MAT 210 TEST 2 REVIEW (Ch 12 and 13)

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1 Class: Date: MAT 0 TEST REVIEW (Ch and ) Multiple Choice Identify the choice that best completes the statement or answers the question.. The population P is currently 0,000 and growing at a rate of 7,000 per year. What is the mathematical notation for the rate? P dp dt dp dt dt dp. Evaluate the integral. x x.4 dx x x C x x 0.4 x x.4 + C x x C x x.4. Find the exact location of all the absolute extrema of the function with domain (-0, ). f (x)=x 4 8x (6, 4) - absolute minimum (6, 4) - relative minimum (6, 0) - absolute maximum (-6, 4) - relative maximum (6, 4) - absolute minimum 4. A general hyperbolic demand function has the form to follow. q = k p r (r, k - nonzero constants) Obtain a formula for the elasticity of demand at a unit price of p. E = k E = rp k E = rkp E = -kp E = r

2 5. The estimated monthly sales of Mona Lisa paint-by-number sets is given by the formula p 5p q = 50e where q is the demand in monthly sales and p is the retail price in yen. At what price will revenue be a maximum? Round the answer to the nearest hundredth yen 0.56 yen 8.74 yen 0.99 yen. yen 6. Evaluate the integral. x 6 ( x +6x + 7) dx (x +6x + 7) x +6x +7 (x +6x + 7) (x +6x + 7) (x +6x + 7) 7. Evaluate the integral. (x 78x) dx 9 8. The HMS Dreadnaught is 80 miles north of Montauk and steaming due north at 40 mph, while the USS Mona Lisa is 00 miles east of Montauk and steaming due east at an even 60 mph. How fast is their distance apart increasing? 77 miles per hour 69 miles per hour 7 miles per hour 60 miles per hour 44 miles per hour 9. Find the exact location of all the relative and absolute extrema of the function with domain (-, ). g (x) = 4 x 4 x + x + (0, ) - absolute minimum (0, ) - relative maximum (0, ) - relative maximum (0, ) - absolute maximum 5 (0, ) - absolute minimum 5 0. Find the coordinates of all relative and absolute extrem x g(x) = x 7 ( 9, 9), (0, 0) (0, 0) ( 9,.5), (9,.5) (0, 0), (9,.5) ( 9,.5), (9,.5), (0, 0),46,068 5,500 7,568 44

3 . Evaluate the integral. x 9 dx x C x C x C x C x C. The velocity of a particle moving in a straight line is given by v = e t + t. Find an expression for the position s after a time t. s(t) = e t +6t + C s(t) = e 4t + t + C s(t) =e t +t + C s(t) = e t + t + C s(t)=e t ++C. For a rectangle with perimeter to have the largest area, what dimensions should it have? Find f (x) if f(9) = -7 and the tangent line at (x, f(x)) has slope 8e x +6. f (x)=8e x 6x 8e f (x)=8e x +6x 8e 9 6 f (x)=8e x +6x +8e f (x)=8e x +6x 8e f (x)=8e x +6x +8e The consumer demand curve for Professor Stefan Schwartzenegger dumbbells is given by q = (9 8p) where p is the price per dumbbell and q is the demand in weekly sales. Find the price Professor Schwartzenegger should charge for his dumbbells in order to maximize revenu $8 $04 $4 $40 $0 6. Evaluate the integral. x( x )8 dx (x ) 0 (x )9 0 (x ) 9 (x )0 9 0 (x ) 0 (x )9 + 0 x (x ) 0 +(x ) 9 x (x ) 9 (x )0 9 90

4 7. Calculate the left Riemann sum for the function over the given interval using the given values of n. f(x)=6x over [0, ], n =4 Left Sum = 5 Left Sum = - 5 Left Sum = 4 Left Sum = 7 Left Sum = Evaluate the integral. xe 9x +9 dx 8e 9x +9 e 9x +9 e9x +9 9 xe9x +9 8 e9x A rather flimsy spherical balloon is designed to pop at the instant its radius has reached 5 cm. Assuming the balloon is filled with helium at a rate of 0 cm /s, calculate how fast the diameter is growing at the instant it pops. The volume of a sphere of radius r is V = 4 πr cm/s 0.55 cm/s 0.0 cm/s 0.44 cm/s 0.8 cm/s. Find the exact location of all the relative and absolute extrema of the function with domain (-, ). f (x) =e -8x8 (, 8) - absolute maximum (, 0) - absolute maximum (0, ) - absolute maximum (0, ) - absolute minimum (0, 8) - relative minimum 9. Combined SAT scores in the United States could be approximated by T(t) = -0.00t t 9.9t ( 0 t 4) in the years Here t is the number of years since 969, and T is the combined SAT score average for the United States. Find all points of inflection of the graph of T, and interpret the result. (0, 0) (7.90, 96.0) (0, 0), (4, 9.69) (4, 9.69) (7.64, 96.0), (8.6, 96.0) 4

5 . A baseball diamond is a square with side 90 ft. A batter at the home base hits the ball and runs toward first base with a speed of ft/se At what rate is his distance from third base increasing when he is halfway to first base? 4. Evaluate the integral. (( 5x 4)e5x 8x +4xe x )dx e 5x 8x +4e x x e 5x 8x +8e x e 5x 8x +4e x x e 5x 8x +4e x e 5x 8x +4e x 5. Evaluate the integral. 9.4 ft/sec 0.5 ft/sec 7.0 ft/sec 47.0 ft/sec 4.8 ft/sec. The volume of paint in a right cylindrical can is given by V =t t, where t is time in seconds and V is the volume in cm. How fast is the level rising when the height is cm? The can has a height of 6 cm and a radius of cm. [Hint: To get h as a function of t, first solve the volume V = πr h for h.] 7 cm/s cm/s 4 cm/s cm/s 5 cm/s ( x..)dx x. 9.9.x + C x. 9.9.x + C x. 9.9.x x C x x + C 6. Minimize S = x + y with xy = 6 and both x and y > 0. S = 6 S = 4 S = 8 S = 8 S = 6 5

6 7. A race car has a velocity of v(t) = 500( e.4t ) ft/s t seconds after starting. Use a Riemann sum with n = 0 to estimate how far the car travels in the first seconds. (Round your answer to the nearest whole number.),069 feet 980 feet 49 feet,7 feet,58 feet 8. Evaluate the integral. x +5 x 6 dx 4x 4 x 5 + C 4x 4 + x 5 + C 4x 4 + x 5 + C 4x 4 x 5 + C 5x 4 x 4 + C 9. A company finds that the number of new products it develops per year depends on the size if its annual R&D budget, x (in thousands of dollars), according to the following formul n =-4+0x +4x 0.x Find n"() You have been hired as a marketing consultant to Johannesburg Burger Supply, In, and you wish to come up with a unit price for its hamburgers in order to maximize weekly revenu To make life as simple as possible, you assume that the demand equation for Johannesburg hamburgers has the linear form q = mp + b where p is the price per hamburger, q is the demand in weekly sales, and m and b constants are certain constants you must determin Your market studies reveal the following sales figures: when the price is set at $5 per hamburger, the sales amount to,900 per week, but when the price is set at $7 per hamburger, the sales drop to zero. Now estimate the unit price in order to maximize annual revenue and predict what the weekly revenue will be at that pric p = $5.50, R = $,96.50 p = $6.50, R = $4,7.50 p = $.50, R = $7,76.50 p = $.50, R = $5, p = $4.50, R = $6,.50. Find the exact location of all the relative and absolute extrema of the function with domain ( 0, ). f (x) =x ln x 4 ( e,-4 ) - relative maximum e ( e,-4 ) - absolute minimum e ( 4 e,- ) - absolute minimum 4e ( e,- ) - relative minimum 4e ( e, 4 ) - absolute minimum e 6

7 . Minimize F = x + y with x +5y = 0. F = 65 F = 50 F = 5 F = 0 F = 650. The graph of the derivative of a function f is shown. Determine the x-coordinates of all stationary and singular points of f. (Assume that f (x) is defined and continuous everywhere in [-0, ].) 5. There are presently N = 500 cases of Bangkok flu, and the number is growing by 0 new cases every month. Rewrite the rate in mathematical notation. dn dt = 6.67 dn = dt 50 dn dt = 0 dn = 0 N = 0 6. The radius of a circular puddle is growing at a rate of 8 cm/s. How fast is the area growing at the instant when it equals 0 cm? 7 cm /s cm /s 6 cm /s 005 cm /s 7 cm /s 7. The graph of the derivative f ' (t) of f(t) is shown. Compute the total change of f(t) over the interval [,6]. x = - x = x = 0 x = - x = I want to fence in a rectangular vegetable patch. The fencing for the east and west sides costs $5 per foot, while the fencing for the north and south sides costs only $ per foot. I have a budget of $00 for the project. What is the largest area I can enclose? 00 square feet 5 square feet 0 square feet 6.5 square feet 50 square feet f(6) f() = f(6) f() = 6 f(6) f() = 4 f(6) f() = 8 f(6) f() = 7

8 8. A general quadratic demand function has the form to follow. q = ap + mp + t (a, m, t are constants, m 0) Obtain a formula for the elasticity of demand at a unit price p. E =- p + t ap +m E = ap + mp + t E = -mp p(ap + m) E =- ap + mp + t E = ap + mp + t 9. The graph of a function f(x) =x (x 4) is given. Find the coordinates of all points of inflection of this function (if any). (-, -80), (, -80) (-, 80), (, -80) (0, 0), (- 4, 0), ( 4, 0) (-, -44), (, -44) (0, 0) 8

9 40. The way Professor Waner drives, he burns gas at the rate of e t gallons each hour, t hours after a fill-up. Find the number of gallons of gas he burns in the first hours after a fill-up. Please round the answer to the nearest gallon. gallons 6 gallons gallons 7 gallons 4 gallons 4. The base of a 0-foot ladder is being pulled away from a wall at a rate of feet per secon How fast is the top of the ladder sliding down the wall at the instance when the base of the ladder is 8 feet from the wall? 9 ft/s 6 ft/s 6 ft/s ft/s 48 ft/s 4. Evaluate the integral..8 ex dx. -.7 (e -0. e -.7 ) (e -0. e -.7 ) ( e -0. e -.7 ) (e.8 e -.7 ) (e e -.7 ) 4. Evaluate the integral. x dx 8 8 log () 8 8 log () 8 8ln () 8 8ln() A packaging company is going to make closed boxes, with square bases, that hold 79 cubic centimeters. What are the dimensions of the box that can be built with the least material? cm cm 79 cm cm cm 45. Calculate the total area of the region bounded by the line y =9x +7, the x axis, and the lines x =5 and x = 4.,64 7, ,670 7,794 9

10 46. Calculate the left Riemann sums for the function over the given interval, using the given values of n. (When rounding, round answers to four decimal places.) f(x) =e x over [ 5,5], n =5 Left sum = 4.69 Left sum = 4.69 Left sum = Left sum =, Left sum = Decide on what substitution to use, and then evaluate the given integral using a substitution..5 5x +9dx.8( 5x + 9) + C.8( 5x + 9) + C.75 5x +9 + C.8 5x +9 + C.75 5x +9 + C 48. A general linear demand function has the form to follow. q = kp + a (k and a constants, with k 0) Obtain a formula for the elasticity of the demand at a unit price p. E = a + kp p E =- p + a E = kp a p E =- k + a kp E =- kp + a 49. Calculate d y dx. y =6x - + lnx 6 x 4 + x 6 x 4 x 6x 4 4 x x 6 x 5 x 50. The Chocolate Box Co. is going to make open-topped boxes out of 5'' 4'' rectangles of cardboard by cutting squares out of the corners and folding up the sides. What is the largest volume box it can make this way? 6 cubic inches 55 cubic inches 8 cubic inches 7 cubic inches 45 cubic inches 5. My aunt and I were approaching the same intersection, she from the south and I from the west. She was travelling at a steady speed of 0 mph, while I was approaching the intersection at 50 mph. At a certain instant in time, I was 5 of a mile from the intersection, while she was 0 of a mile from it. How fast were we approaching each other at that instant? 44 miles per hour 48 miles per hour 5 miles per hour 46 miles per hour 54 miles per hour 0

11 5. Evaluate the integral. 54. Evaluate the integral. 8 58ex dx x 4x 4dx 58 e 7 + e 58e 5 58 e 8 e 58 e 9 e 4 e 8 e 5. A model rocket has upward velocity v(t) = 60t ft/s, t seconds after launch. Use a Riemann sum with n = 0 to estimate how high the rocket is seconds after launch feet 40 feet 84.8 feet 6.8 feet 5. feet (4x ) (4x ) 4x 6 (4x ) 4x (4x ) 4x (4x ) 4x The graph of the second derivative, f ''(x), is given. Determine the x-coordinates of all points of inflection of f(x), if any. (Assume that f(x) is defined and continuous everywhere in [-0, 0].) x = 6., x = 6. x = 5., x = 0, x = 5. x = 5., x = 5. x = 0 x = 9, x = 0, x = 9

12 56. The consumer demand curve for tissues is given by q = (94 p) where p is the price per case of tissues and q is the demand in weekly sales. Determine the elasticity of demand E when the price is set at $. Round the answer to the nearest hundredth Sketch the graph of the function, labeling all relative and absolute extrema and points of inflection, and vertical and horizontal asymptotes. Check your graph using technology. 4 k(x) = 4 5 x (x ) 5

13 59. In 965 the economist F.M. Scherer modeled the number, n, of patens produced by a firm as a function of the size, s, of the firm (measured in annual sales in millions of dollars). He came up with the following equation based on a study of 448 large firms. n = s.85s +.s Find d n ds. s = 58. Find f (x) if f(0) = 0 and the tangent line at (x, f(x)) has the slope x ( x + 0) (x + 0) x (x + 0) (x + 0) x A fried chicken franchise finds that the demand equation for its new roast chicken product, "Roasted Rooster", is given by p = 56 q.5 where p is the price (in dollars) per quarter-chicken serving and q is the number of quarter-chicken servings that can be sold per hour at this pric Express q as a function of p, and find the elasticity of demand when the price is set at $4 per serving

14 6. Fair Weather Airlines will accept only bags for which the sum of the length and width is 7 inches, while the sum of length, height, and twice the width is 54 inches. What is the largest volume of the bag that it will accept?,458 cubic inches 79 cubic inches,645 cubic inches,87 cubic inches,96 cubic inches 6. The velocity of a particle moving in a straight line is given by v = t( t + 8) 4 +9t. Find an expression for the position s after time t. s( t )= ( t + 8) 5 0 s( t )= ( t + 8) 5 0 s( t )= ( t + 8) 5 5 s( t )= ( t + 8) 5 0 s( t )= t( t + 8) t + C +9t + C + 9t + C + 9t + C + 9t + C 6. Calculate the total area of the region bounded by the line y = 7x, the x axis, and the lines x = and x =. 67,469,498 4, Calculate the Riemann Sum for the integral using n = x dx Give the answer correct to two decimal places Evaluate the integral. 5 0x x dx 0 ( ln 6 - ln ) 5 ( ln + ln 6 ) 5 ( ln - ln 6 ) 0 ( ln + ln 6 ) 0 ( ln - ln 6 ) 66. Calculate d y dx. y = e -(x 4) x -(x 4) e (-x + 4) (x 4) e -x e x -(x 4) (x 4)e -x 4

15 67. Evaluate the integral. x( x + 9). dx ( x + 9). 8.4 ( x + 9) x( x +9) ( x + 9). 6.4 ( x + 9) Evaluate the integral. v v dv v ln( v )+C v ln v + C v- + 6 ln v + C v - 6 ln v + C v- + 6 ln( v )+C 69. A cylindrical bucket is being filled with paint at a rate of 8 cm /min. How fast is the level rising when the bucket starts to overflow? The bucket has a height of 40 cm and a radius of 0 cm cm/min 0.75 cm/min 0.09 cm/min 0.59 cm/min 0.05 cm/min 70. Calculate the Riemann Sum for the integral using n = 5. 0 e x dx -0,64.8, ,749.69, The position s of a point (in feet) is given as a function of time t (in seconds). s = 64 t +5t ; t =4 Find its acceleration as a function of t. Find its acceleration at the specified tim 6 - t t + 0t ft/sec, 8 ft/sec - t t + 5t ft/sec, 8 ft/sec -6t t + 0t ft/sec, 8 ft/sec - t + 0t ft/sec, 0 ft/sec - 6 t + 5t ft/sec, 0 ft/sec 5

16 7. Find f (x) if f(4) = and the tangent line at (x, f(x)) has slope x. f (x) = x +7 f (x) = x +7 f (x) = x C f (x) = x 7 f (x) = x + C 6

17 MAT 0 TEST REVIEW (Ch and ) Answer Section MULTIPLE CHOICE. ANS: D PTS:. ANS: D PTS:. ANS: E PTS: 4. ANS: E PTS: 5. ANS: B PTS: 6. ANS: C PTS: 7. ANS: B PTS: 8. ANS: C PTS: 9. ANS: A PTS: 0. ANS: C PTS:. ANS: C PTS:. ANS: D PTS:. ANS: D PTS: 4. ANS: B PTS: 5. ANS: A PTS: 6. ANS: C PTS: 7. ANS: A PTS: 8. ANS: E PTS: 9. ANS: B PTS: 0. ANS: A PTS:. ANS: C PTS:. ANS: A PTS:. ANS: D PTS: 4. ANS: E PTS: 5. ANS: A PTS: 6. ANS: C PTS: 7. ANS: A PTS: 8. ANS: A PTS: 9. ANS: A PTS: 0. ANS: C PTS:. ANS: B PTS:. ANS: E PTS:. ANS: E PTS: 4. ANS: D PTS: 5. ANS: C PTS: 6. ANS: E PTS: 7. ANS: C PTS: 8. ANS: D PTS: 9. ANS: A PTS:

18 40. ANS: C PTS: 4. ANS: A PTS: 4. ANS: A PTS: 4. ANS: C PTS: 44. ANS: D PTS: 45. ANS: B PTS: 46. ANS: B PTS: 47. ANS: A PTS: 48. ANS: E PTS: 49. ANS: B PTS: 50. ANS: A PTS: 5. ANS: D PTS: 5. ANS: C PTS: 5. ANS: D PTS: 54. ANS: B PTS: 55. ANS: E PTS: 56. ANS: E PTS: 57. ANS: C PTS: 58. ANS: C PTS: 59. ANS: D PTS: 60. ANS: E PTS: 6. ANS: E PTS: 6. ANS: D PTS: 6. ANS: D PTS: 64. ANS: D PTS: 65. ANS: E PTS: 66. ANS: A PTS: 67. ANS: B PTS: 68. ANS: C PTS: 69. ANS: E PTS: 70. ANS: D PTS: 7. ANS: A PTS: 7. ANS: D PTS: