1 REVIEW TEST 4

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

3 Sorry, that answer is incorrect. Remember the power rule for integration – Please click here to try again.

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.

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

6 Not quite, that answer is incorrect. Remember the power rule for integration – Please click here to try again.

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.

8 3. Find the following integral A. B. C. E. None of the above D.

9 Too bad, that answer is incorrect. Remember the power rule for integration – Please click here to try again.

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.

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

12 Not quite, that answer is incorrect. Remember the rule for exponential functions for integration – Please click here to try again.

13 Terrrrriffffic, the correct answer is e x + c, Using the rule for exponential expressions for integration. Please click here for the next question.

14 A. ln x + c B. e x + c C. D. x + c E. None of the above 5. Find the following integral

15 Not quite, that answer is incorrect. Remember the rule for logarithmic functions for integration – Please click here to try again.

16 Terrrrriffffic, the correct answer is ln x + c, since Please click here for the next question.

17 6. Now, let’s try something really interesting. A. B. ln x – ex + c C. 3ln x – 3ex + c D. E. None of the above

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.

20 7.Let’s go back to the power rule. A. B. D. E. None of the above C.

21 That answer is incorrect. Try changing the radical to exponential form! Please click here to try again. Then use the power rule.

22 Grrrrreat, the correct answer is Please click here for the next question. And using the power rule for integration yields -

23 8. Find the following integral A. -2 x -1 / 2 + c B.2 x 1 / 2 + c2 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. Then use the power rule.

25 WOW, the correct answer is indeed 2x ½ + c. Please click here for the next question.

26 9. Let’s complete that idea. A. 2x 2/3 + 3x 3/2 + c B. -6x -1/2 + 3x -4/3 + c -6x -1/2 + 3x -4/3 + c C. 2x 3/2 + 3x 2/3 + cD. -2x 3/2 - 3x - 2/3 + c E. None of the above

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.

OK. Let’s try some involving substitution techniques. A.B. D. E. None of the above 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.

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.

A. B. 5 (2x 5 – 4x + 7) 4 (20x 3 ) + c D. E. None of the above 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.

34 Grrrrreat, the correct answer is Please click here for the next question. Let u = 2x 5 – 4x + 7, then du = 10x 4 – 4 which is 2 (5x 4 – 2). SO the original problem becomes

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

36 Too bad that answer is incorrect. Please click here to try again. First, rewrite the problem - Let u = x 2 – 3, then du = 2x dx which means that we need a factor of 2 in the problem or

37 Yes, you are an integrating machine. Please click here for the next question. First, rewrite the problem Let u = x 2 – 3, then du = 2x dx which means that we need a factor of 2 in the problem or

B. C.D. E. None of the above A.

39 No that answer is incorrect. You must substitute correctly! Please click here to try again. Let u = 2x 3, then du = 6x 2 dx.

40 Correctamundo, the answer is Please click here for the next question. Let u = 2x 3, then du = 6x 2 dx, then

A. 2 (2x – 1) + cB.½ (2x – 1) + c½ (2x – 1) + c C. ½ ln | 2x – 1| + cD. 2x – 1 + c E. None of the above

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.

43 Yes, the answer is ½ ln | 2x – 1| + c Please click here for the next question. Let u = 2x - 1, then du = 2 dx, then,

44 15.Now that you have indefinite integrals down cold let’s try some definite integrals. A. 5B C. – 8.67D E. None of the above

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”.

A. 3B C. 2.02D E. None of the above

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”.

A. 3B C D E. None of the above

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 Please click here for the next question. I used the “Calc” menu and “integrate”.

Find the area between y = 4x – x 2 and the x-axis A. 4 B C D 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 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.

Find the area between y = 3 – 2x 2 and the x-axis A. 4 B C D 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 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 – and the limits of your definite integral.

Find the area between y = 3 – 2x 2 and y = 2x 2 – 4x. A B C. 5.33D 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.

62 APPLICATIONS Now, let’s try some application problems

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. 0.10B C. 0.25D 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.

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. This answer will be used in the next problem.

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.

The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = ( x)/x A. Find the total cost function. A. C (x) = xB.C (x) = 20 – 0.002xC (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. Solve for C.

70 Yes, the answer is C (x) = x Multiply the average cost by x. Please click here for the next question.

The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = ( x)/x A. Find the total profit function. A. P (x) = xB.P (x) = 15x – x 2 – 100P (x) = 15x – x 2 – 100 C. P (x) = 25x – x 2 D. P (x) = 25x – x E. None of the above

72 Too bad that answer is incorrect. Please click here to try again. P = R – C

73 Yes, the answer is P (x) = 15x – x 2 – 100, since P (x) = R - C = (20 x – x 2 ) – ( x) = 15x – x 2 – 100 Please click here for the next question. You will need this answer for the next question.

The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = ( x)/x for 0 < x < 7000 B. Find the maximum profit. A. $27,500 B. $28,025 $28,025 C. $28,575 D. $29,525 E. None of the above

75 No that answer is incorrect. P = R – C P = 15x – x Please click here to try again. Now find the maximum profit.

76 Yesaroonie, the answer is P (x) = 15x – x 2 – 100, since P (x) = R - C = 20 x – 0.002x 2 – ( x) = 15x – 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.

The “Screaming Green Machine” t-shirts have the following revenue and average cost equations. R(x) = 20x – 0.002x 2 and C(x) = ( x)/x C. Find the price that yields the maximum profit. A. $11.00 B. $11.50 $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) = Please click here for the next question.

“Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x and C(x) = x for 0 < x < 7000 A. Find the average cost function. A. C = x ( x) B. C = ( x) / x C = ( x) / x C. C = 100x + 5x D. C = 100x + 5x - 15x – 0.003x E. None of the above

81 Sorry that answer is incorrect. Please click here to try again. Remember the average cost equation.

82 Shazzammm! The correct answer is C = ( x) / x since, Please click here for the next question.

“Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x and C(x) = x for 0 < x < 7000 B. Find the revenue function. A. R(x) = x ( x) B. R(x) = 20x – 0.003x 2 R(x) = 20x – 0.003x 2 C. R(x) = 10x – 0.003x 2 D. R(x) = 20x – 0.003x 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 ) + ( x) = 20x – 0.003x 2 Please click here for the next question. You will need this for the next question.

“Rolling Stones” t-shirts have the following profit and cost equations. P(x) = 15x – 0.003x and C(x) = x for 0 < x < 7000 B. Find the price-demand function. A. p (x) = x B. p (x) = 20x – 0.003x 2 p (x) = 20x – 0.003x 2 C. p (x) = 20 – 0.003x D. p (x) = ( x) / 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.

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 C D E. None of the above

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 Please click here for the next question.

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 ) B. N = ln (1 + t 2 ) N = ln (1 + t 2 ) C. N = 6000 / (1 + t 2 ) D. N = ln (1 + t 2 ) E. None of the above

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 = ln (1 + t 2 ) Please click here for the next question. Integrate the original equation and use the point (0, 6000) to get the equation for N. Now substitute the point. You will need this answer for the next problem.

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 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) = ln (1 + t 2 )

97 Well done! The correct answer 2,479 bacteria. Please click here for the next question. N (t) = ln ( ) = - (900) (3.912) = = 2479

98 That was a lot of review. I hope you found it helpful. Good luck on the test!