1 review review test 2. 2 1. find the derivative for y = 3x 2 + 5x - 7 a. y’ = 3x + 5c. y’ = 6x...

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REVIEWTEST 2

2

1. Find the derivative fory = 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

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 fory = 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


6


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!


7

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 .

8

3. Find the derivative fory = (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)


9

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)

10

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’ .

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4. Find the derivative for

A.

B. C.

D. y’ = (5x + 2)(2x) – (x 2 + 3) (5)


2x5

3xy

2

2

2

)2x5(

)5)(3x()x2)(2x5('y

2

2

)2x5(

15x4x5'y

2

2

)2x5(

)5)(3x()x2)(2x5('y

12

Sorry, that answer is incorrect.


You need to use the quotient rule.

The quotient rule is –

If

Then

)x(B

)x(Ty

2])x(B[

)x('B)x(T)x('T)x(B'y

13

Great, the correct answer is


2

2

)2x5(

15x4x5'y

OR

2

2

)2x5(

)5)(3x()x2)(2x5('y

You used the quotient rule correctly!!

)x(B

)x(Ty

2])x(B[

)x('B)x(T)x('T)x(B'y

If then

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A. y’ = - 2 x 1/2 B. y’ = 4 x – 3/2

C. y’ = - 2 x – 3/2 D. y’ = 4 x 1/2


x

4y

15

Not quite, that answer is incorrect.

Change the equation to remove the radical to –

y = 4x – 1/2

and use the power rule.


16

Hey great, the correct answer is y’ = - 2 x – 3/2


Nice work on the power rule and negative exponents.

17


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)


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


dx

duunu

dx

d 1nn

19

OK, the correct answer is

y’ = 5 (x 2 – 3x + 6) 4 (2x – 3)


dx

duunu

dx

d 1nn

Good use of the chain rule with

u = x 2 – 3x + 6 !!

20


A. y’ = x (x 2 + 3) – 1/2

B. y’ = (x 2 + 3) – 1/2

D. y’ = 1/2 (x 2 + 3) - ½ (2x)


3xy 2

C. 3+xx2='y 2

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 2 + 3


dx

duunu

dx

d 1nn

22

Terrrrriffffic!!


3x

x'y

2

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

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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


24


You need to take the derivative of the derivative!


25

Yes Yes Yes, the correct answer is y” = 12x - 6.


y’ = 6x 2 - 6x – 5 and

y” = 12x - 6

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APPLICATIONS

Now, let’s try some application problems

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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) = 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


4x

x20.0)x(C

2 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.


29

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.

30

10. 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.

4x

x20.0)x(C

2 for 0 x 10.

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The drug concentration after three hours is decreasing by about 0.0059 ml/cm 3 for the next hour.


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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


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.


34

Yes, the answer is P’ (x) = 6

from your graphing calculator.



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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.

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The profit when selling the next textbook (#41) will increasing by about $6.

37

13. 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) = $4 B. P (x) = $3

C. P (x) = $3.50 D. P (x) = $2


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.


x

400x2.0x22

x

xP)x(P

2

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Yes, the answer is


4$50

200

50

400)50)(2.0()50)(22(

50

)50(P)50(P

2

OR

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14. 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) = $ - 0.04

C. P’ (x) = $ 0.40

D. P’ (x) = $ - 0.40


for 0 x 100.

41


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.


x

400x2.0x22

x

xP)x(PofitPrAvg

2

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Yesaroonie, the answer is – 0.04.


x

400x2.0x22

x

xP)x(PGraph

2


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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.

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The average profit is decreasing by about $0.04 for the sale of the 51st calculator.

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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)


46

Sorry that answer is incorrect.

To find the revenue function use R (x) = xp, where p is the price-demand function.


47

You have what it takes.

The correct answer is R (x) = xp

= x (6000 - x)/30



48


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


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 (6000 - x)/30


50

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


51


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


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),


53

Great work.

The marginal revenue equation is

R’ (x) = 200 – x/15 and

R’ (1500) = 200 – 1500/15

= 200 - 100

= $100



54


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.

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At a sales rate of 1500, the revenue will be about $100 on the sale of the next gadget.

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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


57

Too bad, that is incorrect.

The profit function is found by subtracting cost from revenue, or

P (x) = R (x) – C (x)


58

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



59


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


60

Sorry that is incorrect.

The marginal profit is the derivative of the profit and the profit was found in the previous problem.


61

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

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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


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.


x

)x(C)x(C

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.


65

Terrific!

The correct answer $91.44.


Did you use your calculator to find that?

x

)x(C)x(C

x

2000x2x2

0 < x < 100 and 0 < y < 200.

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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.

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.


68

That us close but incorrect.

You gave the maximum profit – you need the x value where the maximum profit occurs.


69Please click here for the next question.

Yes indeedie calculator speedie! X = 6.

Note P(x) = R – C = (x(270 – 10x))-(100 + 70x)

Save this info for #30 and #31.

70


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


71

Too too too bad that is incorrect.From the previous problem use the profit formula and use your calculator to maximize the profit.


72


Yes indeedie calculator speedie! X = 6.

Save this info for #31.

73


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


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.


75


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!

1 review review test 2. 2 1. find the derivative for y = 3x 2 + 5x - 7 a. y’ = 3x + 5c. y’ = 6x...

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