AP Calculus 2 Summer Review Packet

Transcription

1 AP Calculus Summer Review Packet This review packet is to be completed by all students enrolled in AP Calculus. This packet must be submitted on the Monday of the first full week of class. It will be used as the First Assessment for the AP Calculus Course

2 Name AP Calculus Evaluate each of the following its ( 5 11) Let 5, 3 f( ), Find, f( ) f( ) f( ) 3 18, sin cos 4

3 cos 1. sin 3 0 sin8. sin 3. tan 4. 1 sin sin cos 6. sin( h) sin h 0 h 7. h 0 (3 ) 9 h h h h h 0 h0 h h 30. Is the function 7, f ( ) 9, 33, continuous? Eplain. 31. Determine where the function f ( ) sec is discontinuous. 3. Is the function f ( ) tan continuous on the interval,? 33. For what value(s) of k is the function f( ) , 4 k 1, 4 continuous at = 4?

4 34. For what value(s) of k is the function Continuous at = -3? 61, 3 f ( ) k 5 k, 3 6, At what point is the removable discontinuity for the function 54 f( )? 6 36.

5 Find the derivative of each of the following functions f( ) f ( ) ( 4 3)( 1) 39. f ( ) ( 1) f ( ) f( ) 3 1) 4. f( ) y y

6 45. y f( ) Find dy d at =1 if t y and t t Find du dv at v = if 3 1 u and. v Find the derivative of each trigonometric function. 49. y sin 50. y cos 51. y tan sec 5. y sin3

7 1 sin 53. y 54. y sec tan 1 sin 55. sec y 1 tan Use implicit differentiation to find the derivative of each equation. 56. cos y sin sin y cos y y y y y Find dy d if 16 16y y 1 at (1,1).

8 61. Find dy d if 1/ 1/ y y at (1,1). 6. Find dy d if sin y y sin at, Find d y d if 4y Find d y d if sin 1 cos y.

9 d y 65. Find d if y 4. Find the derivative of each of the following functions f ( ) ln( 8) 67. f ( ) ln(3 3 ) 68. f ( ) ln(cos3 ) f ( ) e cos 70. f log4 71. f( ) 4 e 3 ( ) log 1( ) 7. y ln( e ) 73. log( ) y 10

10 74. Find the equation of the tangent to the graph of y 3 at 1.. f ( h) f ( ) Use the it definition of derivative f '( ) to find the derivative of h 0 h each of the following functions. 75. y f ( ) Find the equation of the tangent to the graph of 1 f ( ) at 3. 7

11 78. Find the coordinates where the tangent to the graph of -ais. y 8 3 is parallel to the 79. Find the equation of the tangent and normal to the graph of y ( 4 4) at. 80. Ma wants to make a bo with no lid from a rectangular sheet of cardboard that is 18 inches by 4 inches. The bo is to be made by a cutting a square of side from each corner and folding up the sides. Find the value of that maimizes the volume.

12 81. A rectangle is to be inscribed in a semicircle of radius 4 with one side on the semicircle s diameter. What os the largest area this rectangle can have? y y The range of a projectile is R v o sin, where vo is the initial velocity, g is the g acceleration due to gravity and is a constant, and is the firing angle. Find the value of that maimizes the projectile s range If the position function of a particle is ( t) t 8t t, t 0, find the values of t where the particle is changing direction.

13 Find the linearization L( ) f ( a) f '( a)( a) at = a for each of the following functions at the given value of a f ( ) 3 at a. 85. f a ( ) 9 at f ( ) at a f a ( ) 9 at f ( ) at a 1

14 89. f ( ) ln( 1) at a Oil spilled from a tanker spreads in a circle hose circumference increases at a rate of 40 ft/sec. How fast is the area of the spill increasing when the circumference of the circle is 100 feet A spherical balloon is increasing at a rate of 7 in / sec. How fast is the radius of the balloon increasing when the radius is 3 inches?

15 9. Cars A and B leave a town at the same time. Car A heads due south at a rate of 80 km/hr and car B heads due west at a rate of 60 km/hr. How fast is the distance between the cars increasing after 3 hours? 93. A cylindrical tank with a radius of 6 meters is filling with fluid at a rate of m / sec. How fast is the height increasing? 94. The side of an equilateral triangle is increasing at a rate of 7 in/sec. How fast is the triangle s area increasing when the sides of the triangle are each 18 inches?

16 Evaluate each of the following integrals analytically d 96. d d (5 3 6) d (3 16 ) d 100. (cos 5sin ) d 101. sec (sec tan ) d 10. (sec ) d ( 3) d e d ln( 3 ) e d 106. ln e d

17 107. Use a Riemann sum to find the area under the curve using 4 subintervals. y from =1 to = 108, Find the eact area under the curve graphing utility. y from =1 to =. Check using a 109. Find the area bounded by the function f ( ), the -ais, over the closed interval [0, 3]. Check using a graphing utility. (Hint: Sketch the graph of the function over the given interval.)

18 Evaluate each of the following definite integrals analytically. Check you answer using a graphing utility / cos d 111. / 9 0 d ( ) d d 114. / sin d 115. / 5 0 e d