Final Exam Review / AP Calculus AB

Transcription

1 Chapter : Final Eam Review / AP Calculus AB Use the graph to find each limit. 1) lim f(), lim f(), and lim π - π + π f y ) lim f(), - lim f(), and + lim f y Find all points where the function is discontinuous and classify the type of discontinuity. )

2 Find numbers a and b, or k, so that f is continuous at every point. 4) f() =, < -5 a + b, , > - Find the limit, if it eists. 5) lim ) lim Determine the limit algebraically, if it eists ) lim 0 Find the limit, if it eists ) lim 11-7 Find the limit. 9) lim

3 Find the limit, if it eists. 10) lim Find numbers a and b, or k, so that f is continuous at every point. 11) f() =, if 7 + k, if > 7 Find all points where the function is discontinuous and classify the type of discontinuity. 1) 1) 14)

4 Chapter : 15. Find an equation of the tangent line to the graph of f ( ) = when = If 5 y e + =, then y = 17. What is the slope of the tangent line to y + y = 6at the point (,1)? 18. Find dy d for y = ln Find y given y y + = If = then f () e = f( ) ln 1. Find y given + y = y. A particle starts at time t = 0 and moves along the -ais so that its position at any time t 0 is t ( ) = t 6t + 9t+ 1. During what time intervals is the particle moving to the left?

5 . For what values of is ( ) f = + increasing? 4. ln y ( e ) 1 =. Find ( 1) y. 5. Write the equation of the line tangent to f ( ) = 1 at =. 6. If y y 5 + =, find dy 1,1. d at ( ) 7. Given the position function s t t = + 5, what is the instantaneous rate of change at t =? 5 8. Given f ( ) ln ( e 1) = +, find ( 0) f. 9. Find an equation of the tangent line to the graph of f ( ) = when = Find the equation of the tangent line to the graph of y = 1 at the point (,1 ).

6 1. If y ( 5) 5 ( ) 4 = + +, then dy d = Chapter 4:. Given ( ) = 9 find the absolute maimum value on the closed interval [ ] f 0,6.. Find all open intervals on which f ( ) = 5+ 4 is increasing? 4. What is the tangent line approimation to y = 1 for values of near 9? 5. Find all points of inflection: ( ) ( ) f = 4 6. Find all open intervals on which ( ) f = is decreasing? 7. The radius of a sphere is measured to be inches. If the measurement is correct to within.01 inch, use differentials to estimate the propagated error in the volume of the sphere.

7 f = ln , then a horizontal tangent line eists at =? 8. Given the curve ( ) ( ) 9. Given f ( ) 4 ( ) =, for what values is the graph concave downwards? 40. Find all inflection points for ( ) 4 f = A rock is projected vertically upward with an initial velocity of 96 ft sec and moves according to the rule s t t = How many seconds will it be before it reaches its maimum height? 4. ( ) f = has a point of inflection at =? 4. A cylindrical terrarium hanging from the ceiling leaks sand at the rate of 5cm min. The sand falls to the floor forming a conical pile. The radius and the height of the cone are in the ratio :. How fast is the height of the pile increasing when the radius is 9cm?

8 44. An archaeologist wishes to enclose a rectangular plot of land in front of an ancient wall and divide the plot into 6 smaller rectangles ( rows of ). She has 00 feet of fencing with which to enclose the area, and she plans to use the ancient wall as one side where no fencing will be needed. Find the dimensions of the rectangle that will produce the maimum area. 45. A leaky cylindrical oil can has a diameter of 4 inches and a height of 6 inches. The can is full of oil and is leaking at a rate of in hr. The oil leaks into an empty conical cup with a diameter of 8 inches and a height of 8 inches. a. At what rate is the depth of the oil in the conical cup rising when the oil in the cup is inches deep? b. At what rate is the depth of the oil in the conical cup rising at the instant the oilcan is empty? 46. The side of a cube is measured to be inches. If the measurement is correct to within.01 inch, use differentials to estimate the resulting error in the volume of the cube. 47. What is the tangent line approimation to y = 8 for values of near 4?

9 48. Given f ( ) = +, find the -value(s) of the points of inflection on the graph of f ( ). 49. Find all intervals on which the following is concave downward: f ( ) 1 = The radius of a circle is given as 8 cm, with a possible error of measurement equal to 5 mm. Use differentials to estimate the ma. error in the area in cm. 51. Gas is escaping from a spherical balloon at a rate of 0 ft /min. At what rate is the radius changing when the volume is 100π ft? 5. As sand leaks out of a container, it forms a conical pile whose altitude is twice the radius. If, at a certain instant, the radius is 5 cm, use differentials to approimate the change in radius that will increase the volume of the pile by 4 cm.

10 5. A powerhouse is on one side of a 400-foot-wide straight river, and a factory is across the river and 00 feet downstream. An electric cable is to be strung from the powerhouse to the factory. It costs 10 dollars for each foot of cable in the water and 6 dollars for each foot of cable on land. What path would minimize the cost of the cable? 54. A container for a juice drink is in the shape of a rectangular prism with a base whose length is twice its width. If the container is to hold approimately 14.5 cubic inches of juice, find the dimensions that minimize the material used in construction. 55. A hot air balloon is rising at a rate of 400 ft/min from a point 1000 feet from an observer on the ground. At what rate is the angle of elevation for the observer changing at the instant that the angle is 45? 56. Gas is being pumped into a spherical balloon. If at a certain instant, the radius of the balloon is 9 inches, use differentials to approimate the change in radius that will result in the balloon increasing its volume by 5 cubic inches.

11 57. The edge of a cube is measured as 10 cm, with a possible error of measurement equal to 1 mm. Use differentials to estimate the maimum error in the calculated volume. 58. A man is on a boat 1 mile from shore. The nearest phone is on the shore miles from the closest point on shore from the boat. If the man can run at a rate of 6 mph and row at a rate of mph, where should he land the boat to get to the phone in the least amount of time? 59. As sand leaks out of a container, it forms a conical pile whose altitude is twice the base radius. If the sand is falling at a rate of 0 ft min onto the pile, find the rate at which the height is changing at the instant the pile is feet high. Chapter 5: 60. Find the average value of f ( ) = on the interval [ ] 0,. t 61. If ( ) = ( + 5), then f ( t) f t d =?

12 6. The position of an object is given by s= t t+ 8. What is its average velocity for t 5? 1 = on 6. The average value of ( ) f 1,1 is 64. π / sec tan d =? 0 d 65. ( 1+ tdt ) d =? 66. Use a Trapezoidal approimation for d 1 with n= Use the Fundamental Theorem of Calculus to evaluate 1 (1 ) d.

13 68. Evaluate: (4 ) d The graph of f is shown for 1 4. What is the value of 4 1 f ( ) d? (needs graph) 70. The average value of f( ) = 4 on the closed interval [1,] is Chapter 6: d = 7. Find the solution to the differential equation dy d = given that the origin is on the curve. 7. 5e d = e + 1

14 74. Solve the differential equation y e y ye + =, given y and y( ),, = ( ) cos ln d = 76. Solve the differential equation y e y ye =, given y and y( ),, =. 77. cos( ) d = ( 1) + d = 79. e d = 80. The solution to the differential equation dy d = y with initial condition ( ) y 0 = 9is

15 dy 81. y d = + and y ( 0) = 1. Solve for y f ( ) = e d = d = + 5 Chapter 7: 84. An object moves in a straight line with velocity vt ( ) = 6t t. a. How far does it travel in the first seconds? b. What is the total distance travelled by the object in the first seconds? 85. The velocity of an object for 0 t is v( t) = t t ( ft sec) and its initial position is ( ) a. What is the acceleration of the object after seconds? 0 =.

16 b. What is the position of the object after seconds? c. When is the object at rest? d. Find the distance travelled by the object during the first seconds? 86. Let R be the region bounded by the graphs of a. Find the area of R. y = +, y 1 =, = 1, and =. b. Find the volume of the solid created by rotating the region R about the line y =. c. Find the volume of the solid created by rotating the region R about the line =. d. Find the volume of the solid whose base is the region R and cross-sections perpendicular to the -ais are squares.

17 87. Let R be the region bounded by the -ais, the graph of y =, and the line = 4. a. Find the area of R. b. Find the value of h such that the vertical line = h divides the region into two regions of equal area. c. Find the volume of the solid generated when R I s revolved about the -ais. d. The vertical line = k divides the region R into two regions such that when these two regions are revolved about the -ais, they generate solids with equal volumes. Find the value of k. 88. Find the volume of the solid formed by revolving the area bounded by the graphs of and y 6 = + about the line y = 4. y =

18 89. A solid has as its base the region in the y-plane bounded by the graphs of y = and y = 4. Every cross section by a plane perpendicular to the y-ais is a isosceles right triangle with hypotenuse in the y-plane. Find the volume of the solid. 90. Find the volume generated by revolving the region bounded by y = 4 and y = 0 the line =. about 91. Find the volume of the solid formed by revolving the area bounded by the graphs of and = y about the line y = 6. = 4 y 9. A solid has as its base the region in the y-plane bounded by the graph of + y = 4. Every cross section by a plane perpendicular to the y-ais is a semicircle with its diameter in the yplane. Find the volume of the solid. 9. Find the volume generated by revolving the region bounded by line y = 1. y and y 6 = = about the