1nn
10
OK, the correct answer is
Using the power rule twice; once with n = 3 and again with n = - 3.
Please click here for the next question.
c2
x3
4
x3 24
1n
xdxx
1nn
11
4. Ready to step it up a notch?
A. e x + c B. x e x + c
C. D. 2e 2x + c
E. None of the above
dxex
c1x
e 1x
12
Not quite, that answer is incorrect.
Remember the rule for exponential functions for integration –
Please click here to try again.
cedxe xx
13
Terrrrriffffic, the correct answer is e x + c,
Using the rule for exponential expressions for integration.
Please click here for the next question.
cedxe xx
14
A. ln x + c B. e x + c
C. D. x + c
E. None of the above
dxx
1
5. Find the following integral
cx
12
15
Not quite, that answer is incorrect.
Remember the rule for logarithmic functions for integration –
Please click here to try again.
cxlndxx
1
16
Terrrrriffffic, the correct answer is ln x + c,
since
Please click here for the next question.
.definitionbyclndxx
1
17
6. Now, let’s try something really interesting.
dx)e3x
3( x
cex
3 x2 A. B. ln x – ex + c
C. 3ln x – 3ex + c D.
E. None of the above
ce3 x
18
That answer is incorrect. It is a hard one!
You must use both the rule for exponential functions for integration and the rule for logarithmic functions for integration as in the previous two problems.
Please click here to try again.
19
Terrrrriffffic, the correct answer is 3ln x – 3ex + c
since
Please click here for the next question.
ce3xln3dxe3dxx
3dx)e3
x
3( xxx
20
7. Let’s go back to the power rule.
A. B.
D.
E. None of the above
dxx3
c4
x3 34
cx3
1 32
c3
x 34
c3
x 31
C.
21
That answer is incorrect. Try changing the radical to exponential form!
Please click here to try again.
dxxdxx 31
3
Then use the power rule.
22
Grrrrreat, the correct answer is
Please click here for the next question.
c4
x3 34
dxxdxx 31
3
cx4
3
34x
dxx 343
4
31
And using the power rule for integration yields -
23
8. Find the following integral
dxx
1
A. -2 x -1 / 2 + c B. 2 x 1 / 2 + c
C. -2 x 1 / 2 + c D. 2 x -1 / 2 + c
E. None of the above
24
That answer is incorrect. Try changing the radical to exponential form!
Please click here to try again.
dxxdxx
12
1
Then use the power rule.
25
WOW, the correct answer is indeed 2x ½ + c.
Please click here for the next question.
cx2c
21x
dxxdxx
12
1212
1
26
9. Let’s complete that idea.
A. 2x 2/3 + 3x 3/2 + c B. -6x -1/2 + 3x -4/3 + c
C. 2x 3/2 + 3x 2/3 + c D. -2x 3/2 - 3x - 2/3 + c
E. None of the above
dx
x
2x3
3
27
That answer is incorrect.
That answer is incorrect. Try changing the radicals to exponential form! See the previous two
problems.
Please click here to try again.
28
Great, the correct answer is 2x 3/2 + 3x 2/3 + c, since
Please click here for the next question.
dx)x2x3(dx
x
2x3 3121
3
cx3x2c32
x2
23
x3 32233223
29
10. OK. Let’s try some involving substitution techniques.
A. B.
D.
E. None of the above
dx)3x(x 754
c40
)3x( 85
c5
)3x( 85
c8
)3x( 85
c)3x(8 85 C.
30
No that answer is incorrect.
Try substitution. Let u = x 5 – 3 and find du and make an adjustment to x 4 that is appropriate.
Please click here to try again.
dx)3x(x 754
31
Yes! Yes! Yes!, the correct answer is
by letting u = x 5 – 3 and finding du = 5x 4 dx
Please click here for the next question.
The problem then needs adjusting by multiplying and dividing by 5.
c40
)3x( 85
dx)3x(x 754
duu5
1dx)3x(x5
5
1dx)3x(x 7754754
c3x40
1
8
u
5
1 858
32
11.
A.
B. 5 (2x 5 – 4x + 7) 4 (20x 3 ) + c
D.
E. None of the above
dx)2x5()7x4x2( 455
c)7x4x2(12
1 65
c6
)7x4x2( 65
c)2x5(12
)7x4x2(5 445
C.
33
That answer is incorrect. You must substitute correctly!
Let u = 2x 5 – 4x + 7, then du = 10x 4 – 4 which is
2 (5x 4 – 2) . SO the original problem becomes
Please click here to try again.
dx)2x5()7x4x2( 455
dx)2x5(2)7x4x2(2
1 455
duu2
1OR
5
34
Grrrrreat, the correct answer is
Please click here for the next question.
c)7x4x2(12
1 65
Let u = 2x 5 – 4x + 7, then du = 10x 4 – 4 which is
2 (5x 4 – 2) . SO the original problem becomes
c)7x4x2(12
1c
6
u
2
1duu
2
1 656
5
dx)2x5(2)7x4x2(2
1 455
35
12.
A. 2 (x 2 – 3) - ½ + c B. 5 (x 2 – 3) - 1/2 + c
C. 2 (x 2 – 3) – 1/2 + c D. 5 (x 2 – 3) ½ + c
E. None of the above
dx3x
x52
36
Too bad that answer is incorrect.
Please click here to try again.
First, rewrite the problem -
dx3x
x52
Let u = x 2 – 3, then du = 2x dx which means that we need a factor of 2 in the problem or
dx3xx5212
dx3xx22
5dx3xx5
212212
37
Yes, you are an integrating machine.
Please click here for the next question.
First, rewrite the problem
dx3x
x52
Let u = x 2 – 3, then du = 2x dx which means that we need a factor of 2 in the problem or
dx3xx5212
duu2
5dx3xx2
2
5dx3xx5
21212212
c3x5cu521
u
2
5 212212
1
39
No that answer is incorrect. You must substitute correctly!
Please click here to try again.
dxex23x22
Let u = 2x 3 , then du = 6x 2 dx .
40
Correctamundo, the answer is
Please click here for the next question.
c3
e3x2
Let u = 2x 3 , then du = 6x 2 dx , then
dxex23x22
due3
1dxex6
3
1dxex2 ux22x22 33
ce3
1ce
3
1 3x2u
41
14.
A. 2 (2x – 1) + c B. ½ (2x – 1) + c
C. ½ ln | 2x – 1| + c D. 2x – 1 + c
E. None of the above
dx
1x2
1
42
No that answer is incorrect.
Please click here to try again.
No that answer is incorrect. You must substitute correctly!
Let u = 2x - 1 , then du = 2 dx .
dx
1x2
1
43
Yes, the answer is ½ ln | 2x – 1| + c
Please click here for the next question.
duu
1
2
1dx
1x2
2
2
1dx
1x2
1
Let u = 2x - 1 , then du = 2 dx , then,
|1x2|ln2
1c|u|ln
2
1
44
15. Now that you have indefinite integrals down cold let’s try some definite integrals.
A. 5 B. 6.67
C. – 8.67 D. 8.67
E. None of the above
dxx3
1
2
45
Too bad that answer is incorrect.
Please click here to try again.
To find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (3) and subtract the value at the bottom limit (1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
46
Yes, the answer is 8.67
Please click here for the next question.
I used the “Calc” menu and “integrate”.
48
Too bad that answer is incorrect.
To find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (3) and subtract the value at the bottom limit (1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
Please click here to try again.
49
Yes, the answer is 2.02
Please click here for the next question.
I used the “Calc” menu and “integrate”.
51
Too bad that answer is incorrect.
To find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (2) and subtract the value at the bottom limit (- 1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
Please click here to try again.
52
Yes, the answer is 27.23
Please click here for the next question.
I used the “Calc” menu and “integrate”.
53
18. Find the area between y = 4x – x 2 and the x-axis
A. 4 B. 10.67
C. 5.38 D. 7.33
E. None of the above
54
That answer is incorrect.
Please click here to try again.
This area is the definite integral between the x-intercepts of the curve. Graph the equation to find the intercepts and then to find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (3) and subtract the value at the bottom limit (1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
55
WOW, the correct answer is indeed 10.67
Please click here for the next question.
I used the “Calc” menu and “integrate”. Note the curve crosses the x- axis at 0 and 4 the limits of your definite integral.
dxxx44
0
2
56
19. Find the area between y = 3 – 2x 2 and the x-axis
A. 4 B. 4.90
C. 5.89 D. 7.33
E. None of the above
57
That answer is incorrect.
Please click here to try again.
This area is the definite integral between the x-intercepts (You will need a calculator to find the x-intercepts.) of the curve. Graph the equation to find the intercepts and then to find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (3) and subtract the value at the bottom limit (1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
58
WOW, the correct answer is indeed 4.90. That is a tough one because of the x-intercepts.
Please click here for the next question.
I used the “Calc” menu and “integrate”. Note the curve crosses the x- axis at – 1.225 and 1.225 the limits of your definite integral.
dxx23225.1
225.1
2
59
20. Find the area between y = 3 – 2x 2 and y = 2x 2 – 4x.
A. - 5.33 B. 6.67
C. 5.33 D. 4.50
E. None of the above
60
That answer is incorrect.
Please click here to try again.
This area is the definite integral of the difference of the two functions. You will need a calculator to find the x-intercepts. Graph the function that is the difference of the two equations to find the intercepts and then to find the definite integral you have two options-
1. Calculate the indefinite integral and evaluate at the top limit (3) and subtract the value at the bottom limit (1)
OR
2. Use your calculator to do the work. Use the “Calc” menu and integrate.
61
Grrrrreat, the correct answer is 5.33
Please click here for the next question.
I used the “Calc” menu and “integrate”. Note the curve crosses the x- axis at – 0.5 and 1.5 the limits of your definite integral.
dx)x4x2()x23( 25.1
5.0
2
63
21. The Lorenz curve for the distribution of income for students at York College is given by f (x) = x 1.5.
A. Find the index of income concentration.
A. 0.10 B. 0.20
C. 0.25 D. 0.35
E. None of the above
64
Sorry that answer is incorrect.
The index of income concentration is
where f (x) is the Lorenz curve.
Please click here to try again.
1
0dx])x(fx[2
65
Yes, the answer 0.2 using the definition of the index of income concentration and integrating on your calculator.
Please click here for the next question.
20.0)10.0(2dx]xx[21
0
5.1
1
0dx])x(fx[2
This answer will be used in the next problem.
66
22. The Lorenz curve for the distribution of income for students at York College is given by f (x) = x 1.5.
B. Interpret the results of the previous problem.
After you have written a response click here to check your answer.
67
Answers will vary. Remember an index of 0.0 indicates complete equality of income distribution
while and index of 1.00 indicates complete inequality.
This index of 0.20 show a fairly equitable distribution of income.
Please click here for the next question.
68
23. The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = (100 + 5x)/x
A. Find the total cost function.
A. C (x) = 100 + 5x B. C (x) = 20 – 0.002x
C. C (x) = 100x + 5 x 2 D. C (x) = 20x 2 – 0.002x 3
E. None of the above
69
Too bad that answer is incorrect.
Remember the average cost equation.
Please click here to try again.
x
CC
Solve for C.
70
Yes, the answer is
C (x) = 100 + 5x
Multiply the average cost by x.
Please click here for the next question.
71
24. The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = (100 + 5x)/x
A. Find the total profit function.
A. P (x) = 100 + 5x B. P (x) = 15x – 0.002 x 2 – 100
C. P (x) = 25x – 0.002 x 2 D. P (x) = 25x – 0.002 x 2 - 100
E. None of the above
73
Yes, the answer is P (x) = 15x – 0.002 x 2 – 100, since
P (x) = R - C= (20 x – 0.002 x 2 ) – (100 + 5x)= 15x – 0.002 x 2 – 100
Please click here for the next question.
You will need this answer for the next question.
74
25. The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = (100 + 5x)/x
for 0 < x < 7000B. Find the maximum profit.
A. $27,500
B. $28,025
C. $28,575
D. $29,525
E. None of the above
75
No that answer is incorrect.
P = R – C
P = 15x – 0.002 x 2 - 100
Please click here to try again.
Now find the maximum profit.
76
Yesaroonie, the answer is
P (x) = 15x – 0.002 x 2 – 100, since
P (x) = R - C
= 20 x – 0.002x 2 – (100 + 5x)
= 15x – 0.002 x 2 – 100
Now graph it to find the max.
max P = $28,025
Please click here for the next question.
0 < x < 7000
You will need this information for the next problem.
77
26. The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = (100 + 5x)/x
C. Find the price that yields the maximum profit.
A. $11.00
B. $11.50
C. $12.00
D. $12.50
E. None of the above
78
No that answer is incorrect.
Use the x value (sales giving max profit) from the last problem and substitute it into the price equation. Do you remember how to find the price equation. Think REVENUE!
Please click here to try again.
79
Perfect, the answer is $12.50
One way to get this is to plug the x value from the maximum profit (previous problem) into the price equation which comes from the revenue equation.
x = 3750
And from R(x) = 20x – 0.002x 2 the price equation is
p (x) = R (x) / x = 20 – 0.002x
p (3750) = 20 – (0.002) (3750) = 12.50
Please click here for the next question.
80
27. “Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x 2 - 100 and C(x) = 100 + 5x for 0 < x < 7000
A. Find the average cost function.
A. C = x (100 + 5x)
B. C = (100 + 5x) / x
C. C = 100x + 5x
D. C = 100x + 5x - 15x – 0.003x 2 - 150
E. None of the above
81
Sorry that answer is incorrect.
Please click here to try again.
x
CC
Remember the average cost equation.
82
Shazzammm!
The correct answer is C = (100 + 5x) / x
since,
Please click here for the next question.
x
CC
83
28. “Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x 2 - 100 and C(x) = 100 + 5x for 0 < x < 7000
B. Find the revenue function.A. R(x) = x (100 + 5x)
B. R(x) = 20x – 0.003x 2
C. R(x) = 10x – 0.003x 2
D. R(x) = 20x – 0.003x 2 - 200
E. None of the above
84
No that answer is incorrect.
Profit = Revenue – Cost.
How would you find Revenue?
Please click here to try again.
85
Hot Dog!
The correct answer is R = 10x – 0.003x 2
since, Profit = Revenue – Cost and hence
Revenue = Profit + Cost
= (15x – 0.003x 2 - 100) + (100 + 5x)
= 20x – 0.003x 2
Please click here for the next question.
You will need this for the next question.
86
29. “Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x 2 - 100 and C(x) = 100 + 5x for 0 < x < 7000
B. Find the price-demand function.
A. p (x) = - 10 + 0.003x
B. p (x) = 20x – 0.003x 2
C. p (x) = 20 – 0.003x
D. p (x) = (100 + 5x) / x
E. None of the above
87
Not the correct answer.
Remember that R (x) = xp
And use R from the previous problem.
Please click here to try again.
88
Great work.
The correct answer is p (x) = 20 – 0.003x
since R = px and then p = R/x
and using R from the previous problem yields
p (x) = 20 – 0.003x.
Please click here for the next question.
89
30. A water supply is treated with a bactericide. The rate of increase in harmful bacteria t days after the treatment is given by the following where N is the number of bacteria per milliliter.
A. Find the minimum value of dN/dt. 0 ≤ x ≤ 18
A. 900
B. 875
C. - 875
D. - 900
E. None of the above
2t1
t1800
dt
dN
90
Sorry that answer is wrong.
Graph dN/dt, the given equation, on your calculator and find the minimum (y-value).
Please click here to try again.
91
WOW, yes a tough one without a calculator!
The correct answer is - 900
Please click here for the next question.
92
31. A water supply is treated with a bactericide. The rate of increase in harmful bacteria t days after the treatment is given by the following where N is the number of bacteria per milliliter.
B. If the initial count was 6000 bacteria per milliliter, find the equation for N (t).
A. N = 1 / (1 + t 2 ) + 6000
B. N = - 900 ln (1 + t 2 ) + 6000
C. N = 6000 / (1 + t 2 )
D. N = - 1800 ln (1 + t 2 ) + 6000
E. None of the above
2t1
t1800
dt
dN
93
Too bad, that is incorrect.
You need to integrate the initial equation dN/dt to find the equation for N.
Don’t forget the initial condition of 6000 bacteria per milliliter.
Please click here to try again.
94
Well done!
The correct answer is N = - 900 ln (1 + t 2 ) + 6000
Please click here for the next question.
Integrate the original equation and use the point (0, 6000) to get the equation for N.
c)t1(ln900dtt1
t2900dt
t1
t1800 2
22
Now substitute the point.
You will need this answer for the next problem.
95
32. A water supply is treated with a bactericide. The rate of increase in harmful bacteria t days after the treatment is given by the following where N is the number of bacteria per milliliter.
C. Find the bacteria count after 7 days.
A. 6,000
B. 4,335
C. 2,479
D. 1,981
E. None of the above
96
Sorry that is incorrect.
Plug t = 7 into the equation for N that you found in the previous problem.
Please click here to try again.
N (t) = - 900 ln (1 + t 2 ) + 6000
97
Well done!
The correct answer 2,479 bacteria.
Please click here for the next question.
N (t) = - 900 ln (1 + 7 2 ) + 6000= - (900) (3.912) + 6000= - 3521 + 6000= 2479