Math 121 Final Review

Transcription

1 Math Final Review Pre-Calculus. Let f(x) = (x + ) and g(x) = x, find: a. (f + g)(x) b. (f g)(x) c. (f/g)(x) d. (f g)(x) e. (g f)(x). Given f(x) = x + x and g(x) = x 5x 5, find the domain of a. f + g b. f g c. f/g d. f g. Given f(x) = sin x + x + 4x + and g(x) = 7x + x find f g and g f. 4. Find f g for f(x) = x x and g(x) = sin x 5. For f(x) = x 4 x + x, find f( a), f(a ), f( a), and f(a ). 6. If f(x) = x + sec x and g(x) = sin x cos x, find (f g)(x) and (g f)(x) 7. Find the equation of the line that passes through the point (, 5) and is parallel to the line x + y = 8. For the function f(x) = x find the domain and f(5). 9. Find the equation of the line through the point (,5) perpendicular to the line through the points (9, ) and (, 7).. If f(x) = x 5, find f() f(5).. Find the line perpendicular to x + y = 5 through the point (, ).. For f(x) = x x + 4 and g(x) = find f g(x) and g f(x) x Simplify the expressions in Problems through 5.. ln e 4. e 5 ln 5. b log b +log b 5

2 Limits x 4 x 6. lim x x 4 7. lim ( sin θ cos θ cot θ) + θ θ π 8. lim x 4 x cos 9. lim x tan x sin x ( ) x x + x + 4. lim x x + 5 { x. Find c so f(x) = + x 5x + 7 x x + c x < is continuous. { cx x. Find c so f(x) = x + c x > is continuous. { x x 5. Find c so f(x) = cx x > 5 is continuous. 4. Find where the function f(x) = x + 5 x 5 is continuous. 5. Find the points of discontinuity for f(x) = x x and are they removable? ( 6. If lim f(x) = 6 and lim g(x) =, find lim [f(x)] + f(x)g(x) + [g(x)] ) x c x c x c

3 Derivatives Use the definition of derivative to find the derivatives of the functions in problems 7 through. 7. f(x) = x x. 8. f(x) = x f(x) = x f(x) = x +. f(x) = x x. f(x) = x +. In problems through 76, find the derivative of the function indicated with respect to x.. f(x) = sin x cos x. 4. f(x) = + sin x cos x. 5. f(x) = x x x + 6. f(x) = x tan x. 7. f(x) = x 9 tan(x) [ ] (x 8. f(x) = ln + 4) cos x 9. f(x) = e sin x 4. f(x) = ( x / ) 4 4. f(x) = cos x + x 5x x 4. f(x) = sin(x ). 4. f(x) = (x + tan x). 44. f(x) = (x + x 5). 45. f(x) = tan x cos x sin(x ) csc(x ) + cos(5x) 46. f(x) = x 8x + 5x 47. f(x) = ln(ln(5x csc x)) 48. f(x) = e x cos x ln e x sin x 49. f(x) = x + x + sin x 5. f(x) = x + cos x x + 5. f(x) = log 5 (7 ln x ) 5. f(x) = ln x e x 5. f(x) = (cosh x) 54. f(x) = x ln x 55. f(x) = x x 56. f(x) = x sin x 57. g(x) = x e x 58. h(x) = ln(x + x + ) 59. f(x) = x +x x 6. f(x) = ( + x ) 5 6. f(x) = ( + x ( 6. f(x) = +x +x 6. f(x) = x + x ) 4 ) f(x) = ( sin 4x)(cos 7x ) 65. f(x) = ( x )(x + 4) 4/ 66. f(x) = x x 67. f(x) = (x)(cos x 4 x ) / 68. f(x) = (6x + 5)(x 4) 69. f(x) = x (4 + x + x ) 4 7. f(x) = (4 + 5x ) /4 (6 x) 6 7. f(x) = sin(x ) ln(x + x) 7. f(x) = ln(x + ) 7. f(x) = (x + ) f(x) = x ( ) 75. f(x) = sin x + tan x 76. f(x) = + x.

4 Find the derivative of the following functions: 77. y = arcsin (ln (x)) 78. y = arctan e x 79. y = tan(arcsin x) ( 8. y = arcsin x + sin ) x 8. y = x arctan x 8. y = e arcsin x 8. y = e x arctan x In problems 84 through 89, use implicit differentiation for find dy for: 84. x y + x y + xy + x + y + = xy 85. x sin y = ln [ x (9y) ] 86. x sin y cos y = 87. x x y + y 5 = x + y = xy 89. x + xy x + y = 4x y 9. Given the curve xy x y = 6 a. Find dy b. Find the tangent line(s) to the curve where x =. c. Find the x-coordinate of each point on the curve where the tangent line is vertical. 9. Find the equation of the tangent line to x y + xy + y = 5 at (, ). In problems 9 through 99, use logarithmic differentiation to find f (x) for: 9. f(x) = x + x + 5 x + (x + ) 9. f(x) = (x ) 4 ( ) (x + ) 94. f(x) = ln (x 5) /4 x f(x) = (x + x 7) (7x 8x) f(x) = (sin(x + )) (4x + 6x) [ 97. f(x) = ln 7x ] 4 + x ] 98. f(x) = x [x 5x + 4x x + x + (x 99. f(x) = ) + x In Problems to, given f(x), find g (b) where g(x) is the inverse of f(x).. f(x) = x + 4 b =. f(x) = x + b =. f(x) = 4 x + x b =. f(x) = x 5 5x + x b = 8 4. Find the equation of the tangent line to f(x) = x + at the point where x =. 5. Find the equation of the tangent line to y = e x ln(x) at x = 6. Find the equation of the tangent line to f(x) = 4 x + 6x at the point (, 8). 7. Find the equation of the tangent line to y = x x at (, ). 8. Find the values of x for all points on the graph of f(x) = x x + 5x 6 at which the slope of the tangent line is Find the point(s) on the graph of y = x where the tangent line passes through the point (, ).

5 4 Applications of Derivatives 4. Related Rates. A rectangle has a width that is /4 its length. At what rate is the area increasing if its width is cm and is increasing at.5 cm/s?. Air is leaking out of a spherical balloon at the rate of cubic inches per minute. When the radius is 5 inches, how fast is the radius decreasing?. Sand is being emptied from a hopper at the rate of ft /s. The sand forms a conical pile whose height is always twice its radius. At what rate is the radius of the pile increasing when the height is 5ft?. A balloon is feet off the ground and rising vertically at the constant rate of 5 feet per second. An automobile passed beneath it traveling along a straight line at the constant rate of 66 miles per hour. How fast is the distance between them changing one second later? 4. The top of a 5-foot ladder leaning against a vertical wall is slipping down the wall at the rate of one foot per minute. How fast is the bottom of the ladder slipping along the ground when the bottom of the ladder is 7 feet away from the base of the wall? 5. A plane flying parallel to the ground at the height of four kilometers passes directly over a radar station. A short time later, the radar reveals that the plane is 5 km away and the distance between the plane and the station is increasing at the rate of km/hr. (The distance is straight line distance from ground level at the station to the plane four kilometers high.) At that moment, how fast is the plane moving horizontally? 6. A spotlight is on the ground feet from the vertical side of a very tall building. A person six feet tall stands at the spot light and walks directly toward the building at a constant rate of 5 feet per second. How fast is the top of the person s shadow moving down the building when the person is 5 feet away from it? 7. Two balloons are attached so air can flow freely between them. If the radius of balloon A is decreasing at in/min, what is the rate of change of balloon B. The radius of balloon A is in and the radius of balloon B is 5 in. 8. A girl is flying a kite which is feet above the ground. The wind is carrying the kite horizontally away from the girl at a speed of feet/second. How fast must the string be let out when the string from the girl to the kite is 5 feet long (and taut)? 9. A rocket is rising straight up from the ground at the rate of km per hour. An observer km from the launching site is photographing the rocket. How fast is the angle θ of the camera with the ground changing when the rocket is.5 km above the ground?. What is the radius of an expanding circle at the moment when the rate of change of its area is numerically twice as large as the rate of change of its radius?. A woman standing on a cliff is watching a motor boat though a telescope as the boat approaches the shoreline directly below her. If the telescope is 5 feet above the water and if the boat is approaching at feet per second, at what rate is the angle of the telescope changing when the boat is 5 feet from shore?. Suppose water is leaking out of a cone shaped funnel at a rate of in /min. Assume the height of the cone is 6 in and the radius of the cone is 4 in. How fast is the depth of the water decreasing when the level of water is 8 in. deep?

6 4. Graphing. Find where f(x) = x 4 + 4x x is increasing and decreasing. 4. Find where f(x) = x 6x is increasing or decreasing 5. Find where f(x) = 4x 5 5x 4 is increasing or decreasing and classify all critical points. 6. Find the concavity of f(x) = x 4 x + 7. Find the concavity of f(x) = x 4 4x and any inflection points. For the functions in problems 8 to find the domain, range, x-intercepts, y-intercepts, where y is increasing, decreasing, critical points, where y is concave up, concave down, inflection points, vertical asymptotes, horizontal asymptotes, and sketch the graph of y. 8. y = x x + 9. y = x 9. y = (x 9) x 4. y = 7x 4 x + 4x + 4. y = x x 4. Max/Min. Find the maximum and minimum values of f(x) = 4x 8x + 5x for x. 4. What positive number exceeds its cube by the greatest amount? (Hint: x > x only if < x < ). 5. Find two positive numbers whose sum is and whose product is as large as possible. 6. Among all rectangles with corners on the ellipse which has the largest area? x 9 + y 4 = 7. Find the absolute and local maximums and minimums for f(x) = 6x 7 x + on [, ]. 8. Find the maximum and minimum of f(x) = x x on [, ] + 9. Which points on the graph of y = 4 x are closest to (, )? 4. Find the maximum and minimum of f(x) = 6x 4x on the interval [ 4, 5]. 4. A student see Professor Butler standing down stream on the other side of a straight river, km wide. He then remembers that they agreed to go to Baker s Square in hopes that they haven t run out of pie. He wants to reach Prof. Butler as quickly as possible and recalls that he can swim at 6 km/h and run at 8 km/h. Where should the student reach the other side of the shore relative to where he began if Professor Butler is 8 km down stream to minimize time? 4. A donkey owner has 75 ft of fencing. He wants to make a rectangular area for them bound by a wall on one side. What dimensions will enable him to fit the largest number of asses inside the fenced area? 4. Milo has a pet elephant named Tiny. Since his pet elephant has an affinity for trampling the neighbors, Milo has to build a fence to keep Tiny in. He s building the fence up against his house so it only needs to have three sides. Fencing costs $ a foot. The recent price gouging of elephant food has Milo in a financial bind, so he only has $ to spend on the fence. What are the dimensions that give Tiny the most room?

7 44. You want to make an open-topped rectangular box with square base. You want its volume to be cubic centimeters. The material for the base costs cents per square centimeter and the material for the sides costs 7 cents per square centimeter. Find the dimensions of the cheapest box to build. 45. Among the all the pairs of nonnegative numbers that add up to 5, find the pair that maximizes the product of the square of the first number and the cube of the second number. 46. A rectangular yard is to be laid out and fenced in, and divided into enclosures by fences parallel to one side of the yard. If miles of fencing is available, what dimensions will maximize the area? 47. A motorist is stranded in a desert 5 miles from a point A, which is the point on a long straight road nearest him. He wishes to get to a point B on the road that is miles from point A. He can travel at 5 miles an hour in the desert and 9 miles an hour on the road. Find the point where he must meet the road to get to B in the shortest possible time. Assume he travels in the desert in a straight line. 48. A rectangular area with fence all around is to be divide into three smaller areas by running two lengths of fence parallel to one side. If you have 8 yds total of fence, what is the largest area that can be enclosed? 49. You work for an ice cream cone company and you are given the task of finding the greatest possible volume of a cone given a slant height of 4 inches. 5. A certain company owns two buildings. The first building is located on one side of a river that is 5 m wide. The second building is located on the opposite side of the river, but 55 m down stream. The company owns a computer systems on each of the buildings and wants to network the two buildings with cables. Of course, they want to minimize the cost of cabling since it will cost twice as much to lay cable underwater as compared to laying cable above ground. What path should the cabling take in order to minimize the total cost? 5. A peanut vendor sells bags of peanuts. He sells them for $5 a bag. At this price he sells 5 bags a day. The vendor observes that for every $.5 he takes off his price per bag, he sells more bags per day. At what price should the peanut man sell his peanuts to maximize revenue. 5. A window is to be made in the shape of a rectangle surmounted by a semicircle with diameter equal to the width of the rectangle. If the perimeter of the window is feet, what dimensions will emit the most light (i.e. maximize the area)? 5. A greeny bus caries passengers from North side to Schmitt Lecture Hall. The cost to ride is $.5 per person. Market Research reveals that fewer people will ride the bus for each $. increase in fare. What fare should be charged to get the largest possible revenue? 54. An oil can is to be made in the form of a right circular cylinder and will contain cm. What are the dimensions of the can that requires the least amount of material? 55. A rectangular swimming pool is to be built with a 6 foot wide deck at the north and south ends, and a foot wide deck at the east and west ends. If the total area available is 6 square feet, what are the dimensions of the largest possible water area? Be sure to draw a picture of your set-up. 56. A flower bed will be in the shape of a sector of a circle (a pie-shaped region) of radius r and vertex angle θ. Find r and θ if its area is a constant A and the perimeter is a minimum. 57. A cylindrical container is to be produced that will have a capacity of cubic meters. The top and bottom of the container are to be made of a material that costs $ per square meter, while the side of the container is to be made of material costing $.5 per square meter. Find the dimensions that will minimize the total cost of the container. What is this minimum cost?

8 4.4 Newton s Method 58. Use Newton s method to find where cos x = x to 4 decimal places. 59. Use Newton s method to find the root of x x + = between and. (4 decimal places) 6. Find 9 by using Newton s method on f(x) = x It is a dark and stormy night. You are drinking hot coffee as you do your chemistry lab. You are inspecting the graph of your data when a loud clap of thunder startles you. Your coffee spills- a drop falls on to the graph and since the lab print-out is so thin the paper dissolves! There is now a hole right where the graph crosses the x-axis. You need the x-intercept for you experiment. You know the equation is f(x) = x /. Use Newton s method to find the x-intercept between 4. and Linear Approximaitons 6. Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a to estimate Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a to estimate Use a linear approximation L(x) to an appropriate function f(x), with an appropriate value of a to estimate l Hospital s Rule x + x lim x x + x 66. lim x 4 x x lim x x + x + x cos x + x 68. lim x sin x + x 69. lim x x sin x 7. lim x 5x. 7. lim x + x 5x. cos x 7. lim. x x x 7. lim x x. x + 6x lim x x x lim x x x x sin x 76. lim x x sin x 77. lim x x 78. lim x ( x + x + x x + ) 79. lim x + x ln x 8. lim x sin π x x 8. lim x x x + 8. lim x ( + ) x x ( ) x 8. lim (cos x) x 84. lim x ( x e x ) ( ( 85. lim + x ) ln x x 86. lim x ( + x) 4 x )

9 5 Integration 87. Compute the Riemann Sum using the right-hand end-points over the indicated interval divided into n sub-intervals. Also, compute the integral and compare the results. f(x) = x + ; [, ], n = If f(x) = 4 and g(x) =, find Compute the derivative of the following functions: 4f(x) g(x) 89. F (x) = 9. F (x) = x x + t dt + t dt 9. F (x) = 9. F (x) = ln x x e + t dt (t + 4t + 4) 5 dt Compute the following integrals: sin x 9. + cos x sin x cos x (x + )(x ) (x ) 5 x x + x x(x + ) x(x + ) x + x x(ln x) x e x x +. x + 4. x 7x ( cos t + 5 sin 4t + e 7t )dt 5 /x x e x ex + e x ln(sin e x ) tan e x e x x + e x 9 x4 + ( x ) + x 5 sec x tan x x + x + x x x + x + x + 6x cos (4 + x )

10 x ln x cos x sin x 4 π/4 π/4 5 π/ 4 π π/ π/4 (x 9 + x 5 ) x 6 + x x sin x 6 x x 4x + cos x ( x x ) x e x cos x sec x (x + ) 4 6 Applications of Integration 6. Area e e π 8 4 x x x + x( + x) x ln x e x x sin x x + x x x x x x + /6 9x x + x + 9 x 4x 5 x + 6x + cos x 9 + sin x x + x 48. Find the area of the region R bounded by the line y = x + and the parabola y = x Find the area of the region bounded by the graphs of f(x) = x 6x and g(x) = x. 5. Find the area of the region enclosed by y = x and y = x + between x = and x =. 5. Find the area of the region enclosed by y = x + 4/x, the x-axis, x =, and x = 4.

11 5. Find the area of the region enclosed by y = x + 5 and y = x. 5. Find the area of the region enclosed by x = y 4y + and x = y. 54. Find the area of the region enclosed by y = 6x x and y = x x 6. Volume 55. Set-up the integral for the volume of revolution, if the region bounded by y = 4x x, and y = x is spun about x = Set up the integral that represents the volume of the formed by revolving the region bounded by the graphs of y = 5 x and y = about the x-axis. (Do not evaluate the integral.) 57. Find the volume generated when the region bounded by y = x, y =, x = and x = 4 is spun about the x-axis. 58. Using the method of cylindrical shells, find the volume if the solid generated by rotating around the line x = the region bounded by y = x +, y = and x =. 59. Find the volume if the area enclosed by f(x) = x, x = 4, x =, y = is rotated about the line y = 6. Find the volume if the region bounded by y = x, y = x +, and the y-axis in the first quadrant, is rotated about the y-axis. 6. Find the volume of the solid that results when the area of the smaller region enclosed by y = 4x, y =, and x = 4 is revolved about the y-axis. 6. Find the volume of the solid that results when the first quadrant region enclosed by y = x and y = x is revolved about the y-axis. 6. Find the volume of the solid that results when the area enclosed by y = 4x, y =, and x = 4 is revolved about the x-axis. 64. Find the volume of the solid that results when the area of the region enclosed by y = x, x =, and y = is revolved about x =. 65. Find the volume of the solid that results when the area of the region enclosed by y = x, x =, and y = is revolved about the x-axis. 66. Find the volume of the solid that results when the area of the region enclosed by y = 4x and y = x is revolved about the line x = A storage tank is designed by rotating y = x +, x, about the x-axis where x and y are measured in meters. Use cylindrical shell to determine how many cubic meters the tank will hold. 6. Work 68. A spring has a natural length of 4 feet. A force of lbs. is required to compress that spring to a length of.5 feet. How much work is done to stretch the spring from its natural length to 6 feet? 69. A 5 foot chain weighing pounds per foot supports a beam weighing pounds. How much work is done in winding 4 feet of the chain onto a drum? 7. A conical tank has a diameter of 9 feet and is feet deep. If the tank is filled with water of density 6.4 lbs/ft, how much work is required to pump the water over the top?

12 7. A cylindrical tank 8 feet in diameter and feet high is filled with water weighing 6.4 lbs/ft. How much work is required to pump the water over the top of the tank? 7. Set up the integral for the work need to empty a right circular conical tank of altitude ft and radius of base 5 ft has its vertex at ground level and axis vertical. If the tank is full of orange marmalade weighting lb/ft pumping all the orange marmalade over the top of the tank. 7. Water is drawn from a well 5 feet deep using a bucket that scoops up lbs of water. The bucket is pulled up at the rate of ft/s, but it has a hole in the bottom through which water leaks out at a rate of /4 lb/s. How much work is done in pulling the bucket to the top of the well. Neglect the weight of the bucket and rope and work done to overcome friction. 74. You are a secret agent. You and your partner, who is kind of annoying, are trying to escape an evil super villain. His hit men chase you on foot for several dozen miles before your partner collapses and is captured. Even though you are really tried and you don t like your partner that much, you infiltrate the villain s lair to try to rescue him. You find your partner has been knocked out and secured at the bottom of a circular pool of water foot in radius and 9 feet in height and the pool is filled with water (density ρ= 6.4 lb/ft ). Since you are so tired and you wouldn t really be that upset if your partner died, you decide that you are only willing to expend million ft pounds of work. Calculate the amount of work required to lift all the water out of the pool to decide if your partner will be rescued. 75. A foot chain weighing 5 pounds per foot is lying coiled on the ground. How much work is required o raise one end of the chain to a height of feet? 76. You are at an ice cream shop. You order your favorite ice cream, chocolate chip cookie dough. You haven t eaten all day, and begin to gobble down the ice cream. If the ice cream cone is feet long and has a radius of foot, how much work is required to eat all the ice cream. (The density of ice cream is ρ = 55.5lb/ft.)

13 Answers. a. x + x + 4x + 4 b. x 5 + 4x 4 + 4x Math Final Review c. x + 4 x + 4 x d. x 6 + 4x + 4 e. (x + 4x + 4). a. all reals. b. all reals. c. x, 5/ d. all reals. (f g)(x) = sin(7x + x) + (7x + x) + 4(7x + x) + and (g f)(x) = 7( sin x + x + 4x + ) + ( sin x + x + 4x + ) 4. (f g)(x) = tan x 5. f( a) = a 4 a a, f(a ) = a 4 a + a, f( a) = a a + a /, and f(a ) = a 8 a 4 + a. 6. (f g)(x) = (sin x cos x) + sec(sin x cos x), (g f)(x) = sin(x + sec x) cos(x + sec x) 7. x + y = 7 8. Domain x ±, f(5) = 4 9. y = (x ) y = x. f g(x) = x + x 4 g f(x) = [ x + x π /. c =. k = 4 ]

14 . c = Continuous everywhere except where x = 5 5. x = is removable, x = is not removable f (x) = x 8. f (x) = x 9. f (x) = (x + 5). f (x) =. f (x) = ( x). f (x) = x +. f (x) = cos x sin x 4. f (x) = cos x sin x ( cos x) 5. f (x) = (x + )(x ) (x x )(x ) (x + ) 6. f (x) = x sec x + tan x 7. f (x) = x 9 sec (x) + 9x 8 tan(x) 8. f (x) = 6x x tan x 9. y = cos xe sin x 4. f (x) = ( 4) 4. f (x) == 9 sin x + 4. f (x) = x cos(x ) ( x / ) 5 ( x 4/ ) 4. f (x) = (x + tan x)( + sec x) 44. f (x) = (x + x 5) 99 (x + ) 45. f (x) = sin x cos x 5 sin(5x) ( x)( x / 5) ( x 5x)( x ) x 46. f (x) = ( x / 4x )( + 5x ) ( x 8x )(x) ( + 5x ) 47. f (x) = x + csc x cot x (5x csc x) ln(5x csc x) 48. f (x) = (x sin x + x cos x)e x cos x + x sin xe x cos x (x cos x x sin x) 49. f (x) = x + x + cos x 5. f (x) = ( x + )(x sin x) (x + cos x)( x ) ( x + )

15 5. f (x) = ln 5 (ln 7)( x ) 5. f (x) = ex x (ln x)xex (e x ) 5. f (x) = (cosh x) sinh x 54. f (x) = + ln x 55. f (x) = x [ x(ln )] x 56. f (x) = x cos x + sin x 57. g (x) = x e x + xe x 58. h (x) = (x + x + ) 59. f (x) = 6. f (x) = 5x( + x ) / ( 6. f (x) = x + x 6. f (x) = 4 ) ( ) (+x) (+x) 5 x 6. f (x + ) x (x) = x [ ] + x x f (x) = ( sin 4x)( 4x sin 7x ) + ( cos 4x)(cos 7x ) 65. f (x) = ( x ) 4 (x + 4)/ () + (x + 4) 4/ ( x ) 66. f (x) = x( x ) + x ( x ) ln 67. f(x) = (cos x 4 x ) / + (x) (cos x 4x ) / ( sin x (4 x (ln 4)()) 68. f (x) = (x)(x 4) + (6x + 5)(6x) 69. f (x) = (4 + x + x ) 4 () (x)4(4 + x + x ) ( + x) (4 + x + x ) 8 7. f (x) = 4 (x)(4 + 5x ) /4 (6 x) 6 (4 + 5x ) /4 6(6 x) 5 7. f (x) = 7. f (x) = x x + 7. f (x) = x(x + ) f (x) = x ( x ) / 75. f (x) = ( sin x + tan x 76. f (x) = x + x 77. y = x (ln x) ln(x + x)(x cos(x )) sin(x ) x + x + x (ln(x + x)) ) [ ] ( + tan x)( cos x) (sin x)(sec x) ( + tan x)

16 78. y = ex + e x 79. y = ( x ) / 8. y = csc x cot x x 8. y = x 4 + x arctan x + x 8. y = earcsin x x 8. y = e x x + x 4 + ex arctan x 84. dy = xy xy y + y x y + x + xy + x dy = sin y x y x cos y dy = sin y x cos y + sin y dy xy x = 5y 4 x dy = y x y x dy = x + y 6x 8xy xy y + 4x 9. a. dy = x y y xy x b. At (, ) the y =, at (, ) then y + = (x ) c. x = y = (x ) 9. f (x) = ( x + x + 5 ( x + ) x + + x + + ) 5 x + [ 9. f (x) = x + 8x ] [ ] (x + ) x (x ) f (x) = x + + 9x 4(x 5) x f (x) = (7x 8x) 5 )()(x + x 7)(6x + ) (x + x 7) (5)(7x 8x) 4 (x 8) (7x 8x) 96. f (x) = (4x + 6x) (sin(x + )) cos(x + ) (sin(x + )) (4x + 6x)(x + 6) (4x + 6x) 4 [ ] 97. f (x) = 4 4x 7x + 9x x [ ] [ 98. f (x) = x x 5x + 4x ( x + x + )(6x x + 4) (x 5x + 4x)( ] x / + ) x + x + ( x + x + ) +x [ ] x 5x + 4x x + x +

17 (x 99. f (x) = ) ( ) + 4x x (x ) y = 6x 8 5. y (e 4 ln 4) = 78.7(x ) 6. y = 5x 7. y = x 8., 9. x =, 6. 8 cm /s. dr dt = π. dr dt = 5π 4. 5 ft/sec 4. Bottom is slipping away at 4 7 ft/min km/hr ft per second in/min ft/sec 9. rad/hr. r = π. 5 radian per second. dy dt =.6in/min.. Inc (, ) (, ) dec: (, ) (, ) 4. Inc: x >, dec: x <. 5. Inc: (, ) (, ), Dec: (, ) C.P. (, ) max (, ) min. 6. Concave up: (, ) (, ), concave down (, ). 7. Con up: (, ) (, ), Con dn: (, ) I.P. (, ) and (, 6). max. = (at x = ), min. = -7 (at x = ) 4. x = 5. and

18 6. Corners at x = ± and y = ± 7. (, 6) abs min, ( 4 /7,.7) local max, ( 4 /7,.7) local min, (, 6) abs max. 8. Maximum is.5 at x =, the minimum is at x =. 9. (, 5 ) and (, 5 ) 4. Max: min: -/ 4. He should reach shore 9 7 km down stream from where he started by by X. X First is, second is 46. X miles A 48., in m downstream. 5. $ r = π $. 54. r = 4π and h = 5 π by r = A, θ = 57. r = 8π x = f(x) = x, x = 56 so f(x) = x, a = 6 so f(x) = x, a = 5 so

19 π e e and F (x) = + x 9. F (x) = + x 9. F (x) = + (ln x) ( x ) 9. F (x) = (x + 4x + 4) cos x + C 94. sin x + C (x ) x + x + C C x. ( + x)/ 4 5 ( + x)5/ + C

20 . ln x + C. e x + C. x ln x + + C 4. ( 7x ) / + C 5. sin t 5 4 cos 4t + 7 e7t + C 6. 5/x ln 5 + C 7. e x + + C 8. (ln(sin e x )) + C 9. ln + e x + C. 9 x4 + 4 ln 9 + C. x 4x 4 + C. ln tan x + C. 4 x 4/ x + C 4. ( + x) + C 5. ln x + x + + C 6. tan(4 + x ) + C 7. ln(ln x) + C 8. cos 5 x 5 cos x + C ln (e 6 )

21 .. 4. ln 4 5. ln 6. e / ln 5 4. x ln + C π 8 ln( x + ) ln + C 4. ln(x + 9) + arctan x + C 44. arcsec(x) + C arctan x arctan ( sin x 47. x arctan x + C / /6 5. 9/ π 56. π π + C ) + C (4 x) [ (4x x ) x ] (6 x ) 58. π 59. V =.6π 6. V = 4π 6. 98π 5

22 6. 4π π 64. 4π 65. π π π ft-pounds 69. 5, ft lbs 7. 5,6 π 7. 49,9 π ft lbs 7. W = ft pounds. π( x) x ,7 π ft pounds. 75. ft-pounds foot-pounds.