3.1 Maxima and Minima

Transcription

1 Ch. 3 Applications of the Derivative 3.1 Maima and Minima 1 Find Critical Values/Ma/Min from Graph MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find all critical points and find the minimum and maimum value of the function on the given domain. 1) Domain: [-, 3] A) Critical points: -, -,, 3; maimum value: 9; minimum value: 5 B) Critical points: -, ; maimum value: 9; minimum value: 5 C) Critical points: 5, 7, 9; maimum value: 9; minimum value: 5 8 D) Critical points: -, -,, 3; maimum value: -; minimum value: -, ) Domain: [-, 3] A) Critical points: -, -1, 0, 1,, 3; maimum value: 3; minimum value: 1 B) Critical points: -1, 0, 1, ; maimum value: 3; minimum value: 1 C) Critical points: 1, 3; maimum value: -1, 1, 3; minimum value: -, 0, D) Critical points: - 3, - 1, 1, 3, 5 ; maimum value: 3; minimum value: 1 Page 1

2 3) Domain: [-3, 3] A) Critical points: -3,, 3; maimum value: 3; minimum value: B) Critical points: ; maimum value: none; minimum value: none C) Critical points: -3, 3; maimum value: 3; minimum value: D) Critical points: , 3; maimum value: 3; minimum value: -3 ) Domain: [-, ] A) Critical points: -, 0,, ; maimum value: 7; minimum value: 0 B) Critical points: 0, ; maimum value: ; minimum value: 0 C) Critical points: -, ; maimum value: 7; minimum value: 0 D) Critical points: 0,, 7; maimum value: ; minimum value: -, Page

3 Find Critical Values/Ma/Min from Equation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identrif the critical points and find the maimum and minimum value on the given interval I. 1) f() = ; I = [-18, 0] A) Critical points: -18, -9, 0; maimum value 81; minimum value 0 B) Critical points: -9; maimum value 18; minimum value 0 C) Critical points: -18, 0, 9; maimum value 81; minimum value 0 D) Critical points: -18, 0, 81; minimum value 0 ) f() = + ; I = - 3, 1 A) Critical points: - 3, - 1, 1 ; maimum value 5 ; minimum value - 1 B) Critical points: - 1; minimum value - 1 C) Critical points: 0, ; maimum value 8; minimum value 0 D) Critical points: - 3, - 1, 1 ; maimum value - 3 ; minimum value - 1 3) f() = ; I =(-3, 5) A) Critical points: -, ; no maimum value; minimum value -1 B) Critical points: -, ; maimum value 0; minimum value -1 C) Critical points: -3, -,, 5; maimum value 9; minimum value -1 D) Critical points: -3, -,, 5; maimum value 9; minimum value 13 ) f() = ; I =[-3, 5] A) Critical points: -3, -,, 5; maimum value ; minimum value -15 B) Critical points: -, ; maimum value 17; minimum value -15 C) Critical points: -, ; no maimum value; minimum value -15 D) Critical points: -3, -,, 5; maimum value ; minimum value ) f(r) = ; I =[-3, ] r + A) Critical points: -3, 0, ; maimum value 1 ; minimum value 1 38 B) Critical points: -3, ; maimum value 1 11 ; minimum value 1 38 C) Critical points: -3, 0, ; maimum value 1 ; minimum value 1 11 D) Critical points: 0; maimum value 1 ; minimum value 0 Page 3

4 ) r(θ) = cos θ; I = - π, π 3 A) Critical points: - π, 0, π ; maimum value ; minimum value 1 3 B) Critical points: - π, 0, π ; maimum value ; minimum value 3 C) Critical points: 0; maimum value ; no minimum value D) Critical points: - π,, π ; maimum value ; minimum value 1 3 7) g() = - 7 ; I = [5, 10] A) Critical points: 5, 7, 10; maimum value 3; minimum value 0 B) Critical points: 5, 7, 10; maimum value 17; minimum value -7 C) Critical points: 7; no maimum value; minimum value 0 D) Critical points: 5, 10; maimum value 3; minimum value 8) g(t) = t/3; I = [-1, 8] A) Critical points: -1, 0, 8; maimum value ; minimum value 0 B) Critical points: -1, 0, 8; maimum value 1; minimum value 0 C) Critical points: 0; no maimum value; minimum value 0 D) Critical points: -1, 8; maimum value ; minimum value 3 Page

5 3 Sketch Graph of Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch the graph of a function with the given properties. 1) f is differentiable, has domain [, 8], reaches a maimum of 8 (attained when = 8) and a minimum of -1 (attained when = ). Additionall, = and = are stationar points A) B) C) D) Page 5

6 ) f is continuous but not necessaril differentiable, has domain [1, 5.5], reaches a maimum of 8 (attained when = ) and a minimum of (attained when = ). Additionall, = and = 5 are the onl stationar points A) B) C)

7 3. Monotonicit and Concavit 1 Find Monotonic Intervals Given f() MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Montonicit Theorem to find where the function is increasing and where it is decreasing. 1) f() = 7-5 A) Increasing on (-, ) B) Decreasing on (-, ) C) Increasing on -, 5 7, decreasing on 5, D) Increasing on (-, -5], decreasing on [-5, ) 7 ) f() = 1-1 A) Increasing on [1, ), decreasing on (-, 1] B) Increasing on (-, -1], decreasing on [-1, ) C) Increasing on [-1, 1], decreasing on (-, -1] [1, ) D) Increasing on (-, ) 3) g() = A) Increasing on [1, ), decreasing on (-, 1] B) Increasing on [0, ), decreasing on (-, 0] C) Increasing on (-, 1], decreasing on [1, ) D) Increasing on (-, ) ) f() = ( + )( - ) A) Increasing on [, ), decreasing on (-, ] B) Decreasing on (-, ) C) Increasing on (-, -] [, ), decreasing on [-, ] D) Increasing on [-1, ), decreasing on (-, -1] 1 5) h(t) = t + 1 A) Increasing on (-, 0], decreasing on [0, ) B) Increasing on [0, ), decreasing on (-, 0] C) Increasing on (-, 1], decreasing on [1, ) D) Increasing on (-, ) ) G() = A) Increasing on (-, 0], decreasing on [0, ) B) Increasing on [0, ), decreasing on (-, 0] C) Increasing on (-, 7], decreasing on [7, ) D) Increasing on (-, ) Page 7

8 7) f() = 3 - A) Increasing on -, ,, decreasing on - 3 3, 3 3 B) Increasing on 3 3,, decreasing on -, C) Increasing on -, - 3 3, decreasing on 3 3, D) Decreasing on (-, ) 8) h(z) = 108z - z3 A) Increasing on [-, ], decreasing on (-, -] [, ) B) Increasing on (-, -] [, ), decreasing on [-, ] C) Increasing on (-, ], decreasing on [, ) D) Increasing on [-3, 3], decreasing on (-, -3] [3, ) 9) h(t) = cos t, 0 t π A) Increasing on [π, π], decreasing on [0, π] B) Increasing on π, 3π, decreasing on 0, π 3π, π C) Increasing on [0, π], decreasing on [π, π] D) Increasing on [1, ], decreasing on [0, 1] Find Intervals of Concavit and Inflection Points MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Concavit Theorem to determine where the given function is concave up and where it is concave down. Also find all inflection points. 1) f() = A) Concave up for all ; no inflection points B) Concave down for all ; no inflection points C) Concave up on (8, ), concave down on (-, 8); inflection point (8, 5) D) Concave up on (-, 8), concave down on (8, ); inflection point (8, 5) ) G(w) = w + 1w + 15 A) Concave down for all w; no inflection points B) Concave up for all w; no inflection points C) Concave up on (-, ), concave down on (-, -); inflection point (-, -1) D) Concave up on (-, -), concave down on (-, ); inflection point (-, -1) Page 8

9 3) q() = A) Concave up on (0, ), concave down on (-, 0); inflection point (0, 8) B) Concave up on (-, 0), concave down on (0, ); inflection point (0, 8) C) Concave down for all ; no inflection points D) Concave up for all ; no inflection points ) T(t) = t - t3 A) Concave up on (-, 0), concave down on (0, ); inflection point (0, 0) B) Concave up on (0, ), concave down on (-, 0); inflection point (0, 0) C) Concave down for all t, no points of inflection D) Concave up on (-, 0) (1, ), concave down on (0, 1); inflection points (0, 0), (1, ) 5) f() = A) Concave up on (-, ), concave down on (-, -); inflection point (-, 108) B) Concave up on (-, -), concave down on (-, ); inflection point (-

10 3 Sketch Graph Given Function MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine where the graph of the function is increasing, decreasing, concave up, concave down. Then sketch the graph. 1) f() = A) B) C) D) Page 10

11 1 ) g() = + 9 A)

12 3) F() = A) B) C) D)

13 ) H() = 1/3( - 5) A) B) C) D)

14 5) G() = A) B) C) D) Page 1

15 ) f() = A) B) C) D) Page 15

16 7) g() = A) B) C) D) Page 1

17 8) f() = A) B) C) D) ) h() = Page 17

18 A) B) C) D) Page 18

19 10) f() = A) B) C) D) Page 19

20 Sketch Graph Given Characteristics MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch the graph of a continuous function f on the given domain that satisfies all conditions. 1) f has domain [0, ]; f(0) = 0; f() = ; increasing and concave down on (0, ) A) B) C) D) Page 0

21 ) f has domain [-, ]; f(-) = 0; f() = ; fʹ() < 0 on (-, ); fʹ() > 0 on (-, -) (, ) fʹʹ() < 0 on (-, 0); fʹʹ() > 0 on (0, ) A) B) C) D) Page 1

22 5 Tech: Analze Graph MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use a graphing calculator to solve the problem. 1) Let f() = sin + cos on the interval I = (0, 7). Use the graph to estimate where fʹ() < 0 on I. A) (0.51, 3.1) (5.77, 7) B) (0, 0.51) (3.1, 5.77) C) (0.51, 3.1) D) (0, 5.77) ) Let f() = sin + cos on the interval I = (0, 7). Use the graph to estimate where fʹʹ() < 0 on I. A) (0, 1.77) (.5, 7) B) (0, 3.1) (3.1, 7) C) (1.77,.5) D) (0, 5.77) Solve Apps: Monotonicit and Concavit MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Translate into the language of a derivative of distance with respect to time. s is the position of the car at time t. The speed of the car is porportional to the distance it has traveled. A) ds = ks, k is a constant dt ds B) = k, k is a constant dt C) ds = kt, k is a constant dt D) d s dt = ks, k is a constant ) Translate into the language of a derivative of distance with respect to time. s is the position of the car at time t. The car is slowing down. A) d s dt < 0 C) d s dt = k, k is a constant B) d s dt > 0 ds D) dt = 0 3) Translate into the language of a derivative of the number of businesses with respect to time. N is the number of businesses in the downtown district after time t. The number of businesses downtown is decreasing at a slower and slower rate. A) dn dt < 0, d N dt > 0 C) dn dt > 0, d N dt < 0 dn B) < 0, d N dt dt < 0 dn D) < 0, d N = k, k is a constant dt dt Page

23 ) Water is poured into a empt funnel like barrel at a stead constant rate. The diameter of the barrel is 1 inches on the bottom and inches at the top. The height h of the water h is a function of time t. What can ou sa about d h dt? A) d h dt < 0 C) d h dt = 0 B) d h dt > 0 D) d h dt = k, k is a constant 5) Water is poured into a empt funnel like barrel at a stead constant rate. The diameter of the barrel is inches on the bottom and 1 inches at the top. The height h of the water h is a function of time t. What can ou sa about d h dt? A) d h dt > 0 C) d h dt = 0 B) d h dt < 0 D) d h dt = k, k is a constant 7 *Know Concepts: Monotonicit and Concavit SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the problem. 1) If f() is a differentiable function and fʹ(c) = 0 at an interior point c of fʹs domain, and if fʹʹ() > 0 for all in the domain, must f have a local minimum at = c? Eplain. ) Sketch a continuous curve = f() with the following properties: f() = 3; fʹʹ() > 0 for > ; and fʹʹ() < 0 for <. 3) Can anthing be said about the graph of a function = f() that has a second derivative that is alwas equal to zero? Give reasons for our answer. ) What can ou sa about the inflection points of the quartic curve = a + b3 + c + d + e, a 0? Give reasons for our answer. 5) For > 0, sketch a curve = f() that has f(1) = 0 and fʹ() = - 1. Can anthing be said about the concavit of such a curve? Give reasons for our answer. Page 3

24 ) The accompaning figure shows a portion of the graph of a function that is twice-differentiable at all ecept at = p. At each of the labeled points, classif ʹ and ʹʹ as positive, negative, or zero. 3.3 Local Etrema and Etrema on Open Intervals 1 Find Critical Points/Ma/Min MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Identif the critical points. Then use the test of our choice to decide which critical points give a local maimum value and which give a local minimum value. Give these values. 1) f() = A) Critical points: 0, ; local maimum f(0) = 9; local minimum f() = -99 B) Critical point: 0; local maimum f(0) = 9 C) Critical points: 0, 3; local maimum f(0) = 9; local minimum f() = -5 D) Critical points: -3, 3; local maimum f(-3) = 117; local minimum f(3) = -5 ) f(θ) = cos θ, 0 < θ < π A) No critical points; no local maima or minima on the interval 0, π B) Critical points: 0, π ; local maimum f(0) = 1; local minimum f π = 0 C) Critical point: π 8 ; local maimum f π 8

25 ) h() = - A) Critical point: - 3 ; local minimum f - 3 B) Critical point: - 3 ; local minimum f - 3 = = C) No critical points; no local minima or maima D) Critical point: 0; local minimum f(0) = 0 5) f() = ( - )3 A) Critical point: ; no local minima or maima B) Critical points:, 3; local minimum f() = 0; Local maimum f(3) = 1 C) No critical points; no local minima or maima D) Critical points:, 3; local minimum f() = 0 Find Global Ma/Min Values MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find, if possible, the (global) maimum and minimum values of the given function on the indicated interval. 1) f() = - on [-3, ] A) Maimum value f() = ; minimum value f(-3) = - 5 B) Maimum value f() = ; minimum value f(-3) = - 1 C) Maimum value f(-3) = ; minimum value f() = - 1 D) Maimum value f(-) = ; minimum value f(3) = - 5 ) g() = on [, 7] A) Maimum value g 13 = 1 ; minimum value g(7) = g() = 0 B) Maimum value g 15 = 1 ; minimum value g(7) = g() = 0 C) Maimum value g 13 = 337 ; minimum value g(7) = g() = 0 D) Maimum value g 15 = 5 ; minimum value g(7) = g() = 0 3) f() = sin + π on 0, 7π A) Maimum value f(0) = 1; minimum value f(π) = -1 B) Maimum value f 5π = 1; minimum value f π = -1 C) Maimum value f π = 1; minimum value f 5π = -1 D) Maimum value f 7π = 1; minimum value f 5π = -1 Page 5

26 ) h() = csc on - π, 3π A) No maimum valuet; no minimum value B) Maimum value h(π) = -1; minimum value h(0) = 1 C) Maimum value h(π) = 1; minimum value h(π) = -1 D) Maimum value h(-π) = 0; minimum value h(π) = -1 5) F() = - on 1, 3 A) Maimum value F(3) = - 9 ; minimum value F 1 = -8 B) Maimum value F(3) = - 9 ; minimum value F - 1 = -8 C) Maimum value F 1 = - ; minimum value F(-3) = -8 9 D) Maimum value F 1 = ; minimum value F(3) = -8 9 ) F() = 3 on [-, 7] A) Maimum value F(7) = 3; minimum value F( 3 -) =0 B) Maimum value F(-7) = 3; minimum value F(0) =0 C) Maimum value F(7) = 3; minimum value F(-7) = -3 D) Maimum value F(0) =0; minimum value F(7) = 3 7) g() = 1 + on [-3, 3] A) Maimum value g(3) = ; minimum value g(-3) = B) Maimum value g(3) = ; minimum value g(-3) = C) Maimum value g(-3) = ; minimum value g(3) = D) Maimum value g(-3) = ; minimum value g(3) = 8) H() = - 7 on [-, 3] A) Maimum value H(0) = ; minimum value H(3) = -57 B) Maimum value H(0) = ; minimum value H(-) = - C) Maimum value H(0) = 1; minimum value H(3) = - D) Maimum value H(0) = 7; minimum value H(3) = -9 Page

27 9) h(t) = cos t - π 3 on 0, 7π A) Maimum value h π 3 = 1; minimum value h π 3 B) Maimum value h - π 3 = 1; minimum value h π 3 C) Maimum value h π 3 = 1; minimum value h - π 3 = -1 = -1 = -1 D) Maimum value h - π 3 = 1; minimum value h - π 3 = -1 3 Find Critical Points Given f ʹ() MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The first derivative f ʹ is given. Find all values of that make the function a local minimum and a local maimum. 1) fʹ() = ( + 8)( + ) A) Local minimum at = -; local maimum at = -8 B) Local minimum at = ; local maimum at = 8 C) Local minimum at = - and 0; local maimum at = -8 D) No local etrema ) fʹ() = ( - 7)( + 8) A) Local minimum at = -8 B) Local minimum at = -8; local maimum at = 7 C) Local maimum at = 7 D) Local minimum at = -8; local maimum at = -7 and 7 *Sketch Graph Given Characteristics SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch a graph of a function with the given properties. If it is impossible indicate this and justif our answer. 1) f is differentiable, has domain [-, ], and has two local maima and two local minima on (-, ) Page 7

28 ) f is continuous, but not necessaril differentiable, has domain [-, ], and has two local maima and one local minimum on (-, ) ) f has domain [-3, 5], but is not necessaril continuous, and has two local maima and no local minimum on (-3, 5) Practical Problems 1 Solve Apps: Optimization I MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) Find two numbers whose product is -9 and the sum of whose squares is a minimum. A) 3 and -3 B) 3 and - C) 5 and D) 3 and - 3 Page 8

29 ) How close does the curve = come to the point, 0? (Hint: If ou minimize the square of the distance, 3 ou can avoid square roots.) A) The distance is minimized when = 1 ; the minimum distance is 5 1 units. B) The distance is minimized when = 1 3 ; the minimum distance is 3 units. C) The distance is minimized when = ; the minimum distance is 3 units. D) The distance is minimized when = ; the minimum distance is 5 1 units. 3) A carpenter is building a rectangular room with a fied perimeter of 100 feet. What are the dimensions of the largest room that can be built? What is its area? A) 5 ft b 5 ft; 5 ft B) 50 ft b 50 ft; 500 ft C) 5 ft b 75 ft; 1875 ft D) 10 ft b 90ft; 900 ft ) A piece of molding 18 centimeters long is to be cut to form a rectangular picture frame. What dimensions will enclose the largest area? Round to the nearest hundredth, if necessar. A) cm b cm B) 1.9 cm b 1.9 cm C) 33. cm b 33. cm D) 1.9 cm b cm 5) A compan wishes to manufacture a bo with a volume of cubic feet that is open on top and is twice as long as it is wide. Find the width of the bo that can be produced using the minimum amount of material. Round to the nearest tenth, if necessar. A). ft B) 5. ft C) 3. ft D). ft ) From a thin piece of cardboard 50 inches b 50 inches, square corners are cut out so that the sides can be folded up to make a bo. What dimensions will ield a bo of maimum volume? What is the maimum volume? Round to the nearest tenth, if necessar. A) 33.3 in. b 33.3 in. b 8.3 in.; in.3 B) 1.7 in. b 1.7 in. b 1.7 in.; 9. in.3 C) 33.3 in. b 33.3 in. b 1.7 in.; 18,518.5 in.3 D) 5 in. b 5 in. b 1.5 in.; in.3 7) A compan is constructing an open-top, square-based, rectangular metal tank that will have a volume of 53 cubic feet. What dimensions ield the minimum surface area? Round to the nearest tenth, if necessar. A).7 ft b.7 ft. b. ft B) 3.8 ft b 3.8 ft. b 3.8 ft C) 10.3 ft b 10.3 ft. b 0.5 ft D) 5. ft b 5. ft. b 1.8 ft 8) A 3-inch piece of string is cut into two pieces. One piece is used to form a circle and the other to form a square. How should the string be cut so that the sum of the areas is a minimum? Round to the nearest tenth, if necessar. A) Square piece = 1.9 in., circle piece = 10.1 in. B) Circle piece = 1.9 in., square piece = 10.1 in. C) Square piece = 0 in., circle piece = 3 in. D) Square piece = 5.8 in., circle piece = 5.3 in. Page 9

30 9) A private shipping compan will accept a bo for domestic shipment onl if the sum of its length and girth (distance around) does not eceed 11 inches. What dimensions will give a bo with a square end the largest possible volume? A) 19 in. b 19 in. b 38 in. B) 38 in. b 38 in. b 38 in. C) 19 in. b 38 in. b 38 in. D) 19 in. b 19 in. b 95 in. 10) A farmer decides to make three identical pens with 88 feet of fence. The pens will be net to each other sharing a fence and will be up against a barn. The barn side needs no fence. What dimensions for the total enclosure (rectangle including all pens) will make the area as large as possible? A) 11 ft b ft B) 11 ft b 11 ft C) ft b ft D) 1.7 ft b ft Solve Apps: Optimization II MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A rectangular field is to be enclosed on four sides with a fence. Fencing costs $5 per foot for two opposite sides, and $ per foot for the other two sides. Find the dimensions of the field of area 850 square feet that would be the cheapest to enclose. A) 18. ft at $5 b.1 ft at $ B).1 ft at $5 b 18. ft at $ C) 7.9 ft at $5 b 11.7 ft at $ D) 11.7 ft at $5 b 7.9 ft at $ Page 30

31 ) The velocit of a particle, in feet per second, is given b v = t - t + 7, where t is the time (in seconds) for which it has traveled. Find the time at which the velocit is at a minimum. A) 1 sec B) 3.5 sec C) sec D) 7 sec 3) Supertankers off-load oil at a docking facilit shore point 5 miles offshore. The nearest refiner is 9 miles east of the docking facilit. A pipeline must be constructed connecting the docking facilit with the refiner. The pipeline costs $300,000 per mile if constructed underwater and $00,000 per mile if over land. 5 mi 9 mi Locate point B to minimize the cost of construction. A) Point B is.7 miles from Point A. B) Point B is.50 miles from Point A. C) Point B is 3.51 miles from Point A. D) Point B is 5. miles from Point A. ) A highwa must be constructed to connect Village A with Village B. There is a rudimentar roadwa that can be upgraded 50 miles south of the line connecting the two villages. The cost of upgrading the eisting roadwa is $300,000 per mile, whereas the cost of constructing a new highwa is $500,000 per mile. Find the combination of upgrading and new construction that minimizes the cost of connecting the two villages. Note that due to geographical restrictions it is not possible to connect the villages in a 150 mile straight line. A) $85,000,000 B) $8,83,0 C) $83,71,95 D) $87,83,0 Page 31

32 5) A highwa must be constructed to connect town A with town B. There is an eisting roadwa that can be upgraded 5 miles north of the line connecting the two towns. The cost of upgrading the eisting roadwa is $00,000 per mile, whereas the cost of constructing new a new highwa is $300,000 per mile. Find the combination of upgrading and new construction that minimizes the cost of connecting the two towns. Clearl define the location of the new highwa. A) Construct 7.08 miles of new road and upgrade miles of eisting road. From each of the two towns, construct 33.5 miles of new road (on a diagonal) to the old road. Upgrade the middle miles of eisting road. B) From town A, construct 5 miles of new road to the old road. Upgrade the old road for 00 miles, then construct 5 miles directl to town B. C) Construct a new road (00 miles) directl from town A to town B. D) Construct new road from town A to the 100 mile mark on the old road. From that point, construct new road directl to town B. ) A trough is to be made with an end of the dimensions shown. The length of the trough is to be feet long. Onl the angle θ can be varied. What value of θ will maimize the troughʹs volume? θ θ A) 30 B) 5 C) 8 D) 3 Page 3

33 7) A rectangular sheet of perimeter 7 centimeters and dimensions centimeters b centimeters is to be rolled into a clinder as shown in the figure. What values of and give the largest volume? A) = 9 cm; = 9 cm B) = 10 cm; = 7 cm C) = 11 cm; = 5 cm D) = 8 cm; = 11 cm 8) A long strip of sheet metal 1 inches wide is to be made into a small trough b turning up two sides at right angles to the base. If the trough is to have maimum capacit, how man inches should be turned up on each side? A) 3 in. B) in. C) in. D) 5 in. 9) A fence 8 feet high runs parallel to a tall building and 7 feet from it. Find the length of the shortest ladder that will reach from the ground across the top of the fence to the wall of the building. 8 feet 7 feet A).9 ft B) 35 ft C) 7.9 ft D) 5.9 ft Page 33

34 SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. 10) You are planning to close off a corner of the first quadrant with a line segment 19 units long running from (, 0) to (0, ). Show that the area of the triangle enclosed b the segment is largest when =. 3 Solve Apps: Optimization III MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A window is in the form of a rectangle surmounted b a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits onl one-third as much light per unit area as clear glass does. The total perimeter is fied. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame. A) width height = 1 + π B) width height = π C) width height = 3 + π D) width height = 1 + π Page 3

35 ) The strength S of a rectangular wooden beam is proportional to its width times the square of its depth. Find the dimensions of the strongest beam that can be cut from a 13-inch-diameter clindrical log. (Round answers to the nearest tenth.) 13ʺ A) w = 7.5; d = 10. B) w = 8.5; d = 9. C) w =.5; d = 11. D) w = 8.5; d = 11. 3) The stiffness of a rectangular beam is proportional to its width times the cube of its depth. Find the dimensions of the stiffest beam than can be cut from a 13-inch-diameter clindrical log. (Round answers to the nearest tenth.) 13ʺ A) w =.5; d = 11.3 B) w = 7.5; d = 1.3 C) w = 5.5; d = 1.3 D) w = 7.5; d = 10.3 ) At noon, ship A was 1 nautical miles due north of ship B. Ship A was sailing south at 1 knots (nautical miles per hour; a nautical mile is 000 ards) and continued to do so all da. Ship B was sailing east at 7 knots and continued to do so all da. The visibilit was 5 nautical miles. Did the ships ever sight each other? A) No. The closest the ever got to each other was.3 nautical miles. B) Yes. The were within nautical miles of each other. C) No. The closest the ever got to each other was 7.3 nautical miles. D) Yes. The were within 3 nautical miles of each other. 5) The altitude h, in feet, of a jet that goes into a dive and then again turns upward is given b h = 1t3-198t , where t is the time, in seconds, of the dive and turn. What is the altitude of the jet when it turns up out of the dive? A) 151 ft B) 1700 ft C) 10 ft D) 1588 ft Page 35

36 ) A small frictionless cart, attached to the wall b a spring, is pulled 10 centimeters back from its rest position and released at time t = 0 to roll back and forth for seconds. Its position at time t is s = 1-10 cos πt. What is the cartʹs maimum speed? When is the cart moving that fast? What is the magnitude of of the acceleration then? A) 10π 31. cm/sec; t = 0.5 sec, 1.5 sec,.5 sec, 3.5 sec; acceleration is 0 cm/sec B) 10π 31. cm/sec; t = 0 sec, 1 sec, sec, 3 sec; acceleration is 0 cm/sec C) 10π 31. cm/sec; t = 0.5 sec,.5 sec; acceleration is 1 cm/sec D) π 3.1 cm/sec; t = 0.5 sec, 1.5 sec,.5 sec, 3.5 sec; acceleration is 0 cm/sec Solve Apps: Least Squares MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) A compan builds pole buildings. The onl construct the outer shell and hire subcontracters for an interior framing and finishing. The total number of hours of labor required to build the outer shell and the area in square feet of the building is given for 5 buildings. Building Size (sq. ft) Labor (hrs) Find the least squares line through the origin. A) = B) = C) = D) = ) A compan builds pole buildings. The onl construct the outer shell and hire subcontracters for an interior framing and finishing. The total number of hours of labor required to build the outer shell and the area in square feet of the building is given for 5 buildings. Building Size (sq. ft) Labor (hrs) Find the least squares line through the origin and use it to estimate the hours of labor required for 1500 square foot building. A) 13 hours B) 11 hours C) 177 hours D) 159 hours Page 3

37 5 Solve Apps: Cost/Revenue MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) If the price charged for a cand bar is p() cents, then thousand cand bars will be sold in a certain cit, where p() = How man cand bars must be sold to maimize revenue? 1 A) 791 thousand cand bars B) 791 cand bars C) 158 thousand cand bars D) 158 cand bars ) If the price charged for a bolt is p cents, then thousand bolts will be sold in a certain hardware store, where p = 9 -. How man bolts must be sold to maimize revenue? 3 A) 178 thousand bolts B) 178 bolts C) 35 thousand bolts D) 35 bolts 3) The price P of a certain computer sstem decreases immediatel after its introduction and then increases. If the price P is estimated b the formula P = 130t - 500t + 500, where t is the time in months from its introduction, find the time until the minimum price is reached. A) 9. months B) 19. months C) 1.5 months D) 38.5 months ) The cost of a computer sstem increases with increased processor speeds. The cost C of a sstem as a function of processor speed is estimated as C = 7S - S , where S is the processor speed in MHz. Find the processor speed for which cost is at a minimum. A) 0. MHz B) 8. MHz C) 0.3 MHz D) 3. MHz 5) Find the optimum number of batches (to the nearest whole number) of an item that should be produced annuall (in order to minimize cost) if 70,000 units are to be made, it costs $3 to store a unit for one ear, and it costs $30 to set up the factor to produce each batch. A) 3 batches B) batches C) 3 batches D) batches ) Find the number of units that must be produced and sold in order to ield the maimum profit, given the following equations for revenue and cost: R() = C() = + 3. A) units B) 7 units C) 5 units D) 9 units 7) Find the number of units that must be produced and sold in order to ield the maimum profit, given the following equations for revenue and cost: R() = 5 C() = A) 1850 units B) 3700 units C) 3150 units D) 300 units Page 37

38 8) Suppose c() = ,000 is the cost of manufacturing items. Find a production level that will minimize the average cost of making items. A) 11 items B) 1 items C) 10 items D) 13 items 9) Suppose a business can sell gadgets for p = dollars apiece, and it costs the business c() = dollars to produce the gadgets. Determine the production level and the sale price per gadget required to maimize profit. A) 11,50 gadgets at $ each B) 111 gadgets at $8.89 each C) 10,000 gadgets at $ each D) 13,750 gadgets at $11.50 each 10) A baseball team is tring to determine what price to charge for tickets. At a price of $10 per ticket, it averages 35,000 people per game. For ever increase of $1, it loses 5,000 people. Ever person at the game spends an average of $5 on concessions. What price per ticket should be charged in order to maimize revenue? A) $.00 B) $.00 C) $1.00 D) $.00 Page 38

39 3.5 Graphing Functions Using Calculus 1 Sketch Graph MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Make an analsis using calculus and sketch the graph. 1) f() = A) B) C) D) Page 39

40 ) g() = + 1 A) B) C) D) Page 0

41 3) h() = A) B) C) D) Page 1

42 ) P() = 1/3( - 3) A) B) C) D) Page

43 5) F() = + 13 A) B) C) D) Page 3

44 ) f() = + cos, 0 π A) B) C) D) Page

45 7) g() = A) B) C) D)

46 8) h() = A) B) C) D) Page

47 ( - 1)( - ) 9) f() = ( + )( - ) A) B) C) D) Page 7

48 10) f() = cos A) B) C) D) Page 8

49 *Sketch Graph Given Characteristics SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Sketch a graph of a function f that has the given properties. 1) (a) f() = 3 (b) fʹʹ() > 0 for > (c) fʹʹ() < 0 for <

50 3) (a) Defined for all real numbers (b) Increasing for -3 < < 3 (c) Decreasing for - < < -3 and 3 < < (d) Concave downward for 0 < < (e) Concave upward for - < < 0 (f) fʹ(-3) = fʹ(3) = 0 (g) Inflection point at (0, 0) ) (a) Continuous everwhere ecept for a vertical asmptote at = 0 (b) fʹ() < 0 for - < < 0 (c) fʹ() < 0 for 0 < < (d) fʹʹ() < 0 for all Page 50

51 3 Tech: Linearization MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the linear approimation to the curve at the indicated point. Graph both functions. 1) = ( - )5 + at inflection point A) = B) = C) = ( - ) + D) = Page 51

52 Sketch Graph Given Derivative MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Sketch a possible graph of f() using fʹ(). 1) fʹ() = (5 - ) and f(0) = 0 A)

53 ) fʹ() = -/3( - ) and f(0) = 0 A) B) C) D) Page 53

54 3) fʹ() = ( - 1)( - 3) and f(0) = 1 A)

55 5 Sketch Graph Given Graph of Derivative MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. The graphs of the first and second derivatives of a function = f() are given. Select a possible graph of f that passes through the point P. (NOTE: Vertical scales ma var from graph to graph.) 1) fʹ fʹʹ P P A) B) C) D) Page 55

56 ) fʹ fʹʹ P

57 3) fʹ fʹʹ P P A) B) C) D) Page 57

58 ) fʹ fʹʹ P P A) B) C) D)

59 5) fʹ fʹʹ P P A) B) C)

60 ) Is it possible for a twice differentiable function to satisf the following properties. Justif our answer. Fʹ() < 0, while Fʹʹ() > 0 3. The Mean Value Theorem for Derivatives 1 Determine if Mean Value Theorem Applies MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Decide whether the Mean Value Theorem applies to the given function on the given interval. 1) f() = 1/3; [-1, ] A) Yes B) No ) g() = 3/; [0, 3] A) Yes B) No 3) h(t) = t( - t); [-1, 5] ) f(θ) = A) Yes B) No cos θ θ, -π θ < 0 0, θ = 0 A) Yes B) No Find c in fʹ(c) =(f(b) - f(a))/(b - a) MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Mean Value Theorem and find all possible values of c on the given interval. 1) f() = ; [-, -1] A) c = - 3 = -1.5 B) c = ± 3 = ±1.5 C) c = 0, - 3 = 0, -1.5 D) c = -, -1 ) f() = + 8 ; [, ] A) c =.83 B) c = ± ±.83 C) c = D) c =, 3) f() = 1/3; [0, 1] A) c = 1 3/ B) c = C) c = D) c = 1 3/ Page 0

61 3 *Solve Apps: Mean Value Theorem SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) It took 7 seconds for the temperature to rise from 8 F to 17 F when a thermometer was taken from a freezer and placed in boiling water. Although we do not have detailed knowledge about the rate of temperature increase, we can know for certain that, at some time, the temperature was increasing at a rate of 5 9 F/sec. Eplain. ) A trucker handed in a ticket at a toll booth showing that in hours he had covered 1 miles on a toll road with speed limit 5 mph. The trucker was cited for speeding. Wh? 3) A marathoner ran the. mile New York Cit Marathon in.5 hrs. Did the runner ever eceed a speed of 9 miles per hour? *Know Concepts: Mean Value Theorem SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Provide an appropriate response. 1) The function f() = -3, 0 < 1 is zero at = 0 and = 1 and differentiable on (0, 1), but its derivative on 0, = 1 (0, 1) is never zero. Does this eample contradict Rolleʹs Theorem? ) Suppose that g(0) = 5 and that gʹ(t) = - for all t. Must g(t) = -t + 5 for all t? 3) Decide if the statement is true or false. If false, eplain. The points (-1, -1) and (1, 1) lie on the graph of f() = 1. Therefore, the Mean Value Theorem insures us that there eists some value = c on (-1, 1) for which fʹ() = 1 - (-1) 1 - (-1) = 1. ) Show that the function f() = has eactl one zero on the interval (-, 0). 5) Show that the function r(θ) = cot θ + 1 θ + 5 has eactl one zero on the interval (0, π). ) Use the Mean Value Theorem to show that s = /t decreases on an interval over which it is defined. 7) Let f have a derivative on an interval I. fʹ has successive distinct zeros at = 1 and = 5. Prove that there can be at most one zero of f on the interval (1, 5). Page 1

62 3.7 Solving Equations Numericall 1 Use Bisection Method to Approimate Root MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Bisection Method to approimate the real root of the equation on the given interval. The answer should be accurate to two decimal places. 1) = 0; [0, 1] A) 0.3 B) 0.5 C) 0.50 D) 0.3 ) cos = 0; [-, -1] A) -1.5 B) C) D) -1.5 Use Newtonʹs Method to Approimate Root MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the indicated root of the given equation b using Newtonʹs method. 1) = 0 (between 1 and ) A) B) C) D) ) = 0 (between -1 and 0) A) B) C) D) ) = 0 (the negative root) A) B) C) D) ) = 0 (the larger positive root) A) B) C) D) ) = 0 (between 0 and 1) A) B) C) D) ) = 0 (all real roots) A) , B) C) , D) ) = 0 (all real roots) A) , , B) , , C) , D) ) 3 - sin = 1 (onl root) A) B) C) D) Page

63 3 Approimate Ma/Min Value MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Approimate the values of that gives the maimum and minimum values of the function on the indicated intervals. 1) f() = ; [0, ] A) Minimum f(1.8858) ; maimum f() = B) Minimum f(1.7505) -.330; maimum f(0) = C) Minimum f( ) ; maimum f(0) = 0 D) Minimum f(1.88) ; no maimum ) f() = cos ; [0, 3π] A) Minimum f(7.8719) ; maimum f(.15375) B) Minimum f(7.8719) ; maimum f(3π) = 0 C) Minimum f(0) = 0; maimum f(.15375) D) Minimum f(7.5) ; maimum f(.5).1883 Use Fied-Point Algorithm to Solve Equation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Use the Fied-Point Algorithm with 1 as indicated to solve the equation to five decimal places. 1) = 3 - sin ; 1 = A) B) C).1500 D) No solution ) = cos ; 1 = 1 A) B) C) D) No solution 3) = ; 1 = 1 A) B).899 C).79 D) No solution ) = + ; 1 = A).7305 B) 3 C).777 D) No solution 5 Solve Apps: Numerical Methods MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. Use a numerical method to approimate the solution. 1) The altitude h (in m) of a rocket is given b h = -t3 + 90t + 00t + 30, where t is the time (in s) of the flight. When does the rocket hit the ground? A) s B).37 s C) s D) 7.59 s Page 3

64 ) The capacitances (in μf) of three capacitors in series are C, C , and C If their combined capacitance is 1.50 μf, their individual values can be found b solving the equation 1 C + 1 C C +.50 = 1.50 Find these capacitances. A) 1.17 μf,.7 μf, 3.7 μf B) 1.15 μf,.715 μf, μf C) μf,.59 μf, 3.59 μf D) 1.19 μf,.9 μf, 3.9 μf 3) The length of a rectangular bo is.0000 inches more than its width, and its height is inch more than its width. If the volume of the bo is eactl inches3, what is the width of the bo? A) inches B) 71 inches C) inches D) 9.0 inches 30 ) A dome in the shape of a spherical segment has a volume of eactl,000,000 ft3. If the radius of the dome is ft, what is its height? (The volume of a spherical segment is given b V = 1 πh(h + 3r).) A) ft B) ft C) ft D) ft 5) An oil storage tank has the shape of a right circular clinder with a hemisphere at each end. If the volume of the tank is ft3 and the (clindrical) length is ft, find the radius r (of each end). A) ft B) ft C) 5.31 ft D) ft ) A tin can has the propert that the sum of its volume (in cubic inches) and surface area (in square inches) is eactl If the length of the can is equal to its diameter, find its radius. A) 1.7 inches B) π 78,9 inches C) 1.57 inches D) 50,000 inches 3.8 Antiderivatives 1 Find General Antiderivative MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the general antiderivative F() + C for the function. 1) f() = 3-9 A) C B) C C) 3 + C D) 3 + C ) f() = A) C B) C C) C D) C 3) f() = 8 7 /7 A) /7 + C B) / + C C) /7 + C D) 11/ + C Page

65 ) f() = 9 - A) 3/ - + C B) 93/ - + C C) 3/ - + C D) 93/ - + C 5) f() = 7 + A) 7 8/7 + + C B) 1 8/7 + + C C) 7 8 7/8 + + C D) 1 7/8 + + C ) f() = - 18 A) 3 + C B) + C C) C D) C 7) f() = 1 + π5 A) + π5 + C B) + π5 + C C) + π5 + C D) + + C 1 8) f() = A) / + C B) / + C C) / + C D) / + C 9) f() = A) C B) C C) C D) C Evaluate Indefinite Integral MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Evaluate the indefinite integral. 1) t + t 3 dt A) 3 t 3 + t + C B) 8t C C) 1t t + C D) 3 t 3 + t + C ) (3 + + ) d A) C B) C C) C D) C 3) + d A) 1 3/ C B) 3 8 3/ C C) C D) 1 3/ C Page 5

66 ) + A) - d + C B) C C) - + C D) C 5) d (7 + 3) 5 A) (7 + 3) - + C B) (7 + 3) - + C C) (7 + 3) - + C D) (7 + 3) - + C ) (7-5) d A) ( 7-5) 5 + C B) (7-5) 5 + C C) ( 7-5) C D) ( 7-5) C 7) d A) 1 ( + 5) 3/ + C B) 3 ( + 5) 3/ + C C) - 1 ( + 5) -1/ + C D) 8 ( + 5) 3/ + C 8) d A) 8 15 (1 + 3) 5/ + C B) 8(1 + 3) 5/ + C C) 3 5 (1 + 3) 5/ + C D) (1 + 3) -3/ + C 9) sin t ( + cos t) dt A) 1 5( + cos t)5 + C B) 1 ( + cos t)5 + C C) 1 7( + cos t)7 + C D) 5 ( + cos t)5 + C 3 Find Function Given Second Derivative MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find f() given fʹʹ(). Your answer will involve two arbitrar constants. 1) fʹʹ() = + 5 A) f() = C1 + C B) f() = C1 + C C) f() = C1 + C D) f() = 0 Page

67 ) fʹʹ() = 8 A) f() = / + C1 + C B) f() = 5/ + C1 + C C) f() = - -3/ + C1 + C D) f() = 83/ + C1 + C *Know Concepts: Antiderivatives MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Provide an appropriate response. 1) If we use u = + as a substitution to find ( + ) d, then which of the following would be a correct result? A) u du B) D) None of the above u du C) (u - ) du ) Suppose that we are using substitution to find an antiderivative. If, after making the substitution, we find that there is still an -term left in the integrand, what should we do? A) Go back to the equation relating and u, solve for, and substitute in the integrand. B) This would never happen if we made the correct substitution. C) Use an alternative method, because substitution will never give the antiderivative. D) None of the above 3) Is (5 - ) d = (5 - ) 5 + C? 5 A) Yes B) No ) Is (8 + 7) d = (8 + 7) C? A) Yes B) No 5) Is (5 + )3 d = 1 (5 + ) + C? A) Yes B) No SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. ) Can (7 + ) d be integrated with u = 7 +? Eplain our answer. 7) Is 5 d =? Eplain our answer. Page 7

68 3.9 Introduction to Differential Equations 1 Verif Solution to Differential Equation MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Determine if the function is a solution of the differential equation. 1) d - = 0; = C d A) Yes B) No ) d d - = 0; = 1 - A) Yes B) No 3) d d + = 0; = C 1 cos + C sin A) Yes B) No ) d d + = 1; = sin (5 + C) A) Yes B) No Solve Initial Value Problem MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Find the particular solution that satisfies the given condition. 1) d = - 3; curve passes through (1, 9) d A) = B) = C) = D) = - 3 ) d d = + ; curve passes through (0, 30) A) = B) = C) = D) = ) d d = 1 +, > 0; curve passes through (1, 3) 3 A) = B) = C) = D) = ) d d = -3/; curve passes through (1, 1) A) = 81/ - 7 B) =- 3-7/ - 7 C) = 1/ - 1 D) = 81/ + 8 Page 8

69 5) d d = 0(5-1) 3 ; curve passes through (1, -3) A) = 1 (5-1) B) = 1 (5-1) - 3 C) = (5-1) - 59 D) = 1 (5-1) ) d d = ; = 8 at = 0 A) = + B) = + C) = + 8 D) = - - 7) du dt = u 3(t - t3); u = 1 at = 0 A) u = 1 t - t + 1 B) u = 1 t - t C) u = 1 t - t + 1 3/ D) u = 3 3t - 3t Solve Apps: Differential Equations MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Solve the problem. 1) The rate of ependiture on a particular machine is given b Mʹ() = , where is time measured in ears. Maintenance costs for the second ear are $10. Find the total maintenance function. A) M() = 5( + 5) 3/ + 5 B) M() = 15( + 5) 3/ + 5 C) M() = 5( + 5) 3/ + 15 D) M() = 15( + 5) 3/ + 15 ) The work W (in joules) done b a force F (in newtons) moving an object through a distance (in meters) is given b W = F d. Find a formula for W, if F = k and k is a constant. A) W = k + C B) W = k + C C) W = k + C D) W = k + C 3) The current (in amperes) in an inductor of inductance L (in henries) is given b i = 1 L V dt, where V is the voltage (in volts) and t is the time (in seconds). Find a formula for i, if V = 8t(t - ). A) i = 1 L (t - 8t) + C B) i = L(t - 8t) + C C) i = 1 L (t - t) + C D) i = 1 L (t - 8t) + C ) The rate at which an assembl line workerʹs efficienc E (epressed as a percent) changes with respect to time t is given b Eʹ(t) = 80 - t, where t is the number of hours since the workerʹs shift began. Assuming that E(1) = 88, find E(t). A) E(t) = 80t - t + 10 B) E(t) = 80t - t + 88 C) E(t) = 80t - t + 10 D) E(t) = 80t - t + 1 Page 9

70 5) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bodʹs position at time t. v = -1t + 7, s(0) = 7 A) s = -t + 7t + 7 B) s = -t + 7t - 7 C) s = -1t + 7t + 7 D) s= t + 7t - 7 ) Given the velocit and initial position of a bod moving along a coordinate line at time t, find the bodʹs position at time t. v = cos π t, s(0) = 0 A) s = π sin π t B) s = π sin π t C) s = π sin π t + π D) s = sin t 7) A ball is thrown upward from the surface of the earth with an initial velocit of 3 feet per second. What is the maimum height it reaches? A) 1 ft B) ft C) 0 ft D) 9 ft 8) A ball is thrown upward from the surface of the earth with an initial velocit of 18 feet per second. What is the impact velocit when the ball returns to the ground? A) -18 ft/sec B) -10 ft/sec C) -9 ft/sec D) -3 ft/sec Page 70