CALCULUS

CALCULUS - Transcript

AP Calculus AB 1998 Free Response Questions

These materials are intended for non commercial use by AP teachers for course and exam preparation permission for any other use must be sought from the Advanced Placement Program Teachers may reproduce them in whole or in part in limited quantities for face to face teaching purposes but may not mass distribute the materials electronically or otherwise These materials and any copies made of them may not be resold and the copyright notices must be retained as they appear here This permission does not apply to any third party copyrights contained herein

These materials were produced by Educational Testing Service ETS which develops and administers the examinations of the Advanced Placement Program for the College Board The College Board and Educational Testing Service ETS are dedicated to the principle of equal opportunity and their programs services and employment policies are guided by that principle The College Board is a national nonpro t membership association dedicated to preparing inspiring and connecting students to college and opportunity Founded in 1900 the association is composed of more than 4 200 schools colleges universities and other educational organizations Each year the College Board serves over three million students and their parents 22 000 high schools and 3 500 colleges through major programs and services in college admission guidance assessment nancial aid enrollment and teaching and learning Among its best known programs are the SAT the PSAT NMSQT and the Advanced Placement Program AP The College Board is committed to the principles of equity and excellence and that commitment is embodied in all of its programs services activities and concerns









For further information contact www collegeboard com Copyright 2002 by College Entrance Examination Board All rights reserved College Board Advanced Placement Program AP and the acorn logo are registered trademarks of the College Entrance Examination Board



1998 AP Calculus AB Free Response Questions CALCULUS AB Section II Time 1 hour and 30 minutes Number of problems 6 Percent of total grade 50
A GRAPHING CALCULATOR IS REQUIRED FOR SOME PROBLEMS OR PARTS OF PROBLEMS ON THIS SECTION OF THE EXAMINATION REMEMBER TO SHOW YOUR SETUPS AS DESCRIBED IN THE GENERAL INSTRUCTIONS

1 Let R be the region bounded by the x axis the graph of y a Find the area of the region R



x and the line x 4

b Find the value of h such that the vertical line x h divides the region R into two regions of equal area c Find the volume of the solid generated when R is revolved about the x axis d The vertical line x k divides the region R into two regions such that when these two regions are revolved about the x axis they generate solids with equal volumes Find the value of k

Copyright

1998 College Entrance Examination Board All rights reserved

1998 AP Calculus AB Free Response Questions 2 Let f be the function given by f x 2xe2x a Find lim f x and lim f x
x x

b Find the absolute minimum value of f Justify that your answer is an absolute minimum c What is the range of f d Consider the family of functions de ned by y bxebx where b is a nonzero constant Show that the absolute minimum value of bxebx is the same for all nonzero values of b

Copyright

1998 College Entrance Examination Board All rights reserved

1998 AP Calculus AB Free Response Questions

v t

90 80
Velocity feet per second

t seconds 0 5 10 15 20 25 30 35 40 45 50

v t feet per second 0 12 20 30 55 70 78 81 75 60 72

70 60 50 40 30 20 10
O t

5 10 15 20 25 30 35 40 45 50
Time seconds

3 The graph of the velocity v t in ft sec of a car traveling on a straight road for 0 t 50 is shown above A table of values for v t at 5 second intervals of time t is shown to the right of the graph a During what intervals of time is the acceleration of the car positive Give a reason for your answer b Find the average acceleration of the car in ft sec2 over the interval 0 t 50 c Find one approximation for the acceleration of the car in ft sec2 at t 40 Show the computations you used to arrive at your answer
50

d Approximate
0

v t dt with a Riemann sum using the midpoints of ve subintervals of

equal length Using correct units explain the meaning of this integral

Copyright

1998 College Entrance Examination Board All rights reserved

1998 AP Calculus AB Free Response Questions 4 Let f be a function with f 1 4 such that for all points x y on the graph of f the slope is 3x2 1 given by 2y a Find the slope of the graph of f at the point where x 1 b Write an equation for the line tangent to the graph of f at x 1 and use it to approximate f 1 2 dy 3x2 1 c Find f x by solving the separable di erential equation with the initial dx 2y condition f 1 4 d Use your solution from part c to nd f 1 2

Copyright

1998 College Entrance Examination Board All rights reserved

1998 AP Calculus AB Free Response Questions 5 The temperature outside a house during a 24 hour period is given by F t 80 10 cos t 0 t 24 12

where F t is measured in degrees Fahrenheit and t is measured in hours a Sketch the graph of F on the grid below

100 Degrees Fahrenheit


90 80 70

0

b Find the average temperature to the nearest degree Fahrenheit between t 6 and t 14 c An air conditioner cooled the house whenever the outside temperature was at or above 78 degrees Fahrenheit For what values of t was the air conditioner cooling the house d The cost of cooling the house accumulates at the rate of 60 05 per hour for each degree the outside temperature exceeds 78 degrees Fahrenheit What was the total cost to the nearest cent to cool the house for this 24 hour period

Copyright

1998 College Entrance Examination Board All rights reserved





60

6

12 Time in Hours

18

24

1998 AP Calculus AB Free Response Questions 6 Consider the curve de ned by 2y 3 6x2 y 12x2 6y 1 dy 4x 2xy a Show that 2 dx x y2 1 b Write an equation of each horizontal tangent line to the curve c The line through the origin with slope 1 is tangent to the curve at point P Find the x and y coordinates of point P

END OF EXAMINATION
Copyright

1998 College Entrance Examination Board All rights reserved