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1 MATHEMATICS 57 Final Eamination Review Problems. Let f 5. Find each of the following. a. fa+b b. f f. Find the domain of each function. a. f b. g +. The graph of f + is the same as the graph of g ecept that it is moved how?. For the functions f and g, find f g, g f, and the domains of each. a. f and g b. f + and g 5. Find the intercepts of the following functions. Also, determine whether the graphs of the functions are symmetric with respect to the y-ais or the origin. a. f + + b. g c. h 6. For each of the following, find f. c, if < a. f 7, if ; c b. f, if > { +5, if - 7, if < - ; c - 7. Calculate the following its. a. + b. 5 c d e f g. h. i. + j. k. l m. sin n. π o. sec tan 8+ tan π p. +h tan π h h sin π q. +h sin π 6 6 h h r. h cos+h cos h

2 8. For each of the following, define P such that the given function is continuous at. 8 a. f 9, if P+9, if 7 b. g 9, if P, if 9. Determine the intervals on which the functions defined below are continuous. { { 8 7, if +, if - a. f b. g - 6, if > 5, if > -. Identify all asymptotes of the following. a. y b. y + c. y + d. y Giveaspecific eample toshow thatitispossible for a ftoeist iffaisundefined.. Determine y when if y Determine f 5 if f +7.. Determine y when if y Determine y when if y Find the equation of the tangent line to the curve defined by y when. 7. At what point,y is the tangent line to the curve y 5+8 parallel to the line y 7? 8. Find the equation of the line tangent to the graph of +y +y at the point,-. 9. Let y. Find y when.. If f +, find f.. Determine f if f +.. Determine f if f 5.. Determine y at if y.. Let y [cos π]. Find y at π Determine f π tan if f +cos. 6. If f sincos, find f π If y + y Find y at,.

3 8. Let f and f 6. Determine the intervals on which f is increasing. 9. Determinetheintervalsonwhichf isdecreasingiff + andf Determinetheintervalsonwhichf isconcaveupwardiff Determine the intervals on which f is concave downward if f f 6. - and. Determine all points of inflection for f Determine all points of inflection for f Let f + +. Find all local etrema for f. 5. Let f 5 +. Find all local etrema for f. 6. For each of the following, find the maimum and minimum values of the given function on the indicated interval. a. f ; [,] b. f ; [-,] 7. A rock thrown from the top of a cliff is st 9+6t 6t feet above the ground t seconds after being thrown. a. Determine the height of the cliff. b. Determine the time it takes the rock to reach the ground. c. Find the velocity of the rock when it strikes the ground. 8. A rock is thrown vertically upward from the roof of a house feet high with an initial velocity of 8 ft/sec. a. What is the speed of the rock at the end of seconds? b. What is the maimum height the rock will reach? 9. What is the maimum area which can be enclosed by ft of fencing if the enclosure is in the shape of a rectangle and one side of the rectangle requires no fencing?. A woman throws a ball vertically upward from the ground. The equation of its motion is given by st -6t +ct, where c is the initial velocity of the ball. If she wants the ball to reach a maimum height of ft, find c.

4 . A rectangular open tank is to have a square base, and its volume is to be 5 yd. The cost per square yard for the base is $8 and for the sides is $. Find the dimensions of the tank in order to minimize the cost of the material.. A power station is on one side of a river which is mile wide, and a factory is mile downstream on the other side of the river. It costs $ per foot to run power lines overland and $5 per foot to run them under water. Find the most economical way to run the power lines from the power station to the factory.. A cardboard bo manufacturer wishes to make open boes from pieces of cardboard in square by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out in order to obtain a bo of the largest possible volume. What is the largest possible volume?. A train leaves a station traveling north at the rate of 6 mph. One hour later, a second train leaves the same station traveling east at the rate of 5 mph. Find the rate at which the trains are separating hours after the second train leaves the station. 5. A street light hangs ft above the sidewalk. A man 6 ft tall walks away from the light at the rate of ft/sec. At what rate is the length of his shadow increasing? 6. A barge is pulled toward a dock by means of a taut cable. If the barge is ft below the level of the dock, and the cable is pulled in at the rate of 6 ft/min, find the speed of the barge when the cable is 5 ft long. 7. Find the values of y and dy if y,, and d.. 8. Use differentials to approimate the maimum possible error that can be produced when calculating the volume of a cube if the length of an edge is known to be.5 ft. 9. Approimate 5 using each of the following. a. Differentials b. A linearization 5. The moment of inertia of an annular cylinder is I.5M R R, where M is the mass of the cylinder, R is its outer radius, and R is its inner radius. If R, and R changes from to., use differentials to estimate the resulting change in the moment of inertia. 5. The range of a shell shot from a certain ship is R sinθ meters, where θ is the angle above horizontal of the gun when it is shot. If the gun is intended to be fired at an angle of π radians to hit its target, but due to waves it actually shot.5 radians too low, use 6 differentials to estimate how far short of its target the shell will fall. 5. For each of the following, determine whether the Intermediate Value Theorem guarantees the equation has a solution in the specified interval. a ; [,] b. 5; [,9] c. cos; [,π] d. tan; [ π, ] π

5 5 5. Find f if f +, f, and f Find f if f ++ and f- -. For problems 55 and 56, let ft be the function defined by the graph shown. ft t 55. Estimate each of the following. Round numbers to the nearest integer. a. The instantaneous rate of change of f at t. b. The average rate of change of f over the interval [,]. c. The intervals where ft is increasing and where it is decreasing. d. The inflection point or points of ft. 56. Find the intervals where f t is increasing and where it is decreasing. Round numbers to the nearest integer. 57. Let vt, as shown in the graph below, be the velocity of a car in meters per second at time t in seconds, where positive velocity means the car is moving forward. Round your answers to the nearest integer.

6 6 vt t a. When did the car stop? b. Approimately how far did the car travel in the time interval 8 to seconds? c. Approimately how far did it travel in the time interval to seconds? d. Approimately how far did it travel in the time interval 8 to seconds? e. At the time seconds, is the car moving forward or backward? Is the driver s foot on the gas or the brake? f. At the time 6 seconds, is the car moving forward or backward? Is the driver s foot on the gas or the brake? 58. Integrate. a. + d b. + +d 8 e. d f. +d i. j. 6 +d d c. d d +6+ d g. h. +5 d d +z - k. d z l. sectancossecd m. sin cos d n. tansec d

7 7 59. Differentiate. a. f t 8 dt b. g 6. Find the area of the region bounded by y and y. 6. Find the area of the region bounded by y and y. 6. Find the area of the region bounded by y and y 8. t +6 5 dt 6. Find the volume of the solid generated by revolving the region bounded by y, the -ais, and the line about the -ais. 6. Find the volume of the solid generated by revolving the region bounded by y and y about the -ais. 65. Find the volume of the solid generated by revolving the region in the first quadrant bounded by y, y, and the y-ais about each of the following. a. the y-ais b. y 66. Find the volume of the solid generated by revolving the region bounded by y +, the -ais, and the lines and about the y-ais. 67. A solid has as its base the region in the first quadrant bounded by +y 5. Every plane section of the solid taken perpendicular to the -ais is a square. Find the volume of the solid. 68. A solid has as its base the region in the y-plane bounded by the graphs of y and y. Find the volume of the solid if every cross section by a plane perpendicular to the -ais is a semicircle with diameter in the y-plane. 69. Find the average value of the function f ++ on the interval [-,]. 7. Find the average value of the function f sin on the interval [,π]. 7. Assume that the density of water is 6.5 lb/ft. A cylindrical water tank with a circular base has radius feet and height feet. Find the work required to empty the tank by pumping the water out of the top for each of the following situations. a. The tank is full. b. The tank is half full. 7. A bucket with lb of water is raised feet from the bottom of a well. Find the work done in each of the following cases. a. The weight of the empty bucket is lb and the weight of the rope is negligible. b. The bucket weighs lb and the rope weighs oz/ft. c. The bucket weighs lb, the rope weighs oz/ft, and water is leaking out of the bucket at a constant rate so that only 8 lb of water remain in the bucket when it reaches the top.

8 8 7. Match each numbered item with a lettered item. There are more lettered items than numbered items. Some lettered items do not match any numbered item.. Definition of a f L. a. a f fa.. Definition of a f L. b. For every ε > there is a corresponding number N such that f L < ε whenever > N.. Definition of f L.. Definition of f is continuous at a. 5. The Intermediate Value Theorem. 6. Definition of the derivative of f at a. 7. Definition of a function f being differentiable at a. 8. Theorem relating differentiability and continuity. 9. The power rule for differentiation.. Definition of the differential.. Definition of a function f having an absolute maimum at c.. Definition of a function f having a local maimum at c.. The Etreme Value Theorem.. Definition of a function that is increasing on an interval I. 5. The Mean Value Theorem. 6. Definition of an antiderivative of f on an interval I. c. If f is continuous on the closed interval [a,b], and N is a number strictly between fa and fb, then there eists a number c in a,b such that fc N. fa+h fa d. The it eists. h h e. If f is differentiable at a, then f is continuous at a. f. If f is continuous at a, then f is differentiable at a. g. d d n n n. h. The function g has the property g f for all in I. i. h fa+h fa h j. fc f for all in the domain of f. k. For everyε > there is a corresponding numberδ > such that f L < ε whenever a < < a+δ. l. For every ε > there is a corresponding number δ > such that f L < ε whenever < a < δ. m. There is an open interval I containing c such that fc f for all in I. n. f > for all in I. o. If f is continuous on [a,b], then there are numbers c and d in [a,b] such that fc is an absolute maimum for f in [a,b] and fd is an absolute minimum for f in [a,b]. p. If f is differentiable, dy f d. q. f < f whenever < and and are in I. r. If f is continuous on [a,b] and differentiable on a,b, then there is a number c in a,b such that f c fb fa b a.

9 9 MATHEMATICS 57. a. a +ab+b 5a 5b b.. a. Dom: -,-] [, Answers to Final Eamination Review Problems 6. a. DNE b. 7. a. - b. Dom: -,- -,,,. Left units and down units.. a. f g Dom: [,, g f + Dom: -,-, b. f g Dom: -,, g f + Dom: -,, + 5. a. y-intercept:, -intercept:, Symmetry: none b. y-intercept: none -intercepts: none Symmetry: origin c. y-intercept:, -intercepts: -,,, Symmetry: y-ais b. - c. d. - e. f. DNE g. h. - i. DNE j. k. l. DNE m. n. o. p. q. r. -sin 8. a. 5 b a. -, b. -,-], -,

10 . a. HA: y VA: OA: none b. HA: y VA: - OA: none c. HA: y -, y VA: OA: none d. HA: y VA: OA: none. f a There are infinitely many correct answers y 7.,6 8. y Inc: [, 9. Dec: -,]. CU: -,,,. CD: -,-,,. IP: -,- 56. IP: none,,- 7. Local ma: + at - Local min: at Local ma: at Local min: - at - 5, - at 5 6. a. Ma: Min: -6 b. Ma: 6 Min: 7. a. 9 ft b. 6 sec. c. -8 ft/sec

11 8. a. 6 ft/sec b. 88 ft 9. 5 ft. 8. Dimensions: Overland 5 mile 8 Underwater 5 mile 8. square Volume of 8 in. 5 mph 5. ft/sec 6. 9 ft/min 7. y.7 dy ft 9. a. 7+ b di.m 5. 5 meters 5. a. Yes b. Yes 55. a. b. c. Dec: [,], [,5] Inc: [,] d. IP:, 56. Dec: [, 5] Inc: [,] 57. a. seconds b. 6 meters c. meters d. meters e. Forward Accelerator f. Backward Brake 58. a C b C c d. - +C e. 8 f. + +C c. Yes d. No 5. f g. h C C 5. f + + i C

12 j C k. - +z - +C l. sinsec+c m. - cos +C n. tan +C 59. a. f 8 b. g π 6. 96π a. 8π b. 56π π π π 7. a. 85π foot-pounds b. 9.75π foot-pounds 7. a. 8 foot-pounds b foot-pounds c foot-pounds 7.. l. k. b. a 5. c 6. i 7. d 8. e 9. g. p. j. m. o. q 5. r 6. h

13 MATHEMATICS 57 Solutions to Final Eamination Review Problems. Let f 5. Find each of the following. a. fa+b a+b 5a+b a +ab+b 5a 5b a +ab+b 5a 5b b. f f a +ab+b 5a 5b. Find the domain of each function. a. f f + Dom: -,-] [, b. g + g Dom: -,- -,,,. The graph of f + is the same as the graph of g ecept that it is moved how? Since f g+, the graph of f is the graph of g shifted left units and down units.. For the functions f and g, find f g, g f, and the domains of each. a. f and g f g fg f Dom: [,, g f gf g +

14 Dom: -,-, b. f + and g f g fg f + + Dom: -,, g f gf g Dom: -,, 5. Find the intercepts of the following functions. Also, determine whether the graphs of the functions are symmetric with respect to the y-ais or the origin. a. f + + f y-intercept:, intercept:, f f- Neither even nor odd. Symmetry: none b. g g undefined y-intercept: none No solution. -intercepts: none g g So g is odd. Symmetry: origin c. h h y-intercept:, -intercepts: -,,, h- - h So h is even. Symmetry: y-ais

15 5 6. For each of the following, find f. c, if < a. f 7, if ; c, if > f f So f does not eist. b. f { +5, if - 7, if < - ; c - f f f Calculate the following its. a. - + b c. + 5 Divide the leading coefficients e If > 7, then 7 > f DNE g. If >, then >. +9 d. 7 7 If < 7, then 7 <.

16 6 h. k If <, then <. If -, then i. - DNE + j. + + k By the Squeeze Theorem, l If - < < -, then 9 <. 9 DNE

17 7 m. sin If >, then - sin. - By the Squeeze Theorem, sin n. π π π cos π sin π o. sec tan cos sin cos cos sin o. 8+ Let f f f 8+ - f tan π p. +h tan π h h Let f tan. tan π +h tan π h h f sec f π sec π sin π q. +h sin π 6 6 h h Let f sin. sin π +h sin π 6 6 h h f cos f π 6 cos π 6 r. h cos+h cos h Let f cos. cos+h cos h h f -sin f π f π 6 f

18 8 8. For each of the following, define P such that the given function is continuous at. 8 a. f 9, if P+9, if 7 b. g 9, if P, if f f 5 P +9 5 g P 9 P 5 9. Determine the intervals on which the functions defined below are continuous. { { 8 7, if +, if - a. f b. g - 6, if > 5, if > - f - g- f g + f g 5 8 f - f f So f is continuous at. Hence, f is continuous on -,. f DNE So g is not continuous at -. Note that g is left continuous at -. Hence, g is continuous on -,-] and -,.

19 9. Identify all asymptotes of the following. a. y HA: y VA: OA: none b. y HA: y VA: - + OA: none + c. y

20 HA: y -, y VA: - OA: none d. y HA: y VA: OA: none + +. Giveaspecific eample toshow thatitispossible for a ftoeist iffaisundefined. f f DNE There are infinitely many correct answers.. Determine y when if y 7 5. y y

21 y - y Determine f 5 if f +7. f +7 f +7- f f 5 9. Determine y when if y 8 8. y y y y 7 5. Determine y when if y +8. y +8 y y -+ y Find the equation of the tangent line to the curve defined by y when. y 5+8 y 5 y 6 y y 6 y 7. At what point,y is the tangent line to the curve y 5+8 parallel to the line y 7? y 5+8 y y 6,6

22 8. Find the equation of the line tangent to the graph of +y +y at the point,-. +y +y +y +yy +y yy +y - y y y + - y y - y y + y - y y y + y 7 9. Let y. Find y when. y y - y - y y y - 7 y - -. If f +, find f. f f f + + f. Determine f if f +. f + f f -. Determine f if f 5.

23 f 5 - f f. Determine y at if y. y - y + - y - -5 y - 8. Let y [cos π]. Find y at π 6. y cos π. y cos π-sin π 5. Determine f π tan if f +cos. f tan +cos f sec +cos tan-sin +cos f π π 6 y If f sincos, find f π 6. f sincos f coscos+sin-sin f π 6 f π 6 -

24 7. If y + y Find y at,. y + y +9 9 y y + y y +9 y y y - +y 9 y y - +y 9 y - +y 9 y y y Let f and f 6. Determine the intervals on which f is increasing. f Inc: [, 9. Determinetheintervalsonwhichf isdecreasingiff + andf f + Dec: -,] Determinetheintervalsonwhichf isconcaveupwardiff f f f + f + f + CU: -,,,. Determine the intervals on which f is concave downward if f f 6. f 6 f 6 [+ ] CD: -,-,, - and. Determine all points of inflection for f f f + f + f + CD: -, CU: -,-,, IP: -,- 56,,- 7

25 5. Determine all points of inflection for f f f 7 +9 f +9 f 7 CD: never CU: -, IP: none. Let f + +. Find all local etrema for f. f + + f Dec: [-,-+ ] Inc: -,- ], [-+, Local ma: + at - Local min: at Let f 5 +. Find all local etrema for f. f 5 + f 5 f 5 f Dec: -,- 5], [, 5] Inc: [- 5,], [ 5, Local ma: at Local min: - at - 5, - at 5 6. For each of the following, find the maimum and minimum values of the given function on the indicated interval. a. f ; [,] f f f + f ma f -9 f -6 min Note that - / [,]. b. f ; [-,] f f f +

26 6 f- 9 f min f- 6 ma Note that / [-,]. 7. A rock thrown from the top of a cliff is st 9+6t 6t feet above the ground t seconds after being thrown. a. Determine the height of the cliff. s 9 9 ft b. Determine the time it takes the rock to reach the ground. -6t +6t+9 t t t 6 6 sec t+t 6 c. Find the velocity of the rock when it strikes the ground. st -6t +6t+9 vt -t+6 v ft/sec 8. A rock is thrown vertically upward from the roof of a house feet high with an initial velocity of 8 ft/sec. a. What is the speed of the rock at the end of seconds? st -6t +8t+ vt -t+8 v 6 6 ft/sec b. What is the maimum height the rock will reach? vt -t+8 -t+8 s ft t

27 9. What is the maimum area which can be enclosed by ft of fencing if the enclosure is in the shape of a rectangle and one side of the rectangle requires no fencing? 7 A - + A A -+ A5 5 ma A A - 5 A 5 ft. A woman throws a ball vertically upward from the ground. The equation of its motion is given by st -6t +ct, where c is the initial velocity of the ball. If she wants the ball to reach a maimum height of ft, find c. st -6t +ct vt -t+c -t+c t c s c -6 c +c c - c 6 + c c 6 c 6 c 8. A rectangular open tank is to have a square base, and its volume is to be 5 yd. The cost per square yard for the base is $8 and for the sides is $. Find the dimensions of the tank in order to minimize the cost of the material. h h C 6 h h h h C 6 h h C 6 5 h 5 h 5 C 8 + h C C 8 + Dec:,5] Inc: [5, Absolute min: 6 at 5 Length of base side: 5 Height: 5 Dimensions: 5 5 5

28 8. A power station is on one side of a river which is mile wide, and a factory is mile downstream on the other side of the river. It costs $ per foot to run power lines overland and $5 per foot to run them underwater. Find the most economical way to run the power lines from the power station to the factory. Recall: mile 58 feet C C C C C 58 C 58 Solve C C 9 C C 8 6 min Overland 5 8 mile Underwater 5 8 mile

29 . A cardboard bo manufacturer wishes to make open boes from pieces of cardboard in square by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out in order to obtain a bo of the largest possible volume. What is the largest possible volume? 9 }{{} {}}{ } {{ } V V 8 + }{{} V 96+ V 8+ V 6 V V6 V 8 ma square Volume of 8 in. A train leaves a station traveling north at the rate of 6 mph. One hour later, a second train leaves the same station traveling east at the rate of 5 mph. Find the rate at which the trains are separating hours after the second train leaves the station. Let t be the amount of time the second train has traveled. d 55t+ 5t +t+6 6t+6 d t d t d 5 d 6t+6 +5t d 6t +7t+6+5t 5 mph Alternatively, d 565t +7t+6 d 5 5t +t+6 d 55t +t+6 d 5 5t +t+6-5t+ y d

30 +y d +yy dd d 5d 65 d 5 5 mph d 5. A street light hangs ft above the sidewalk. A man 6 ft tall walks away from the light at the rate of ft/sec. At what rate is the length of his shadow increasing? 6 y y 6 +y y +y y y y y ft/sec 6. A barge is pulled toward a dock by means of a taut cable. If the barge is ft below the level of the dock, and the cable is pulled in at the rate of 6 ft/min, find the speed of the barge when the cable is 5 ft long. b c b + c bb cc 8 b 5 6 b 9 9 ft/min 7. Find the values of y and dy if y,, and d.. y y y dy d d. dy.7 8. Use differentials to approimate the maimum possible error that can be produced when calculating the volume of a cube if the length of an edge is known to be.5 ft. V V dv d d.5 dv.6.6 ft

31 9. Approimate 5 using each of the following. a. Differentials y y dy d 9 d dy b. A linearization f f 9 f9 7 f 9 L 9+7 f5 L The moment of inertia of an annular cylinder is I.5M R R, where M is the mass of the cylinder, R is its outer radius, and R is its inner radius. If R, and R changes from to., use differentials to estimate the resulting change in the moment of inertia. I.5M R I.5MR M I MR di MR dr R dr. di M. di.m 5. The range of a shell shot from a certain ship is R sinθ meters, where θ is the angle above horizontal of the gun when it is shot. If the gun is intended to be fired at an angle of π radians to hit its target, but due to waves it actually shot.5 radians too low, use 6 differentials to estimate how far short of its target the shell will fall. R sinθ R cosθ R 6cosθ dr 6cosθdθ θ π 6 dθ -.5 dr 6cos π -.5 dr dr -5 5 meters 5. For each of the following, determine whether the Intermediate Value Theorem guarantees the equation has a solution in the specified interval. a ; [,] f f - < f 5 >

32 Yes. b. 5; [,9] f f < 5 f9 6 > 5 Yes. c. cos; [,π] f > fπ - π < Yes. d. tan; [ π, ] π f tan No. Note that f is not continuous on [ π, ] π. f cos 5. Find f if f +, f, and f -. f + f ++C f - 5+C - C -8 f + 8+C f + 8+C C 7 6 f f Find f if f ++ and f- -. f ++ f + ++C f C - C f + +

33 For problems 55 and 56, let ft be the function defined by the graph shown. ft t 55. Estimate each of the following. Round numbers to the nearest integer. a. The instantaneous rate of change of f at t. Note that the slope of the tangent line at the point,8 is approimately. So f. b. The average rate of change of f over the interval [,]. f - f - c. The intervals where ft is increasing and where it is decreasing. Dec: [,], [,5] Inc: [,] d. The inflection point or points of ft. IP:, 56. Find the intervals where f t is increasing and where it is decreasing. Round numbers to the nearest integer. Recall that f is increasing where f is concave upward and that f is decreasing where f is concave downward. The function f CD:,5 CU:, The function f

34 Dec: [,5] Inc: [,] 57. Let vt, as shown in the graph below, be the velocity of a car in meters per second at time t in seconds, where positive velocity means the car is moving forward. Round your answers to the nearest integer. vt t a. When did the car stop? v seconds b. Approimately how far did the car travel in the time interval 8 to seconds? Let st be the position at time t. s s8 8 vt dt 8 area under the curve 6 6 meters c. Approimately how far did it travel in the time interval to seconds? Let st be the position at time t. s s vt dt meters area above the curve d. Approimately how far did it travel in the time interval 8 to seconds?

35 5 From time t 8 to time t, the car travels forward 6 meters. From time t to time t, the car travels backward meters. The total distance that the car travels is meters. The displacement of the car is meters. e. At the time seconds, is the car moving forward or backward? Is the driver s foot on the gas or the brake? v > Forward v > Accelerator f. At the time 6 seconds, is the car moving forward or backward? Is the driver s foot on the gas or the brake? v6 < Backward Accelerating Brake v 6 > The above answer may be counterintuitive. When you are driving a car in reverse, the velocity of the car is negative. When you apply the brake, the speed decreases but the velocity increases. This is the reason that the car is accelerating when your foot is on the brake. 58. Integrate. a. + d Let u + and du 6d. + d 6 u du u5 +C + 5 +C b. + +d Let u + and du 8 +d. + +d u du 6 u +C 6 + +C c d d

36 d d. +6+ d d d C - +C e. 8 8 d d f d Let u + and du 6 d. +d udu du u u +C u +C + +C g. +5 d +5 - d C h C d Let u ++ and du +d d u du u- du - u- +C - u +C C

37 i. 6 +d - udu 7 Let u 6 + and du 8 d. 6 +d 6 udu 6 u du 9 u +C 9 u +C C j. d Let u and du -d. Note that u. d - u udu u u du 5 u5 u +C 5 u5 u +C 5 5 +C k. +z - d z Let u +z - and du -z - d. +z - d z -u du - u +C - u +C - +z - +C l. sectancossecd Let u sec and du sectand. sectancossecd cosudu sinu+c sinsec+c m. sin cos d Let u cos and du - sin d. sin -u du - u +C cos d - cos +C n. tansec d Let u tan and du sec d. tansec d

38 8 udu u +C tan +C 59. Differentiate. Use the Fundamental Theorem of Calculus Part I. a. f f 8 t 8 dt b. g t +6 5 dt g Find the area of the region bounded by y and y. +, -, - d Find the area of the region bounded by y and y , - d Find the area of the region bounded by y and y , 8 d Find the volume of the solid generated by revolving the region bounded by y, the -ais, and the line about the -ais. Disk Method π d π d

39 π 6π Shell Method y y 8 π πy 8 y dy y y 5 dy 8 π y y 8 8 π8 96 6π 9 6. Find the volume of the solid generated by revolving the region bounded by y and y about the -ais. First, find the points of intersection. y y y y , Intersection Points:,,, Find the volume using either the washer method or the shell method. Washer Method y y y y π [ π ] d 6 d π 8 5 π 8 96π 5 Shell Method y y y y π πy y y dy y y dy π y y π π Find the volume of the solid generated by revolving the region in the first quadrant bounded by y, y, and the y-ais about each of the following. a. the y-ais Disk Method y y π y dy πydy

40 π y 8π Shell Method y π π d - +8 d π - + π-8+6 8π b. y Disk Method π π d 8 +6 d π π π 5 Shell Method y y π π y ydy y y dy π 8 y y 5 5 π π Find the volume of the solid generated by revolving the region bounded by y +, the -ais, and the lines and about the y-ais. Shell Method π + d π + d π + π [ 9+ +] π π 67. A solid has as its base the region in the first quadrant bounded by +y 5. Every plane section of the solid taken perpendicular to the -ais is a square. Find the volume of the solid. +y 5 y 5 A 5 A d A solid has as its base the region in the y-plane bounded by the graphs of y and y. Find the volume of the solid if every cross section by a plane perpendicular to the -ais is a semicircle with diameter in the y-plane. y y r

41 A π semicircle A π 8 π d π 8 π 8 π Find the average value of the function f ++ on the interval [-,] d [ ] Find the average value of the function f sin on the interval [,π]. π π sind π π -cos π - π 7. Assume that the density of water is 6.5 lb/ft. A cylindrical water tank with a circular base has radius feet and height feet. Find the work required to empty the tank by pumping the water out of the top for each of the following situations. a. The tank is full. Partition the tank into n equal slices. The height of each slice is n ft. The volume of each slice is 9π n ft. The weight of each slice is 565π n lb. Theworkrequiredto lift outthei th slice foot-pounds. n is 565πi n n n i 565πi n 56.5π n 56.5π n n i d 56.5π 85π i n n 85π foot-pounds Alternatively, n n i 565πi n 565π n n 565π n n 85π n n i i nn+ 85π foot-pounds b. The tank is half full. Partition half of the tank into n equal slices. The height of each slice is 5 n ft. The volume of each slice is 5π n ft. The weight of each slice is 8.5π n lb.

42 Theworkrequiredto lift outthei th slice is 8.5π 5i +5 foot-pounds. n n n n i 8.5π n 56.5π n 56.5π 5 5i n +5 n i 5 n +5 d 56.5π π 9.75π foot-pounds Alternatively, 5i n +5 n n i n n n 8.5π n n i n i 6.5π n 5i n πi + 6.5π n n 6.5πi n + n i+ i n i n i 6.5π n 6.5π n 6.5π nn+ +6.5π n n 7.5π + 6.5π 9.75π 9.75π foot-pounds 7. A bucket with lb of water is raised feet from the bottom of a well. Find the work done in each of the following cases. a. The weight of the empty bucket is lb and the weight of the rope is negligible foot-pounds b. The bucket weighs lb and the rope weighs oz/ft. f d foot-pounds c. The bucket weighs lb, the rope weighs oz/ft, and water is leaking out of the bucket at a constant rate so that only 8 lb of water remain in the bucket when it reaches the top. f d foot-pounds

43 7. Match each numbered item with a lettered item. There are more lettered items than numbered items. Some lettered items do not match any numbered item.. Definition of a f L. a. a f fa.. Definition of a f L. b. For every ε > there is a corresponding number N such that f L < ε whenever > N.. Definition of f L.. Definition of f is continuous at a. 5. The Intermediate Value Theorem. 6. Definition of the derivative of f at a. 7. Definition of a function f being differentiable at a. 8. Theorem relating differentiability and continuity. 9. The power rule for differentiation.. Definition of the differential.. Definition of a function f having an absolute maimum at c.. Definition of a function f having a local maimum at c.. The Etreme Value Theorem.. Definition of a function that is increasing on an interval I. 5. The Mean Value Theorem. 6. Definition of an antiderivative of f on an interval I. c. If f is continuous on the closed interval [a,b], and N is a number strictly between fa and fb, then there eists a number c in a,b such that fc N. fa+h fa d. The it eists. h h e. If f is differentiable at a, then f is continuous at a. f. If f is continuous at a, then f is differentiable at a. g. d d n n n. h. The function g has the property g f for all in I. i. h fa+h fa h j. fc f for all in the domain of f. k. For everyε > there is a corresponding numberδ > such that f L < ε whenever a < < a+δ. l. For every ε > there is a corresponding number δ > such that f L < ε whenever < a < δ. m. There is an open interval I containing c such that fc f for all in I. n. f > for all in I. o. If f is continuous on [a,b], then there are numbers c and d in [a,b] such that fc is an absolute maimum for f in [a,b] and fd is an absolute minimum for f in [a,b]. p. If f is differentiable, dy f d. q. f < f whenever < and and are in I. r. If f is continuous on [a,b] and differentiable on a,b, then there is a number c in a,b such that f c fb fa b a.. l. k. b. a 5. c 6. i 7. d 8. e 9. g. p. j. m. o. q 5. r 6. h