Final Exam Fall 2012

8/13/2019 Final Exam Fall 2012

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MATH 1231 Final Exam Fl2012 Name:

NORTHEASTERN UNIVERSITYDepartment of Mathematics

MATH 1231 (Calculus for Business and Economics) Final Exam — Fall 2012

Do not write in these boxes:

pg1 pg2 pg3 pg4 pg5 pg6 pg7 pg8 pg9 Total (100 points)

Name: Instructor:

Instructions:

• Write your name in the blank above. Put your final answers to each question in the designated spaces onthese test pages (you may lose all credit for a problem if you do not).

• SHOW YOUR WORK. If there is not enough room to show your work, use the back of the preceding

page.• Whenever you use nDeriv or fnInt on your calculator, say so. Always tell what function is in

Y 1, Y 2, etc. in your calculator.

• For your convenience, there is a table of formulas at the end of the exam.• On this exam, you may only use one of the following calculators: TI-83, TI-83+, TI-84 or TI-84+.

1. (5 points) Suppose that f (x) is a function and

f (x) = 5x2 − 3x − 8,

i.e., the DERIVATIVE of f (x) is 5x2

− 3x − 8.USE BASIC ALGEBRA to find the x-coordinate of each critical point of f (x). Write down the equationyou solve to find the critical points and SHOW all the steps to solve it.

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2. Circle the number of the correct answer. There is only one correct answer and there is no partialcredit. (4 points each)

(a) An antiderivative of m(x) = 12e−7x + 3(2.2x) is:

(i) −12

7 e−7x +

3(2.2x)

ln(2.2) (ii) −84e−7x + 3 ln(2.2)(2.2x) (iii) 12e−3.5x2 + 3(2.20.5x2)

(iv) 12e−7x + 6.6x

ln(6.6) (v) none of the previous

(b) An antiderivative of h(x) = 6(2x − 3)8 is:

(i) 96(2x − 3)7 (ii) 23

(2x − 3)9 (iii) 43

(2x − 3)9

(iv) 1

3(2x − 3)9 (v) none of the previous

(c) An antiderivative of g(x) = 2

7√

x3is:

(i) 2 ln |x3/2|

7 (ii)

−37√

x5(iii)

−47

x−1/2 + 1 (iv) −x−1/2

7 (v) none of the previous

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3. (4 points) Suppose that f (x) and f (x) are continuous functions on the interval [1, 3]. Assume that

f (3) = 11, f (3) = −15 and 31

f (x) dx = −2. Find the value of f (1). Circle the correct answer. Thereis only one and there is no partial credit.

(i) -13 (ii) 13 (iii) 9 (iv) 2 (v) None of the previous

4. Suppose that N (x) is the number of tablet PCs sold, in thousands, when the selling price of each tabletPC is x hundred dollars.

(a) (2 points) Give the units of N

(x):

(b) (2 points) Write a complete sentence with units explaining the practical meaning of the followingstatement:

dN

dx = −35 at x = 7.

Do not use words such as per, rate, slope, derivative or any term relating to calculus. Be brief (25words or less) and precise.

5. (4 points) The revenue of a candy company is $98,000 when it sells 5500 boxes of candy, and its marginalrevenue is $3.20 per box when it sells 5500 boxes of candy. Use this information to give the best estimateof the company’s revenue when it sells 6000 boxes of candy. Circle the correct answer.

(i) $106,900 (ii) $101,200 (iii) $114,000 (iv) $99,600 (v) $100,000

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6. (4 points) The function g(x) has a critical point at x = 0 and x = 6. The DERIVATIVE of g(x) is equalto x3 − 6x2, i.e., g(x) = x3 − 6x2. Circle the correct statement for each critical point. Show all workincluding the values of any functions you evaluate.

x = 0 is a (i) relative maximum (ii) relative minimum (iii) neither

WORK:

x = 6 is a (i) relative maximum (ii) relative minimum (iii) neither

WORK:

7. (6 points) Find F (x), the specific antiderivative of the function

f (x) = x4 − 3x

+ 4

5,

such that F (1) = 5. Show all your work, especially the equation you must solve. Be sure to write outyour final formula for F (x) clearly and fully.

F (x) =

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8. Find the derivative of each of the following functions. (6 points each)

(a) f (x) = (2x3 − 4x + 8)6(π2 + 2 ln(x))

(b) g(x) = 3 5√

x + 4x−10 + 2x

(c) h(x) = 12.5

1 + 48e−0.5x − 16

3x6

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9. The functionB(t) = 5000(1.035)t

gives the balance in dollars in a savings account t years after January 1, 2000.

(a) (3 points) Find the average rate of change of the balance in the account between January 1, 2005and January 1, 2010. Show work. Give your answer using appropriate units, rounded to two decimalplaces.

.

(b) (3 points) Write a formula involving a definite integral that gives the average balance in dollars inthe account between January 1, 2005 and January 1, 2010. Use proper notation for the definiteintegral, not fnInt.

(c) (2 points) Use your formula in part (b) to find the average account balance (in dollars) betweenJanuary 1, 2005 and January 1, 2010. Show work. If you use fnInt show this. Round your answerto the nearest cent.

(d) (3 points) Write a complete sentence with units explaining the practical meaning of the followingstatement:

105

dB

dt dt = $1114.56.

Do not use any technical terms from calculus like integral or derivative in your answer. Be brief (25words or less) and precise.

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10. (5 points) A company produces bracelets for a fundraiser. The profit, in dollars, from selling bracelets isgiven by the function

P (x) = (x − 10)(500− x ln(3x))when the selling price of each bracelet is x dollars. Carefully enter the function P (x) into Y 1 on yourcalculator. Check that you’ve entered the function correctly by computing P (30). You shouldobtain 7300.114198.

Based on past experience with fundraisers, the company expects the selling price of a bracelet to bebetween $20 and $80. Use nDeriv and your calculator to determine the price of a bracelet that maximizesthe PROFIT. Write the answer with all the decimal places the calculator gives.

Calculator answer:

11. A company makes personalized baseball caps. The company has kept a record of its costs for selectedhourly production levels. This information is given in the table below.

x = number of hats produced 10 20 30 40 50 60Cost in $ 100 200 250 400 750 1400

(a) (3 points) Let C (x) denote the (hourly) cost, in dollars, of making x baseball caps. Use the tableabove to fit the best model for C (x), among the following choices: QUADRATIC, EXPO-NENTIAL, OR CUBIC. Write the formula for C (x) rounding every coefficient to 3 decimalplaces,

(b) (3 points) Use the formula for C (x) from part (a) to find the marginal cost function for producing xbaseball caps. Include the units of the marginal cost function in your answer.

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12. Put the function h(x) = 2x3 − 9x2 + 15x into Y 1 in your calculator. Check that Y 1(3) = 18.(a) (5 points) Use derivatives and algebra to find the inflection point of the function h(x). Give the

exact values of its x and y coordinates. Do not round your answers. Show your work especially theequation you must solve and how you solve it. Use your calculator only for basic arithmetic.

x-coordinate of inflection point:

y-coordinate of inflection point:

(b) (2 points) Graph the function h(x) (from part (a)) using the window:

X min = 0 , X max = 3 , Y min = 0 , Y max = 18.

Carefully sketch what you see below. In your sketch, label the inflection point clearly with theletters “IP”.

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12

18

1 2 3

(c) (2 points) Use the graph you sketched in part (b) to decide which kind of inflection point the graphhas. Circle one of the following:

Point of fastest increase Point of slowest increase

Point of fastest decrease Point of slowest decrease

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13. (6 points) Use the method of u-substitution to evaluate the following integral, i.e., to find the generalantiderivative:

(6x2 − 4)(x3 − 2x + 7)5 dx .

Show all work, especially u and du.

u = , du =

14. A portion of the graph of y = x2 − 16x + 72 is given below.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

10

20

30

40

50

60

70

(a) (2 points) Write the definite integral which represents the area of the shaded region above. Useproper notation. The notation fnInt is not acceptable.

(b) (4 points) Estimate the area (in square units) of the shaded region using the 4 right rectangleapproximation. Circle the number below of the answer closest to the approximation. There isno partial credit.

(i) 112 (ii) 138 (iii) 108 (iv) 176 (v) 136

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List of Formulas

Derivatives: (xn) = nxn−1, (ex) = ex, (ax) = (ln a)ax, (ln x) = 1

x;

(chain rule) (f (g(x))) = f (g(x))g(x), (product rule) (f g) = f g + f g

Approximation Formula for the change in a function using the derivative:

f (x + h) − f (x) ≈ f (x)h

Integrals:

ebx dx =

ebx

b + C ,

1

x dx = ln(|x|) + C ,

xn dx =

xn+1

n + 1 + C (n = −1),

ax dx =

ax

ln(a) + C .

Fundamental Theorem of Calculus If dF

dx = f (x), then

ba

f (x) dx = F (b)

−F (a).

Average value of f(x) = 1

b − a ba

f (x) dx

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Final Exam Fall 2012