[email protected] AB 2004 SCORING GUIDELINES (Form B) Question 2 APR3 - Solutions For 0 I t I 3 1, the rate of change of the number of mosquitoes on Tropical Island at time t days is (3 modeled by R ( t ) = 5ficos - mosquitoes per day. There are 1000 mosquitoes on Tropical Island at time t = 0. (a) Show that the number of mosquitoes is increasing at time t = 6. (b) At time t = 6, is the number of mosquitoes increasing at an increasing rate, or is the number of mosquitoes increasing at a decreasing rate? Give a reason for your answer. (c) According to the model, how many mosquitoes will be on the island at time t = 3 l ? Round your answer to the nearest whole number. (d) To the nearest whole number, what is the maximum number of mosquitoes for 0 I t I 3 l ? Show the analysis that leads to your conclusion. (a) Since R ( 6 ) = 4.438 > 0, the number of mosquitoes is increasing at t = 6. (b) R'(6) = -1.913 Since R'(6) < 0, the number of mosquitoes is increasing at a decreasing rate at t = 6. 1 : shows that R ( 6 ) > 0 2: { 1 : considers R'(6) 1 : answer with reason 2 : { 1 : integral 1 : answer To the nearest whole number, there are 964 mosquitoes. (d) R ( t ) = 0 when t = 0 , t = 2 . 5 ~or, t = 7 . 5 ~ R ( t ) > 0 on 0 < t < 2 . 5 ~ R ( t ) < 0 on 2 . 5 < ~t <7.5~ R ( t ) > 0 on 7 . 5 < ~ t < 31 The absolute maximum number of mosquitoes occurs at t = 2 . 5 ~or at t = 3 1. There are 964 mosquitoes at t = 3 1, so the maximum number of mosquitoes is 1039, to the nearest whole number. 2 : absolute maximum value 1 : integral 1 : answer 2 : analysis 1 : computes interior critical points . 1 : completes analysis ' APR3S Page 1 Copyright O 2004 by College Entrance Examination Board. ,411 rights reserved. Visit apcentral.collegeboard.com(for AP professionals) and www.collegeboard.com/apstudents(for AP students and parents). Final Draft for Scoring AP Calculus AB-5 APR3 - Solutions A cubic polynomial function f is defined b y f(x) = 4x3 + u2+ bz +k where a, b, and k are constants. The function f has a local minimum at x = - 1 , and the graph o f f has a point of inflection at z = - 2 . (a) Find the values of a and b. ( b ) If J1( z )& = 32, what is the value of k ? U - 1 : f I(x) (a) ff(z)= 122' +2m + b ftl(x) = 24s + 2a 1 :f"(x) 5 : . 1 : fl(-1) = 0 1 : ff'(-2) = 0 1:a, b - (b) Ju1(4r3 + 2 4 x 2 + 3 6 x + t)& = Z' + 8 2 + 18x2 + klZ=' = 27 -tk e=O 2 : antidifferentiation < - 1 > each error 4 : . 1 : expression in I; l : k APR3S Page 2 AP Calculus AB-5 / BC-5 APR3 - Solutions Consider the curve given by q2- x3y = 6 . dy 3x2y - y2 (a) Show that - = dx 2 q - x3 ' (b) Find all points on the curve whose scoordinate is 1, and write an equation for the tangent line at each of these points. (c) Find the scoordinate of each point on the curve where the tangent line is vertical. 1 : implicit differentiation dY 1 : verifies expression for dx (b) When x = 1, y2 - y = 6 4 y2 - y - 6 = 0 (Y - 3)(Y + 2) = 0 y = 3, y = - 2 1 1 : y2 - y = 6 1 : solves for y 2 : tangent lines Note: 014 if not solving an equation of the form y2 - y = k Tangent line equation is y = 3 Tangent line equation is y + 2 = 2(x - 1) (c) Tangent line is vertical when 2 q - x3 = 0 1 ~ ( 2 ~ 2 -) = x O givesx= Oor y = -x2 2 There is no point on the curve with scoordinate 0. 1 1 1 When y = - x 2 , -x5 - - x 2 4 2 5 =6 3 I dY equal to 0 1 : sets denominator of dx 1 1 : substitutes y = -x2 or x = f a 2 into the equation for the curve I 1 : solves for x-coordinate 1I APR3S Page 3 Copyright 0 2000 by College Entrance Examination Board and Educational Testing Service. All rights reserved. AP is a registered trademark of the College Entrance Examination Board.

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