1 REVIEW REVIEW TEST 2

2 1. Find the derivative for y = 3x 2 + 5x - 7 A. y’ = 3x + 5C. y’ = 6x C. y’ = 6x + 5D. y’ = 6x E. None of the above

3 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.

4 Yes, the answer is y’ = 6x + 5 Please click here for the next question. Good work on using the power rule! If y = x n, then y’ = n x n – 1.

5 2. Find the derivative for y = 2x – 1 – x – 3 A. y’ = - 2x 0 + 3x – 2 B. y’ = - 2x – 2 – 3x – 4 y’ = - 2x – 2 – 3x – 4 C. y’ = 2x – 2 + 3x – 4 D. y’ = - 2x – 2 + 3x – 4 E. None of the above

6 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.

7 Yes, the answer is y’ = - 2x – 2 + 3x – 4 Please click here for the next question. Good work on using the power rule! If y = x n, then y’ = n x n – 1.

8 3. Find the derivative for y = (2x 2 – x + 1)(x – 2) A. y’ = (4x – 1)(1) B.y’ = 6x x + 3y’ = 6x x + 3 C. y’ = (2x 2 – x + 1)(1) + (x – 2)(4x – 1) D. y’ = (4x – 1)(x - 1) E. None of the above

9 Too bad that answer is incorrect. You need to use the product rule. Please click here to try again. The product rule is – If f (x) = F (x) S (x), Then f ’ (x) = F (x) S’ (x) + S (x) F ’(x)

10 Yes, the answer is y’ = 6x x + 3 or y’ = (2x 2 – x + 1)(1) + (x – 2)(4x – 1) Please click here for the next question. Good work on using the product rule If y = f · s, then y’ = f · s’ + s · f’.

11 4. Find the derivative for A. B. C. D. y’ = (5x + 2)(2x) – (x 2 + 3) (5) E. None of the above

12 Sorry, that answer is incorrect. Please click here to try again. You need to use the quotient rule. The quotient rule is – If Then

13 Great, the correct answer is Please click here for the next question. OR You used the quotient rule correctly!! Ifthen

14 5. Find the derivative for A. y’ = - 2 x 1/2 B. y’ = 4 x – 3/2 C. y’ = - 2 x – 3/2 D. y’ = 4 x 1/2 E. None of the above

15 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.

16 Hey great, the correct answer is y’ = - 2 x – 3/2 Please click here for the next question. Nice work on the power rule and negative exponents.

17 6. Find the derivative for 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 y = (x 2 – 3x + 6) 5

18 Too bad, that answer is incorrect. You need to use the chain rule with u = x 2 – 3x + 6 Please click here to try again.

19 OK, the correct answer is y’ = 5 (x 2 – 3x + 6) 4 (2x – 3) Please click here for the next question. Good use of the chain rule with u = x 2 – 3x + 6 !!

20 7. Find the derivative for A. y’ = x (x 2 + 3) – 1/2 B. y’ = (x 2 + 3) – 1/2 D. y’ = 1/2 (x 2 + 3) - ½ (2x) E. None of the above C.

21 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 Please click here to try again.

22 Terrrrriffffic!! Please click here for the next question. Rewrite the equation without a radical and then use the chain rule - y = (x 2 + 3) ½ and the let u = x Then y’ = x (x 2 + 3) – ½ OR y’ = 1/2 (x 2 + 3) - ½ (2x) OR

23 6. If f (x) = 2x 3 - 3x 2 – 5x + 3, find the second derivative. A. y” = 6x – 6B. y” = 6x 2 C. y” = 12x - 6D. y” = 6x 2 - 6x - 5 E. None of the above

24 No that answer is incorrect. You need to take the derivative of the derivative! Please click here to try again.

25 Yes Yes Yes, the correct answer is y” = 12x - 6. Please click here for the next question. y’ = 6x 2 - 6x – 5 and y” = 12x - 6

26 APPLICATIONS Now, let’s try some application problems

27 9. A drug is injected into the bloodstream. The concentration of the drug after x hours is given by A. Find the marginal concentration after 3 hours (4 decimal places). A. C’ (x) = ml/cm 3 B. C’ (x) = – ml/cm 3 C. C’ (x) = – ml/cm 3 D. C’ (x) = ml/cm 3 E. None of the above for 0  x  10.

28 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.

29 Yes, the answer is C’ (x) = – ml/cm 3 from your graphing calculator. Please click here for the next question. NOTE: You will need this answer for the next question.

A drug is injected into the bloodstream. The concentration of the drug after x hours is given by B. Interpret the results of the previous question. Take a minute and try to write a response. Go to a representative correct answer. for 0  x  10.

31 The drug concentration after three hours is decreasing by about ml/cm 3 for the next hour. Please click here for the next question.

The total profit from the sale of text books is P (x) = 30 x – 0.3 x ≤ 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

33 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.

34 Yes, the answer is P’ (x) = 6 from your graphing calculator. Please click here for the next question. NOTE: You will need this answer for the next question.

B. The total profit from the sale of text books is P (x) = 30 x – 0.3 x ≤ x ≤ 100. B. Interpret P ‘ (40) = 6 Write an interpretation and then click here to see a representative answer.

36 Please click here for the next question. The profit when selling the next textbook (#41) will increasing by about $6.

The total profit (in dollars) from the sale of x calculators is P (x) = 22x – 0.2x 2 – 400 A. Find the average profit per calculator if 50 calculators are produced and sold. A. P (x) = $4B.P (x) = $3P (x) = $3 C. P (x) = $3.50D. P (x) = $2 E. None of the above for 0  x  100.

38 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. Please click here to try again.

39 Yes, the answer is Please click here for the next question. OR

The total profit (in dollars) from the sale of x calculators is P (x) = 22x – 0.2x 2 – 400 B. Find the marginal average profit at a production level of 50 calculators. A. P’ (x) = $ 0.04 B.P’ (x) = $ P’ (x) = $ C. P’ (x) = $ 0.40 D. P’ (x) = $ E. None of the above for 0  x  100.

41 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. Please click here to try again.

42 Yesaroonie, the answer is – Please click here for the next question. NOTE: You will need this answer for the next question.

Write a brief interpretation of the results of the previous problem. Take a minute and write an interpretation. Go to a representative correct answer.

44 Please click here for the next question. The average profit is decreasing by about $0.04 for the sale of the 51st calculator.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x for 0  x  6,000. A. Find the revenue function. A. R (x) = x ( x)/30 B. R (x) = ( x)/(30x) R (x) = ( x)/(30x) C. R (x) = x ( x) D. R (x) = p (x) + C (x) E. None of the above

46 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.

47 You have what it takes. The correct answer is R (x) = xp = x ( x)/30 Please click here for the next question. NOTE: You will need this answer for the next question.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x for 0  x  6,000. A. Find the marginal revenue function. A. R’ (x) = – x/30 B. R’ (x) = p’ (x) – C’ (x) R’ (x) = p’ (x) – C’ (x) C. R’ (x) = 200 – x/15 D. R’ (x) = 200 – x/30 E. None of the above

49 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 ( x)/30 Please click here to try again.

50 Great calculus work. The correct answer is the derivative of R (x) = x ( x)/30 Please click here for the next question. R (x) = x ( 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.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x for 0  x  6,000. C. Find the marginal revenue at x = A. $25 B. $100 $100 C. $75 D. $125 E. None of the above

52 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.

53 Great work. The marginal revenue equation is R’ (x) = 200 – x/15 and R’ (1500) = 200 – 1500/15 = = $100 Please click here for the next question. NOTE: You will need this answer for the next question.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x 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.

55 Please click here for the next question. At a sales rate of 1500, the revenue will be about $100 on the sale of the next gadget.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x for 0  x  6,000. E. Find the profit function. A. P (x) = 260x – x 2 / B. P (x) = C (x) – R (x) P (x) = C (x) – R (x) C. P (x) = 140x – x 2 / D. P (x) = 140x – x/ E. None of the above

57 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.

58 Well done! The correct answer is P (x) = 140x – x 2 / P (x) = R (x) – C (x) = (200x – x 2 /30) – ( x) = 140x – x 2 / Please click here for the next question. NOTE: You will need this answer for the next question.

The price-demand and cost equations for producing gadgets are given by: p (x) = ( x)/30 and C (x) = x for 0  x  6,000. E. Find the marginal profit function. A. P’ (x) = 140x – x 2 / B. P’ (x) = 140 – x/30 P’ (x) = 140 – x/30 C. P’ (x) = 140 – x/15 D. P’ (x) = 140x – x/15 E. None of the above

60 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.

61 Well done! The correct answer P’ (x) = 140 – x/15 Please click here for the next question. Take the derivative of P in the previous problem. P (x) = 140x – x 2 /30 – and P’ (x) = 140 – x /15

A company that makes CD players has the following total cost function. C (x) = x 2 + 2x C. What is the minimum average cost? A. $ B. $ $ C. $ D. $ E. None of the above

63 No that is incorrect. Remember the average cost function is the cost divided by x or 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.

64 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.

65 Terrific! The correct answer $ Please click here for the next question. Did you use your calculator to find that? 0 < x < 100 and 0 < y < 200.

66 23 – 25. The price-demand and cost equations for producing gadgets are given by: p (x) = 270 – 10x and C (x) = x for 0  x  How many gadgets should be produced to maximize profit?. A. 900 B. 10 C. 120 D. 12 E. None of the above.

67 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.

68 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.

69 Please click here for the next question. Yes indeedie calculator speedie! X = 6. Note P(x) = R – C = (x(270 – 10x))-( x) Save this info for #30 and #31.

70 23 – 25. The price-demand and cost equations for producing gadgets are given by: p (x) = 270 – 10x and C (x) = x for 0  x  What is the maximize profit?. A. 900 B. 10 C. 120 D. 12 E. None of the above.

71 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.

72 Please click here for the next question. Yes indeedie calculator speedie! X = 6. Save this info for #31.

73 23 – 25. The price-demand and cost equations for producing gadgets are given by: p (x) = 270 – 10x and C (x) = x for 0  x  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.

74 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.

75 Please click here for the next question. Hey, hey, hey, hey, HEY!!!! Yes you plugged x = 10 into the price equation to get $170 didn’t you. You cleaver devil.

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