A.P. Calculus - ln and e Practice test
1) y = ln("14 x 3 )
2) y = (ln2x )
5
!
!
(
3) y = ln 4 " x 2
)
4) y = ln(ln(cos x ))
!
!
5) y =
x4
1" 2ln x
!
x2 +1
x 2 "1
6) y = ln 4
!
7) y = e"4 x +4
8) y = 2e
2x
3
!
!
9) y = xe 2 " e x
10) y = 4
2x"1
!
!
11) y = " x x "
!
12) x 4 + e xy " y 2 = 20
!
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Stu Schwartz
Find the integral or definite integral as indicated
1
13) #
dx
5" x
!
14)
7x dx
2
"8
#x
!
15)
#
8x 3 " 7x 2 "1
dx
2x
!
x 2 " 5x + 2
dx
x "2
16)
#
18)
" (e
!
17)
"
(ln x) 3
dx
2x
4x
+ e) dx
!
!
19)
" 2e x (1+ e x )4 dx
!
e x + e -x
dx
e x " e"x
20)
#
22)
# "e
!
21)
# (e
x
" e -x ) 2 dx
!
cos x
sin x dx
!
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Stu Schwartz
4
23)
"
1
e2
e x
dx
2 x
24)
2
" x dx
1
!
!
25) Find relative extrema and inflection points of y = xe"x
work is understandable.
Use calculus methods and be sure that your
!
26) Given f ( x ) = e
x
2
a) Find the area of the region R bounded by the line y = e, the graph of f and the y-axis.
!
!
b) Find the volume of the solid generated by revolving R about the x-axis.
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Stu Schwartz
A.P. Calculus - ln and e Practice test - Solutions
!
2) y = (ln2x )
dy
1
3
=
"42x 2 ) =
3 (
dx "14 x
x
5(ln2x )
dy
4 1
= 5(ln2x )
(2) =
dx
2x
x
(
3) y = ln 4 " x 2
!
)
(
dy
=
dx
)
# "2 &
(
$ x'
(1" 2ln x )(4 x 3 ) " x 4 %
(1" 2ln x )
2
6x " 8x ln x
=
2
(1" 2ln x )
!
[
8) y = 2e
!
2x
!
2x " 2 %
dy
4e
= 2e 3 $ ' =
# 3&
dx
3
10) y = 4
!
!
dy
=4
dx
4x3
!
!
3
2x"1
2x"1
(ln 4 )
4 2x"1 (ln 4 )
1
(2) =
2 2x "1
2x "1
12) x 4 + e xy " y 2 = 20
!
dy
= " x +1 x "#1 + x "" x ln"
dx
]
3
2x
dy
= e2 " ex
dx
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x2 +1
x 2 "1
1
ln( x 2 + 1) " ln( x 2 "1)
4
dy 1 # 2x
2x &
= % 2
" 2 (
dx 4 $ x + 1 x "1'
y=
3
dy
= "4e"4 x +4
dx
11) y = " x x "
!
3
dy
1
1
"tan x
=
("sin x ) =
dx ln(cos x ) cos x
ln(cos x )
6) y = ln 4
!
9) y = xe 2 " e x
!
!
x4
1" 2ln x
4
4) y = ln(ln(cos x ))
!
7) y = e"4 x +4
!
!
dy
1
" 2
=
" 2 =
dx 4 " x 2
4"x 2
5) y =
!
5
1) y = ln("14 x 3 )
" dy
dx
dx %
dy
+ e xy $ x
+ y ' ( 2y
=0
# dx
dx
dx &
dx
dy 4 x 3 + ye xy
=
dx 2y ( xe xy
Stu Schwartz
Find the integral or definite integral as indicated
1
13) #
dx
5" x
14)
"ln 5 " x + C
7
ln x 2 " 8 + C
2
!
!
15)
#
#
!
) %$ 4 x
8x 3 " 7x 2 "1
dx
2x
2
"
"
(ln x) 3
dx
2x
u = ln x
!
7x 1 &
" ( dx
2 2x '
du =
!
!
4
1 (ln x )
(ln x ) + C
=
2 4
8
19)
!
x 4
" 2e (1+ e )
u = 1+ e x
!
5
2
1+ e x ) + C
(
5
# (e
!
# (e
2x
x
" e -x ) 2 dx
4x
+ e) dx
e x + e -x
dx
x
" e"x
#e
u = e x " e"x
du = e x dx
2 " u 4 du
21)
" (e
1 4x
e + ex + C
4
20)
dx
division)
x2
" 3x " 4 ln x " 2 + C
2
!
x
4 &
) %$ x " 3 " x " 2 (' dx (long
18)
!
1
dx
x
4
#
x 2 " 5x + 2
dx
x "2
#
4 x 3 7x 2 1
"
" ln x + C
3
4
2
17)
16)
!
7x dx
2
"8
#x
du = (e x " e"x ) dx
du
u
ln e x " e"x + C
#
22) # "e cos x sin x dx
u = cos x,du = "sin x dx
!
" 2 + e"2x ) dx
"# $ e u du
e 2x
e"2x
" 2x "
+C
2
2
!
"#e cosx + C
!
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Stu Schwartz
4
"
23)
1
!
e2
e x
dx
2 x
u = x,du =
24)
2
" x dx
1
1
x = 1,u = 1
2 x
x = 4,u = 2
e2
!
2ln x ]1
2
2lne 2 " 2ln1 = 4 lne " 0 = 4
eu ] = e2 " e
1
!
25) Find relative extrema and inflection points of y = xe"x
work is understandable.
y " = #xe#x + e#x = 0
e#x (1# x ) = 0
$ 1'
x = 1!!!!!!!!Rel.Max at &1, )
% e(
26) Given f ( x ) = e
!
x
!
Use calculus methods and be sure that your
y "" = xe#x # e#x # e#x
e#x ( x # 2) = 0
x =2
$ 1'
Inf. Pt. at & 2, 2 )
% e (
2
!
a) Find the area of the region R bounded by the line y = e, the graph of f and the y-axis.
!
!
2
A=
# (e " e ) dx
x 2
0
2
A = ex " 2e x 2 ] 0 = 2e " 2e " (0 " 2) = 2
b) Find the volume of the solid generated by revolving R about the x-axis.
2
V = " $ (e 2 # e x ) dx
0
]
2
V = "(e 2 x # e x ) = (e 2 + 1)" = 8.389" = 26.355
0
!
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Stu Schwartz