AP CALCULUS BC SUMMER ASSIGNMENT

Transcription

1 AP CALCULUS BC SUMMER ASSIGNMENT Work these problems on notebook paper. All work must be shown. Use your graphing calculator only on problems -55, 80-8, and 7. Find the - and y-intercepts and the domain and range, and sketch the graph. No calculator.. y. y 9. y. y sin, 5. y cos, 6. y tan, 7. y cot, 8. y sec, 9. y csc, 0.. y e. y ln, y,. 7, y,, 0 0 Find the asymptotes (horizontal, vertical, and slant), symmetry, and intercepts, and sketch the graph. No calculator.. y 5. 9 y 6. y 7. y Solve. No calculator ( ) ( ) ( 5) Evaluate. No calculator cos 6. tan 8. sin. sec 9. 5 tan 5. csc 0. 7 sin 6. cos cot Evaluate. No calculator.. tan cos. sec Arc sin. cos sin. sec Arc tan

2 Solve. Give eact answers in radians, 0. No calculator. 5. cos cos 0 6. sin cos 7. sin( ) cos 8. cos( ) 0 9. csc csc 0 0. tan sec. cos 0. tan( ). sin( ) 0 Solve. Show all steps. Give the eact answer and then use your calculator, and give decimal answers correct to three decimal places.. e 7 5. e 5e e e e 9. ln The number of students in a school infected with the flu t days after eposure is modeled by the function 00 P( t). t e (a) How many students were infected after three days? (b) When will 00 students be infected? (c) What is/are the horizontal asymptote(s) of P(t)? 55. Eponential growth is modeled by the function After an hour, the number of bacteria is 00. (a) How many bacteria are there after four hours? (b) After how many hours will there be 8000 bacteria? Use the figure to find the it. No calculator. kt n n0e. A culture contains 500 bacteria when 0 t. 56. f ( ) 57. f ( ) 58. f ( ) 59. f ( ) f ( ) 6. f ( ) 5

3 Evaluate. Show supporting work for each problem (algebraic steps or sketch). No calculator ( 5) , f ( ) (a) f ( ), 5 6 f ( ), (a) f ( ), (b) f ( ) (b) f () (c) f ( ) Use the definition of the derivative to find the derivative. No calculator. f ( ) h0 f ( h) f ( ) (You must know this formula.) h 76. f ( ) f ( ) f ( ) 79. f ( ) Rates and Areas The following graph gives the rate of rainfall (velocity) over a period of time that is collected in a basin. Each square is hr Each y square. inch per pour 80. What does a negative rate mean? 8. How much water (position) is in the basin after 5 hours? hours? hours? 8. At what rate is the rate of rainfall changing (acceleration) at hours?

4 The motion of a projectile follows the position equation s(t) = -6t + 96t What it the average velocity of the projectile on the interval [, 7] 8. Use the definition of the derivative to find the derivative at time t = t At what time does the projectile strike the ground? 86. What is its instantaneous velocity when it strikes the ground? 87. What is the maimum height the projectile reaches? 88. What is the instantaneous velocity at t = seconds? Use the dferentiation rules (power rule, product rule, quotient rule, chain rule, arithmic dferentiation) to find the derivative. Do not leave negative eponents or comple fractions in your answers. No calculator. 89. f ( ) ( ) 6 5 f f ) cos f ( ) 9. f ( ) 5 f ( ) 9. f ( ) sin ( 96. ( ) tan f f ( ) cot 98. f ( ) sec 99. f ( ) csc( ) 0. f ( ) cos 0. f ( 0. f ( ) sec 07. f ( ) e 08. f ) ln 0. f ) lnsin (. 00. f ( ) sin ) tan 0. f ( ) csc f ( ) cot 06. f f ( ) ( 09. f ( ) ln 5 tan ( ). f ( ) e cos e. Given the function f ( ) 7 No calculator. (a) Use dferentiation rules to find f () (b) Write the equation of the tangent line to f at (,5). Leave your equation in point-slope form. dy Use implicit dferentiation to find. No calculator. d. 9 y y 0 5. y 7 y 9 6. sin( y ) 5 Identy all intervals on which the function is increasing, decreasing, concave up, and concave down, and identy all local etrema and inflection points. Sketch the graph. Justy your answers. No calculator. 7. f ( ) 8. f ( ) f ( ) sin cos 0. f ( ). f ( ) e. f ( ) e sin 9 f on,0. Find all global etrema of ( ) Justy your answer. No calculator.

5 8. A particle moving along the y-ais has position equation y( t) cos t. On 0 t 8, identy all points at which the particle is stopped, and identy all intervals on which the particle is moving up, moving down, speeding up, and slowing down. Justy your answers. No calculator. 5. Coffee is draining from a conical filter into a cylindrical coffeepot at the rate of 0 cubic inches per minute. The coffee filter has diameter 6 inches and height 6 inches, and the coffee pot has diameter 6 inches. (a) What is the rate of change of the height of the coffee in the coffeepot at the instant when the coffee is 5 inches deep in the filter? No calculator. (b) What is the rate of change of the height of the coffee in the coffeepot at the instant when the coffee is inches deep in the filter? No calculator. (c) What is the rate of change of the height of the coffee in the filter at the instant when the coffee is 5 inches deep in the filter? No calculator. (d) What is the rate of change of the height of the coffee in the filter at the instant when the coffee is inches deep in the filter? No calculator. 6. A 5-foot ladder leans against the side of a building. The bottom of the ladder is being pushed towards the wall at a rate of feet per second and the ladder remains in contact with the wall. (a) What is the speed that the top of the ladder is sliding up the wall at the instant when the bottom of the ladder is 7 feet from the wall? No calculator. (b) What is the rate of change of the angle formed by the bottom of the ladder and the ground at the instant when the bottom of the ladder is 7 feet from the wall? G decimal answers correct to three decimal places. (c) What is the speed that the top of the ladder is sliding up the wall at the instant when the bottom of the ladder is 5 feet from the wall? No calculator. (d) What is the rate of change of the angle formed by the bottom of the ladder and the ground at the instant when the bottom of the ladder is 5 feet from the wall? Give decimal answers correct to three decimal places.

6 7. Show all steps using Calculus to find an eact answer. Use your calculator, and give decimal answers correct to three decimal places. Use L Hopital s Rule to determine each it. 8. tan ln 0 tan (5) 9. 0 sin () 0. ln. e.. 0 ln( ). 6 Parametrics, Vectors, and Polars Review 5. Two particles move in the y-plane. For any time t 0, the position of particle A is given by = t and y = (t ), and the position of particle B is given by = t/ and y = t/. a. Find the position, velocity, and acceleration vectors for each particle at time t =. b. Find the speed of each particle at time t =. c. Find the line tangent to each graph at time t = d. If the particles collide, find the time and location of collision. 6. Let the two polar graphs, r = and r = ( sinθ) lie on a plane. a. Sketch each curve. b. Find the line tangent to each curve at θ = π/ c. Find the intersection(s) between the two curves.