AP Calculus AB Semester 2 Practice Final

Transcription

1 lass: ate: I: P alculus Semester Practice Final Multiple hoice Identify the choice that best completes the statement or answers the question. Find the constants a and b such that the function f( x) = Ï 8, x Ô Ì ax + b, < x < 8, x ÓÔ is continuous on the entire real line. a =,b = a =, b = 0 a =, b = a =, b = a =, b = Find all values of c such that f is continuous on, ˆ. f( x) = c = 0 c = Ï Ô x, x c Ì ÓÔ x, x > c + + +,, Find an equation of the tangent line to the graph of the function f( x) = x at the point 7,ˆ. y = x + 7 y = x y = x + y = x y = x Find an equation of the line that is tangent to the graph of f and parallel to the given line. f( x) = x, 60x y + = 0 y = 60x y = x + y = 60x + y = 60x and y = 60x + y = 60x and y = 60x +

2 I: Find the x-values (if any) at which the function f( x) = discontinuities are removable? x + x 7x 8 is not continuous. Which of the no points of continuity no points of discontinuity x = (not removable), x = (not removable) x = (not removable), x = (removable) x = ( removable), x = (not removable) 6 Find the derivative of the function f( x) = + cos x. x x 6 x 6 x 6 x 6 x 6 + sinx sinx sinx sinx + sinx 7 Suppose the position function for a free-falling object on a certain planet is given by s( t) = t + v 0 t + s 0. silver coin is dropped from the top of a building that is 78 feet tall. etermine the average velocity of the coin over È the time interval ÎÍ, 4. 4 ft/sec 6 ft/sec ft/sec ft/sec 6 ft/sec 8 Suppose the position function for a free-falling object on a certain planet is given by s( t) = t + v 0 t + s 0. silver coin is dropped from the top of a building that is 60 feet tall. Find the instantaneous velocity of the coin when t = 4. ft/sec ft/sec 6 ft/sec ft/sec 04 ft/sec Use implicit differentiation to find an equation of the tangent line to the ellipse x + y 88 = at, ˆ. y = x + 4 y = x + 4 y = 4x + 6 y = x + 4 y = 7x + 6

3 I: 0 point is moving along the graph of the function y = sin4x such that dx dt = centimeters per second. Find dy dt when x = π. dy dt = cos π dy dt = 4cos 4π dy π = cos dt dy dt = 4cos π dy dt = cos 4π spherical balloon is inflated with gas at the rate of 600 cubic centimeters per minute. How fast is the radius of the balloon increasing at the instant the radius is 70 centimeters? dr = dt 4π cm/min dr = 8π cm/min dt dr = 4π cm/min dt dr = dt 4π cm/min dr = dt 8π cm/min conical tank (with vertex down) is 0 feet across the top and 8 feet deep. If water is flowing into the tank at a rate of 0 cubic feet per minute, find the rate of change of the depth of the water when the water is 0 feet deep. 00π ft/min 00π ft/min 8 00π ft/min 8 0π ft/min 8 000π ft/min ladder 0 feet long is leaning against the wall of a house. The base of the ladder is pulled away from the wall at a rate of 4 feet per second. onsider the triangle formed by the side of the house, the ladder, and the ground. Find the rate at which the area of the triangle is changed when the base of the ladder is 8 feet from the wall. Round your answer to two decimal places. 8.0ft /sec.76ft /sec.66ft /sec 6.ft /sec.4ft /sec 4 Find any critical numbers of the function g( t) = t t, t <

4 I: Locate the absolute extrema of the function f( x) = x È x on the closed interval ÎÍ 0, 4. absolute max: 4, ˆ ; absolute min:, ˆ absolute max: 4, ˆ ; no absolute min absolute max:, ˆ ; absolute min: 4, ˆ no absolute max; absolute min: 4, ˆ no absolute max or min 6 etermine whether Rolle's Theorem can be applied to f( x) = x + x on the closed interval È ÎÍ 0,. If Rolle's Theorem can be applied, find all values of c in the open interval 0, ˆ such that f ( c) = 0. Rolle's Theorem applies; c =, c = 6 Rolle's Theorem applies; c = 6 Rolle's Theorem applies; c = 6 Rolle's Theorem applies; c = 6, c = Rolle's Theorem does not apply 7 etermine whether the Mean Value Theorem can be applied to the function f( x) = x on the closed interval [0,]. If the Mean Value Theorem can be applied, find all numbers c in the open interval (0,) such that f ( c) = f( ) f( 0). 0 MVT applies; MVT applies; MVT applies; 0. MVT applies; MVT does not apply 8 company introduces a new product for which the number of units sold S is S( t) = 600 ˆ where t is the time in + t months since the product was introduced. uring what month does S ( t) equal the average value of S( t) during the first year? November March ugust May September Find the point of inflection of the graph of the function f( x) = sin x È on the interval 0,6π ÎÍ. π,ˆ 0,0ˆ 4π,0ˆ π,0ˆ π,0ˆ 0 Find the points of inflection and discuss the concavity of the function f( x) = x x +. no inflection points; concave down on, ˆ inflection point at x = ; concave up on, ˆ no inflection points; concave up on, ˆ inflection point at x = 0; concave up on,0ˆ; concave down on 0, ˆ inflection point at x = ; concave down on, ˆ 4

5 I: Find the limit. lim x x Find the limit. lim x x ˆ 4x 8 Find the indefinite integral 7tan s + ds. 7tans 6s + 7tans + 8s + 7tans + 6s + 7 tan s + s + 7 tan s + s + 4 n evergreen nursery usually sells a certain shrub after 6 years of growth and shaping. The growth rate during those 6 years is approximated by dh = 0.t + 7, where t is the time in years and dt h is the height in centimeters. The seedlings are 6 centimeters tall when planted (t = 0). Find the height after t years. h(t) = 0.t + 7t + 6 h(t) = 0.t + h(t) = 0.t + 7t + 6 h(t) = 0.t + 6 h(t) = 0.t + t The rate of growth dp of a population of dt bacteria is proportional to the square root of t, where P is the population size and t is the time in days ( 0 t 0). That is, dp = k t. The initial dt size of the population is 400. fter one day the population has grown to 00. stimate the population after 8 days. Round your answer to the nearest integer. P( 8) 800 bacteria P( 8) 76 bacteria P( 8) 78 bacteria P( 8) 66 bacteria P( 8) 68 bacteria 6 ball is thrown vertically upwards from a height of 7 ft with an initial velocity of ft per second. How high will the ball go? Note that the acceleration of the ball is given by a(t) = feet per second per second ft 4.66 ft 4.76 ft 4.44 ft 0.76 ft

6 I: 7 Find the average value of the function f( x) = 48 x over the interval x Find the indefinite integral of the following function and check the result by differentiation. 4x x 4 + dx x x + + x x x + + Find the indefinite integral of the following function and check the result by differentiation. ( x) x dx 0 Find the indefinite integral z cos z dz. cos z + sinz sinz + sinz + sinz + Use the Trapezoid Rule to approximate the value of the definite integral x dx with n = 4. Round your answer to four decimal places Find a function f that has derivative 4x + and with graph passing through the point (,). f( x) = 4x + x + f( x) = x 0 x x 4 x + x x x 4 x x 4 x x f( x) = x + x + f( x) = 4 x f( x) = x + x + 6 6

7 I: Find the indefinite integral x 6x dx. + Find x ln x 4 ˆ dx. 8 ln 6x ln 6x + + ln 6x + + x 6x 4 + x + 8ln 6x Find the indefinite integral lnx x ( lnx) ( lnx) + ( lnx) + ( lnx) + 0 ( lnx) 0 x dx. x ln x 4 ˆ dx = ln x 4 + x ln x 4 ˆ dx = ˆ ln ln x4 4 + x ln x 4 ˆ dx = 4ln ln x 4 ˆ + x ln x 4 ˆ dx = 4 ln x4 + x ln x 4 ˆ dx = 4ln x Find tan7θ dθ. tan 7θ dθ = 7ln sec 7θ + tan 7θ dθ = 7 tan 7θ dθ = 7 tan 7θ dθ = 7 tan 7θ ln cos 7θ + ln cos 7θ + ln sec 7θ + dθ = 7ln cos 7θ + 7 Find the indefinite integral e 6x x 0 dx. 4 e 6x + x e 6x + x ln 6x ˆ + 4 e 6x x 0 e 6x + 7

8 I: 8 Find the derivative of the function f( x) = arcsin( x ). ( x ) + 4x x ( x ) ( x ) + ( x ) Find the indefinite integral. 4 + ( x 4) dx arctan x 4 ˆ + 4arctan x 4 ˆ 4 + arctan x 4 ˆ + 4 arctan x 4 ˆ arctan x 4 ˆ + 40 Solve the differential equation. y = 7x y y = 7x + lny = 7x + y = 7x + lny = 7x + y = 7x + 4 Solve the differential equation. y = x + yˆ ln + y = x + ln + y = x + ln + y = x + 4ln + y = x 4 + ln + y = x + 4 Find the area of the region bounded by the graphs of the algebraic functions. f(x) = x 8 g( x) = x 8 = = 4 = 4 = 6 = 7 8

9 I: 4 Find the volume of the solid generated by revolving the region bounded by the graphs of the equations y = x, y = 0, and x = about the line x = π 00 π 40 0 π 44 Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the line y = 8. 4 Find the volume of the solid generated by revolving the region bounded by the graphs of the equations about the x-axis. y =, y = 0, x =, x = x 7 0 π 0 π 7 0 π 0 π 0 π y = x, y = 7, x = 0 40 π 4 π π 67 6 π 67 π

10 I: P alculus Semester Practice Final nswer Section MULTIPL HOI

11 I: