PRINTABLE VERSION. Practice Final Exam

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1 Page 1 of 25 PRINTABLE VERSION Practice Final Exam Question 1 The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2003 corresponds to x = 0. Year Profits (in millions of dollars) Find the cubic regression model for the data. Which of these is the coefficient of the x 2 term of the cubic regression model? a) b) c) 0.05 d) e) Question 2 The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2001 corresponds to x = 0. Year Profits (in millions of dollars) Find the R 2 value for the cubic regression model. a) 0.99 b) 0.88

2 Page 2 of 25 c) 0.94 d) 0.95 e) 0.90 Question 3 The following table of values gives a company's annual profits in millions of dollars. Rescale the data so that the year 2003 corresponds to x = 0. Year Profits (in millions of dollars) Find the exponential regression model for this data. a) b) c) d) e) f) g) None of the above. Question 4 Evaluate the limit: a) 7 b)

3 Page 3 of 25 c) -3 d) 0 e) Question 5 Evaluate the limit: a) 1 4 b) -1 4 c) 0 d) -4 e) Does not exist Question 6 Find the indicated limit (if it exists). a) Does not exist b) 0 c) 20 d) -10 e) -20

4 Page 4 of 25 Question 7 Find the indicated limit (if it exists) a) 10 3 b) 2 c) Does not exist d) -2 e) 0 Question 8 Find the indicated limit (if it exists). a) 9 b) 7 2 c) -1 2 d) 10 3 e) Question 9 Find using the graph of f(x) given below.

5 Page 5 of 25 a) 0 b) -5 c) 2 d) Does not exist. e) None of the above. Question 10 The graph of the function f is given below. Which of the following statements is true?

6 Page 6 of 25 a) The function is continuous at x = 2. b) The function is discontinuous at x = 2 because f (2) does not exist. c) The function is discontinuous at x = 2 because does not exist. d) The function is discontinuous at x = 2 because even though f (2) exists and exists, the two quantities are not equal. e) None of the above. Question 11 Find the derivative of a) b) c) d)

7 Page 7 of 25 e) Question 12 Suppose Find the average rate of change of f(x) with respect to x in the interval [6, 8]. a) 10 b) -7 c) 22 d) 5 e) 11 Question 13 Give the equation of the tangent line to the graph of at the point where x = 3. a) b) c) d) e) Question 14 A manufacturer has a monthly fixed cost of $270, and a production cost of $48 for each unit

8 Page 8 of 25 produced. The product sells for $72 per unit. Find the break-even quantity. a) 3,750 b) 810,000 c) 6,000 d) 11,250 e) 2,250 Question 15 A leading producer of airplanes finds that the company's weekly cost of manufacturing x airplanes is given by the function where C(x) is given in thousands of dollars. Use the marginal cost function to approximate the cost of producing the 3,001 st airplane. a) $5,100, b) $5,100, c) $5,100, d) $5, e) $5, Question 16 A computer company manufactures a certain variety of flat panel monitor. The demand for this monitor is given by the following equation, where p denotes the unit price and x denotes the quantity demanded. (0 x 5000) Use the marginal revenue function to approximate the actual revenue realized on the sale of the

9 Page 9 of 25 1,000th monitor. a) $ b) $ c) $ d) $ e) $ Question 17 A clothing company manufactures a certain variety of ski jacket. The total cost of producing x ski jackets and the total revenue of selling x ski jackets are given by the following equations (0 x 1000) Use the marginal profit to approximate the actual profit realized on the sale of the 901 st ski jacket. a) $65.00 b) $63.00 c) $62.00 d) $64.00 e) $66.00 Question 18 A company manufactures LED televisions. The total cost of producing x LED televisions can be approximated by the function Find the average cost of producing 120 LED televisions.

10 Page 10 of 25 a) $ per player b) $ per player c) $ per player d) $ per player e) $ per player Question 19 Suppose the demand function for a product is given by p = -0.05x where the function gives the unit price in dollars when x units are demanded. Compute the elasticity of demand, E(p), when the price is $100. a) 1.00 b) 0.75 c) 0.14 d) 0.85 e) 1.24 Question 20 Suppose E(p) = 2 3 when the price of the item is p. Then the demand is a) Elastic b) Unitary c) Inelastic d) None of the above. Question 21

11 Page 11 of 25 If Q(t) = 18.1 e t, find Q(t) when t = 4. a) b) c) d) e) Question 22 At the beginning of an experiment, a researcher has 511 grams of a substance. If the half-life of the substance is 18 days, how much of the substance is left after 18 days? a) grams b) grams c) 0 grams d) grams e) Not enough information is given to answer. Question 23 At the beginning of an experiment, a researcher has 523 grams of a substance. If the half-life of the substance is 16 days, how many grams of the substance are left after 29 days? a) b) c) 0 d) e)

12 Page 12 of 25 Question 24 The demand for a company's product t months after it is introduced on the market can be expressed as where D(t) is the number demanded. How many units should the company expect to be demanded when it is first introduced on the market? a) 4,000 b) 3,825 c) 105 d) 1,500 e) 0 Question 25 Suppose Which of these statements is/are true? I. The domain of the function is not (-, ). II. The range of the function is not (-, ). III. The graph of the function has no asymptotes. IV. The y-intercept is (0, -10). a) Only II and III are true. b) All of the statements are true. c) None of the statements are true. d) Only II, III and IV are true.

13 Page 13 of 25 e) Only I, II and IV are true. f) Only I and III are true. Question 26 Find the critical numbers: a) x = , b) x = -3.45, c) x = 2.09, d) x = 0.54 e) x = -2.09, 1.02 Question 27 The graph shown below is the graph of the first derivative of a function, f(x). State the number of inflection points and the number of relative minima.

14 Page 14 of 25 a) 3 and 3 b) 2 and 2 c) 3 and 1 d) 3 and 2 e) 2 and 3 Question 28 Suppose Find any critical numbers. a) , b) 0 c) , 0,

15 Page 15 of 25 d) , e) , Question 29 Suppose Find intervals on which the function is increasing and intervals on which the function is decreasing. a) Increasing on ( , ) ; decreasing on (-, ) (0.4082, ) b) Increasing on (-, ) (0.4082, ) ; decreasing on ( , ) c) Increasing on (-, ) (0.4134, ) ; decreasing on ( , ) d) Increasing on (0, ) ; decreasing on (-, 0) e) Increasing on ( , ) ; decreasing on (-, ) (0.4134, ) Question 30 Suppose Find any relative extrema. a) Relative maximum at (0.3536, ); relative minimum at ( , ) b) Relative maximum at ( , ); relative minimum at (0.3536, ) c) Relative maximum at (0.3783, ); relative minimum at ( , ) d) Relative maxima at ( , ) and (0.6124, ); relative minimum at (0, 0) e) Relative maximum at (0, 0); relative minima at ( , ) and (0.6124, )

16 Page 16 of 25 Question 31 Suppose Find any values of x for which f''(x) = 0. a) x = , x = 0, x = b) x = , x = 0, x = c) x = , x = d) x = , x = e) x = 0 Question 32 Suppose Find intervals on which the function is concave upward and intervals on which it is concave downward. a) Concave upward on (-, ) (0, ) ; concave downward on ( , 0) (0.6124, ) b) Concave upward on ( , 0) (0.6124, ) ; concave downward on (-, ) (0, ) c) Concave upward on ( , ) ; concave downward on (-, ) (0.3536, ) d) Concave upward on (0, ) ; concave downward on (-, 0) e) Concave upward on (-, 0) ; concave downward on (0, )

17 Page 17 of 25 Question 33 Suppose Find any inflection points. a) ( , ), (0, 0), and (0.3162, ) b) ( , ) and (0.5477, ) c) ( , ), (0, 0), and (0.5477, ) d) ( , ) and (0.3162, ) e) ( , ) and (0.3353, ) Question 34 Suppose Find any asymptotes. a) No horizontal asymptotes ; vertical asymptote at x = 0. b) Horizontal asymptote at y = 1 ; no vertical asymptotes. c) Horizontal asymptote at y = 0 ; no vertical asymptotes. d) Horizontal asymptote at y = 1 ; vertical asymptote at x = 0. e) Horizontal asymptote at y = 0 ; vertical asymptote at x = 0. Question 35

18 Page 18 of 25 The graph of a function, f (x), is given below. Find the absolute maximum value of this function. a) 1 b) -1 c) 0 d) -2 e) 2 Question 36 Suppose you want to fence in a rectangular-shaped field that lies along the straight edge of a river. The side that lies along the river will not need to be fenced. You have 500 feet of fencing material to use. Which of these is a function that expresses the area of the field that can be fenced in under these conditions, where x is the length of one of the two sides of the field that are perpendicular to the river? a) A(x) = x (250 x) b) A(x) = x (500 x) c) A(x) = x (250 2x)

19 Page 19 of 25 d) A(x) = x (500 2x) e) A(x) = x (250x x 2 ) Question 37 Suppose you want to fence in a rectangular-shaped field that lies along the straight edge of a river. The side that lies along the river will not need to be fenced. You have 250 feet of fencing material to use. What is the maximum area that can be fenced in under these conditions? a) b) c) d) e) Question 38 Let f (x) = 4x Compute the Riemann sum of f over the interval [0, 4] using 4 subintervals, choosing the left endpoints of the subintervals as representative points. a) 100 b) 72 c) 60 d) 140 e) 136 Question 39 Use Riemann sums with right endpoints and 20 subdivisions to approximate the area between

20 Page 20 of 25 and the x-axis on the interval [1, 5]. Round the answer to the nearest ten-thousandth. a) b) c) d) e) Question 40 Find the indefinite integral a) b) c) d) e) Question 41 Evaluate a) -738

21 Page 21 of 25 b) c) -234 d) -198 e) -774 Question 42 An efficiency study showed that the rate at which the average worker assembles products t hours after starting work can be modeled by the function where 0 t 4. Determine the number of units the average worker can assemble during the third hour that s/he works during a shift. a) 16 units b) 42 units c) 123 units d) 111 units e) 4 units Question 43 Suppose the velocity of a car can be modeled by the function where t is time given in seconds and v(t) is given in feet per second. Find the total distance traveled by the car from t = 0 to t = 3. a) feet b) feet

22 Page 22 of 25 c) feet d) feet e) feet Question 44 A company estimates that its annual sales during the first t years of operation can be modeled by the function where S is measured in thousands of dollars. What was the company's average annual sales over its first 3 years of operation? a) $19.21 thousand b) $5.59 thousand c) $2.78 thousand d) $8.35 thousand e) $6.40 thousand Question 45 Find the area of the region between f(x) = x 2 9x and g(x) = 6x. a) b) c) d) e)

23 Page 23 of 25 Question 46 Find the area of the region(s) that is/are completely enclosed by the graphs of f (x) = (x 1 ) and g (x) = 3x 3 Round limits of integration to 4 decimal places before integrating. a) b) c) d) e) Question 47 Let Find f ( -7, 8). a) 46 b) 60 c) -448 d) -564 e) -444 Question 48 Find the critical points of

24 Page 24 of 25 a) ( 0, 0 ), ( 1 18, 1 54 ) b) ( 0, 0 ), ( -1 18, ) c) ( 1 18, ) d) ( 0, 0 ) e) ( 0, 0 ), ( 1 18, ), ( 0, ), ( 1 18, 0 ) Question 49 Suppose f xx = 18x, f yy = 8, f xy = f yx = 5 and the critical points for function f are A = (0.6760, ) and B = ( , ) Find the value for D for each critical point and then classify the critical point using the second derivative test. a) D(0.6760, ) = ; relative maximum at (0.6760, ) ; D( , ) = ; saddle point at ( , ) b) D(0.6760, ) = ; relative maximum at (0.6760, ) ; D( , ) = ; saddle point at ( , ) c) D(0.6760, ) = ; saddle point at (0.6760, ) ; D( , ) = ; relative minimum at ( , ) d) D(0.6760, ) = ; relative minimum at (0.6760, ) ; D( , ) = ; saddle point at ( , ) e) D(0.6760, ) = ; relative minimum at (0.6760, ) ; D( , ) = ; saddle point at ( , ) Question 50 Suppose that f ( x, y ) = 4x 3 7xy + 8y 2,

25 Page 25 of 25 (0.2552, ) is a critical point, f xx (0.2552, ) = , and D (0.2552, ) = 49. Which of these statements describes the graph of f at (0.2552, )? a) f has a relative minimum value at f (0.2552, ) = b) f has a saddle point at f (0.2552, ) = c) f has a relative minimum value at f (0.2552, ) = d) f has a relative maximum value at f (0.2552, ) = e) f has a relative maximum value at f (0.2552, ) = f) f has a saddle point at f (0.2552, ) = g) None of the above.

Exam 2 Review (Sections Covered: and )

Exam 2 Review (Sections Covered: 4.1-4.5 and 5.1-5.6) 1. Find the derivative of the following. (a) f(x) = 1 2 x6 3x 4 + 6e x (b) A(s) = s 1/2 ln s ln(13) (c) f(x) = 5e x 8 ln x 2. Given below is the price-demand