AP Calculus AB
Fall Mid-Term Exam
Calculator permitted portion.
v  t   3  4.1cos  0.9t 
76. a  t   v  t   4.1sin  0.9t   0.9 
a  4   3.69sin  3.6   1.632900  C
A   r 2 , C  2 r  20 m,
78.
dr
 0.2 m
sec
dt
dA
dr
dr
 2 r
C
dt
dt
dt
2
dA
m
 20 0.2  4
C
sec
dt
h  x   f  g  x    h  x   f   g  x   g   x 
79. h 1  f   g 1  g  1
 f   1  2    5   2   10  D
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
79. Relative maximums exist when the derivative
goes from positive to negative. This only happens
for f near x = b. →A
dy
 1 y2
dx
d2y
1
dy
81. 2 
 2 y 
2
dx
dx
2 1 y

y
1 y2
1 y2   y  B
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
Free Response
a) Domain : x : x  since x 2  x  1  0 for all x.
b)
c) y  2 and y  2
f   x  changes sign from negative to positive at
x  2, hence there is a absolute minimum at
d) x  2 and the maximum value of f  x  is
constrained by the asymptote at y  2.
f  2   2.309  Range : 2.309  y  2
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
No calculator portion.
dy
 cos  3 x   3  3cos  3 x 
1. y  sin  3 x  
dx
e
e x  cos x  2 x
lim

2
x 0
x  2x
 0
 cos  0   2  0 
0
2
 2 0
0

0
Use L'Hopital
2.
e x  cos x  2 x
e x  sin x  2
lim
 lim
2
x 0
x 0
x  2x
2x  2
1 0  2 1

 C
02
2
x3  2 x 2  3x  4
x3
1 1
 lim 3  lim   C
6. lim 3
2
x  4 x  3 x  2 x  1
x  4 x
x  4
4
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
3 x
f  x   ln  x  4  e
9.

3 x
1
1

3
e
3 x
f  x 
1


3
e





3 x
x4e
x  4  e 3 x
1  3e  
1 3
2

f  0 

 A
3 0 
4 1
5
 0  4  e
3 0
1, 7  ,  2, 2 
16.
7   2  9
m
  3  f  1  C
1   2  3
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
Free Response
for x  2
2 x  1,

f  x  1 2
 2 x  k , for x  2
1
2
2  2  1   2  k  4  1  2  k
2
a) k  3
f  x  is continuous at x  2 if k  3,
since lim f  x   lim f  x  when k  3.
x2
x2
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
f   x   lim
x2
 2 x  1  5
x2
 lim
x2
2  x  2
x2
2
1 2

1 2
x

3

5
x  4



2

f   x   lim 
 lim 2
x2
x2
x2
x2
1 2
1
x

4
 x  2  x  2 


 lim 2
 lim 2
x2
x2
x2
x2
1
 lim  x  2   2
x2 2
b) Since lim f  x  lim f  x ,
 
 
x2
x2
f  x  is differentiable at x  2.
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.
AP Calculus AB
Fall Mid-Term Exam
for x  2
2 x  1,

f  x  1 2
 2 x  k , for x  2
1
2
c)
2  2   1  5;  2   4  6  5  6
2
If k  4, f  x  is not differentiable at x  2,
since f  x  is not continuous at x  2.
AP Calculus Free Response Questions and Solutions 1979-1988. Copyright © 2004, AP Calculus AB 2005 Free Response Questions Form B Copyright ©
2005, and AP Calculus Multiple Choice Question Collection Copyright © 2005 by the College Entrance Examination board. Reprinted with permission. All
rights reserved apcentral.collegeboard.com. This material may not be mass distributed, electronically or otherwise. This publication and any copies made
from it may not be resold.