Final

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Instructor:

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Name: NORTHEASTERN UNIVERSITY Department of Mathematics

MTH U131 (Calculus for Business and Economics)

Final Exam

Fall 2008

Instructions: Put your name and your instructor’s name in the blanks above. Put your final answers to each question in the designated spaces on these test pages. (You may lose all credit for a problem if you don’t.) 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 Y1 , Y2 , 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+. Total Points = 100 1. For each scatter plot below, tell whether it is best described as: Linear, Exponential, Quadratic, Cubic, Logistic or None. The same answer might be used more than once. (1 point each)

a.

b.

c.

d.

2. Suppose that on January 1, 2009 a virus attacks computers on the internet and spreads according to the data given in the following table. x = # of hours after 9 AM Number of infected computers

0 100

0.5 600

1 2900

1.5 7100

2.0 9400

2.5 9900

a) (3 points) Let I(x) denote the number of infected computers x hours after 9 AM. Use the table above to fit the best model for I(x), among the following choices: LINEAR, QUADRATIC, EXPONENTIAL or LOGISTIC. Write the formula for I(x) here rounding each coefficient to 3 decimal places but put the full model into Y1 in your calculator and use the full model for part (b).

b) (2 points) Use the FULL (i.e, EXACT model) for I(x) from part (a) to estimate the number of computers infected with the virus at 10:15 AM. Show work and give your answer rounded to the nearest whole number with units.

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3. Suppose that S(x) is the number of students, in hundreds, who attend a private school when each student’s yearly tuition is x thousand dollars. In parts a), b), and c), of this problem circle the number of the correct answer. There is only one correct answer and there is no partial credit. a) (2 points) Give the units of S (x): (i)

hundreds of students per dollar

(ii)

thousands of dollars per student

(iii)

thousands of dollars per hundred students

(iv)

hundreds of students per thousand dollars

(v)

None of the above

b) (2 points) Which formula can be used to compute the average rate of change of S(x) from x = 10 to x = 12.5? S (12.5) − S (10) 12.5 − 10 S(12.5) − S(10) (ii) 12.5 − 10 12.5 (iii) S (x) dx. (i)

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(iv) S (12.5) − S (10) (v)

100(S(12.5) − S(10)) S(10)

c) (2 points) If S(13) = 4 and S (13) = −0.25, give the best estimate for the number of students at the private school when the yearly tuition is $13,600. . (i) 385 (ii) 415 (iii) 375 (iv) 425 (v) 360 4. A company makes golf tees. a) (2 points) The company’s COST (in dollars) for making golf tees is C(x) = 0.2x3 − 1.9x2 + 5.9x + 145.5, where x is the number of hundreds of golf tees produced. Write the formula for the MARGINAL COST function. Include the unit in your answer.

b) (2 points) Suppose that the company has made 600 golf tees and wants to estimate the additional cost in dollars from making one hundred more golf tees. Which of the following expressions should the company compute? Circle the number of the correct answer. There is only one, and there is no partial credit.)

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C(x) dx

(i)

(ii) C (6)

(iii) C (6)

(iv) C(6)

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(v)

C(6) 6

5. In parts a), b), c), and d) circle the number of the correct answer. (2 points each, no partial credit). a) An antiderivative of h(x) = 5(4.2x) is: (i) 5(4.2x ) ln(4.2)

(ii)

21x ln(21)

(iii) 21x ln(21)

(iv) 5(4.2)x

(iii) −5e−5x

(iv) e−2.5x

(v) None of the above

b) An antiderivative of g(x) = e−5x is: (i) e−5x

(ii)

e−5x −5

c) An antiderivative of m(x) = (i) −4(2x − 5)−3

(i)

3 ln(|x9 |) 4

3 (ii) − x−8 2


d (2x − 5)−2 is: dx

(ii) (2x − 5)−2

d) An antiderivative of k(x) =

2

(iii) (x2 − 5x)−2

(iv)

−1 (2x − 5)−1 2


3 is: 4x9 (iii)

−3 32x8

(iv)

30 4x10


6. (3 points) Let d(t) be the rate of change of the national debt, in billions of dollars per year, t years after January 1, 2001. What is the practical meaning of the following integral?

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d(t) dt 3

Circle the number of the correct answer. (There is only one, and there is no partial credit.) (i)

(ii) (iii)

The average rate of change of the national debt from 1/1/2004 to 1/1/2007 in billions of dollars per year. The change in the national debt from 1/1/2004 to 1/1/2007 in billions of dollars. The change in the rate of change of the national debt from 1/1/2004 to 1/1/2007 in billions of dollars per year.

(iv)

The average value of the national debt in billions of dollars from 1/1/2004 to 1/1/2007.

(v)

None of the above

7. (5 points) Suppose g(x) is a function chosen so that g(1) = 5.43 and 1

value of g(4). (i) 1.78

(ii) 9.08

(iii) -9.08

(iv) -1.78 3

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dg(x) dx = −3.65. Find the dx (v) None of the above

8. Find the derivative of each of the following functions (5 points each). √ 3 a) g(x) = 5 ln(x) − 6x−4 + 3(4.63x) − 15 x7

b) h(x) = (8−5x −

√ x)(ex − x2 )

c) f (x) =

15 + (ln(16))2 1 − 40e0.75x

d) m(x) =

3.65 − ln(2x3 + 10−x ) x6

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9. (7 points) A computer store’s profit function for selling flash drives is P (x) = (10x3 − 100x)e−0.2x , where P (x) is the profit in dollars when the selling price of each flash drive is x dollars and 5 ≤ x ≤ 24. Carefully enter the function P (x) into Y1 on your calculator. Check that you have entered the function correctly by computing P (6). You should obtain 469.8629706. Based on market conditions, the owner of the store knows that she shouldn’t sell a flash drive for more than $24. Use nDeriv and your calculator to determine the price of the flash drive that maximizes PROFIT. Write the answer with all the decimal places the calculator gives. calculator answer: Circle the equation you solved: (i) nDeriv(P (x), x, x) = 0

(ii)nDeriv(P (x), x, x) = P (x)

(iv) nDeriv(P (x), x, x) = 0

(v) nDeriv(P (x), x, x) = 0

(iii) nDeriv(P (x), x, x) = P (x)

Circle the calculator procedure you used: (i) 2nd/Calc/Zero

(ii) 2nd/Calc/Intersect

(iii) SOLV ER

10. (3 points) A campus sandwich shop sells a “super wrap.” Based on a marketing poll, the owner has found the following model for the demand function (i.e., the number of super wraps sold each day): D(x) = 200(.78x) + 4 where the price of each super wrap is x dollars. Each super wrap costs the shop $3.25 to make. The shop has fixed costs of $15 a day for advertising. Write down the formula for P (x), the daily profit function (in dollars). Write the formula out in full. Do not use any abbreviations.

P(x) =

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11. Read this problem carefully. Parts (a) and (b) of this problem are about DIFFERENT functions. a) (9 points) Suppose that f (x) = (4x + 15)(x − 6), i.e., the DERIVATIVE of f (x) is (4x + 15)(x − 6) . (i) Find the x-coordinates of the critical points of f (x). USE ALGEBRA. Show your work especially the equation you must solve and how you solve it.

(ii) Find the x-coordinate of a RELATIVE MAXIMUM of f (x). EXPLAIN how you know it is a relative maximum. SHOW the value(s) of any function(s) you compute.

b) (5 points) (i) Use derivatives and algebra to find the inflection point of the function h(x) = −x3 + 7.5x2 − 40x + 80. Give its x and y coordinates. Show your work especially the equation you must solve and how you solve it.

(ii) Graph h(x) (from part (i)) using the window: Xmin = −2 , Xmax = 6 , Ymin = −80 , Ymax = 240. Sketch what you see below. In your sketch, label the inflection point clearly with its coordinates. Use the graph to decide what kind of inflection point the graph has. Circle one of the following. Point of fastest increase

Point of slowest increase

Point of fastest decrease

Point of slowest decrease

240 160 80

-2

2 -80

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12. (5 points) Evaluate the integral, i.e., find the general antiderivative. Show work. x 4x2 − 3 dx

13. The price in dollars of a certain stock t years after January 1, 2001 can be modeled by the formula: P (t) = 96.50(0.92t). a) (3 points) Find the rate of change of the price of the stock on January 1, 2003. Show work. Give your answer (rounded to two decimal places) with units.

b) (5 points) Write a formula involving a definite integral for the average price of the stock in dollars between January 1, 2003 and January 1, 2007. Use proper notation for the definite integral.

Circle the correct answer for the average price of the stock between January 1, 2003 and January 1, 2007 (rounded to the nearest cent). (i) $69.45

(ii) $70.10

(iii) $69.13

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(iv) $58.51


14a. (2 points) Make a careful sketch of the entire region whose area is given by the definite integral: 1

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(5x − x2 ) dx.

Label the boundary curves (with their equations) and corner points (with their coordinates) of the region. Then shade the region.

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14b. (2 points) The 4 right rectangle approximation (in square units) of the area in part (a) rounded to two decimal places is: (Circle the number of the correct answer). There is no partial credit. (i) 11.33

(ii) 11.65

(iii) 11.95

(iv) 11.85

(v) 11.75

15. (4 points) Find F (x), the specific antiderivative of the function f (x), where f (x) = 144x2 + 16x−2 and F (1/4) = 20. Show all your work especially any equations you solve.

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List of Formulas

1 ; x (chain rule) (f (g(x))) = f (g(x))g (x); (product rule) (f g) = f g + f g

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

Approximation Formula for the change in a function using the derivative: f (x + h) − f (x) ≈ f (x)h Antiderivatives: e x a + C. ax dx = ln(a)

bx

ebx + C, dx = b

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

dF = f (x), then Fundamental Theorem of Calculus: If dx

a

b

xn dx =

xn+1 + C (n n+1

=

−1),

f (x) dx = F (b) − F (a).

The average value of f (x) over the interval from x=a to x= b is given by the following expression: 1 b−a

b

f (x) dx. a

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