Transcript + C

Unit 5
Logarithmic, Exponential and Other
Transcendental Functions
Review Problems
Calculus 5-R
Find the domain of the function: f(x) = ln(3x + 1).
Find the domain of the function: f(x) = 3 + ln(x - 1).
(1, )
1
Review Problems
Match the graph with the correct function
[A] f(x) = ln x
[B] f(x) = ex-1
[C] f(x) = ln(x - 1)
[D] f(x) = ex
2
Review Problems
Sketch the graph: f(x) = ln|x|.
Solve for x: ln(5x + 1) + ln x = ln 4
3
Review Problems
Solve for x: ln(5x - 1) - ln x = 3.
dy/dx for y = ln(5 - x)6
4
Review Problems
Find the derivative: f(x) = ln(x3 + 3x)3
Find the derivative:
5
x2  1
f ( x )  ln
.
3
2
x (2 x  1)
Review Problems
Find the derivative:
Find the derivative:
6
x2 4x  1
f ( x)  ln 3 3 .
( x  5)
f ( x)  ln
x( x 2  5)
x3  5
.
Review Problems
Differentiate: y = ln(ln tan x)
Find y’ y = ln|2x2 - 5|
7
Review Problems
Find y’ if ln xy = x + y
Use logarithmic differentiation to find
dy
( x  2) 1  x
: y
.
3
dx
4x
2
8
Review Problems
Find the slope of the tangent line to the graph of
y = ln x2 at the point where x = e2
Evaluate the integral:
1
X
dx.
Y
Zx
4e
e
ln 4
9
Review Problems
Evaluate the integral:
1
X
dx.
Y
Zax  b
ln|ax + b| + C
Evaluate the integral:
 4x
X
dx.
Y
Zx
e
2
1
-2
10
Review Problems
 7x
X
dx.
Y
Zx
e
Evaluate the integral:
2
1
x  x 1
X
dx.
Y
Z x 1
2
Evaluate the integral:
2
x+
11
ln(x2 + 1) + C
Review Problems
8x  9 x  8
X
dx.
Evaluate the integral: Y
Z x 1
2
2
8x +
X
Y
Z
ln(x2 + 1) + C
Evaluate the integral: 9 x  9 x  9 dx.
2
x 1
2
9x -
12
ln(x2 + 1) + C
Review Problems
ln x
X
Evaluate the integral: Y
dx.
Zx
+C
Evaluate the integral:
ztan 3x dx.
ln|sec 3x| + C
13
Review Problems
Evaluate the integral:
sin
X
Y
Z
2
x  cos x
dx.
sin x
2
-2 cos x + ln|csc x + cot x| + C
csc x
X
Evaluate the integral: Y
dx.
Zcot x
2
ln|tan x| + C
14
Review Problems
1
sin

X
Evaluate the integral: Y
d .
Z cos
1
.
Differentiate: f ( x ) 
2x 4
(4  e )
15
Review Problems
Match the graph shown with the correct function
[A] f(x) = e (x-1)
[B] f(x) = e-(x-1)
[C] f(x) = ex + 1
[D] f(x) = e-x + 1
16
Review Problems
Differentiate:
f ( x)  4  e .
Differentiate:
ye
17
2x
sin x
.
Review Problems
dy
Find:
if xey + 1 = xy
dx
e
X
dx.
Y
Zx
x
Evaluate the integral:
+C
18
Review Problems
Find the slope of the tangent line to the graph of
y = (ln x)ex at the point where x = 2
Evaluate the integral:
zsin x e
cos x
dx.
-ecosx + C
19
Review Problems
Evaluate the integral:
z19e
 t/5
dt .
-95 e-t/5 + C
1
X
Evaluate the integral: Y
Zx e
2 3/ x
dx.
+C
20
Review Problems
dy
Find
if y = 3xx3
dx
3xx2[3 + (ln 3)x]
Differentiate: y = x1-x
x1-x
21
Review Problems
Differentiate y = xx
xx[1 + ln x]
Evaluate the integral:
zx3
x2
dx.
+C
22
Review Problems
Find the area bounded by the function f(x) = 2-x, the
x-axis, x = -2, and x = 1
A certain type of bacteria increases continuously at a
rate proportional to the number present. If there are
500 present at a given time and 1000 present 2 hours
later, how many will there be 5 hours from the initial
time given?
2828
23
Review Problems
A certain type of bacteria increases continuously at a
rate proportional to the number present. If there are
500 present at a given time and 1000 present 2 hours
later, how many hours (from the initial given time) will it
take for the numbers to be 2500? Round your answer
to 2 decimal places.
4.64
A mold culture doubles its mass every three days. Find
the growth model for a plate seeded with 1.6 grams of
mold. [Hint: Use the model y = Cekt where t is time in
days and y is grams of mold.]
1.6e0.23105t
24
Review Problems
The balance in an account triples in 21 years. Assuming
that interest is compounded continuously, what is the
annual percentage rate?
5.23%
The balance in an account triples in 20 years. Assuming
that interest is compounded continuously, what is the
annual percentage rate?
5.49%
25
Review Problems
A radioactive element has a half-life of 50 days. What
percentage of the original sample is left after 85 days?
30.78%
A radioactive element has a half-life of 40 days. What
percentage of the original sample is left after 48 days?
43.53%
26
Review Problems
The number of fruit flies increases according to the law of
exponential growth. If initially there are 10 fruit flies and
after 6 hours there are 24, find the number of fruit flies
after t hours.
y = 10eln(12/5)t/6
Determine whether the function y = 2cos x is a solution
to the differential equation
y   y   0.
No
27
Review Problems
 x2
verify that y  2e
equation y   xy 
4
 Ce
2 x2
xe x y 3 .
is a solution to the differential
Find the particular solution to the differential
equation
y   sin x
y  C  cos x
given the general solution
and the initial condition
I
F
y GJ 1.
H2 K
y = 1 - cos x
28
Review Problems
Find the particular solution to the differential equation
4
dy
2
3
 y (1  x ) given the general solution y 
4
4
x

x
C
dx
and the initial condition y(0) = 5.
Use integration to find a general solution to the
differential equation
y   x x  1.
y = (x + 1)3/2
29
Review Problems
+C
Use integration to find a general solution to the
differential equation
dy
3

.
dx 1  x
y = 3 ln|1 + x| + C
Use integration to find a general solution to the
differential equation xy   2 y  0.
y=
30
Review Problems
Find the general solution to the first-order differential
equation: (4 - x)dy + 2y dx = 0
y = C(4 - x)2
Find the general solution to the first-order differential
equation: x cos2y + tan y dy = 0
dx
x2 + sec2y = C
31
Review Problems
Find the general solution to the first-order differential
equation: y dx + (y - x)dy = 0
y ln|y| + x = Cy
Find the general solution to the first-order differential
equation: e 2 y y  x 3 .

y=
32
Review Problems
Find the general solution to the first-order differential
x2
equation: y 
e
x

y
.
dy
Find the particular solution of the differential equation
 500  y
dx
that satisfies the initial condition y(0) = 7
y = 500 - 493e-x
33
Review Problems
Find the solution to the initial value problem
(e  cos y) y   x,
y
y(-1) = 0
ey + sin y =
(x2 + 1)
Find the solution to the initial value problem
(1  x )(1  y )  xyy , y(1) = 0
2
2
ln(1 + y2) = 2 ln x + x2 - 1
34
Review Problems
A deposit of $1000 is made into a fund with an annual
interest rate of 5 percent. Find the time (in years)
necessary for the investment to double if the interest is
compounded continuously. Round your answer to 2
decimal places.
13.86 years
A deposit of $1000 is made into a fund with an annual
interest rate of 5 percent. Find the time (in years) necessary
for the investment to triple if the interest is compounded
continuously. Round your answer to 2 decimal places.
22.24 years
35
Review Problems
Find the constant k so that the exponential function
y = 3ekt passes through the points given on the graph.
1 5
k  ln  01703
.
3 3
Solve the differential equation:
2 y   y.
y  Ce x/2
36
Review Problems
Find the particular solution to the differential
equation
given the general
y   sin x
solution y  C  cos x and the initial condition
I
F
y GJ 1.
H2 K
y = 1 - cos x
Find the general solution of the differential equation
sin x
y 
.
cos y
37
sin y + cos x = C
Review Problems
Find the general solution of the differential equation
x2
y e

.
x
y
F
I
G
H2 JK
Evaluate: arccos  1
38
x2
y  e C
2
2
3
Review Problems
L
O
F
I
G
J
M
P
H
K
3
N
Q
Find the exact value: cos arctan  2
3 13
13
2 IO
L
F
arccos G
 J
Find the exact value: sin M
P
H
K
7
N
Q
3 5
7
39
Review Problems
3 IO
L
F
Find the exact value: cos M
arctan G
 J
P
H
K
10
N
Q
10 109
109
Find the exact value: sin(arctan 3)
3 10
10
40
Review Problems
L
O
F
I
G
J
M
P
H
K
74
N
Q
Find the exact value: sin arccos  5
7
74
Find the exact value:
L
3 IO
F
sin M
arccos G

J
P
H
K
10 Q
N
1
10
41
Review Problems
Write an algebraic expression for tan[arcsin x].
x 1 x2
1 x2
Differentiate: f ( x)  arcsin 1  4 x 2 .

42
2x
x 1 4x2
Review Problems
Differentiate: f ( x)  arcsin 1  64 x .
2

8x
x 1  64 x 2
Differentiate: y = arctan ex
ex
2x
1 e
43
Review Problems
Differentiate: h(t) = arccos t2
 2t
1 t4
x
Differentiate: g ( x )  arcsec .
2
2
x x2  4
44
Review Problems
cos x
X
Evaluate the integral: Y
dx.
Z9  sin x
2
X
Evaluate the integral: Y
Z2 x
F
I
G
J
+
C
H K
1
sin x
arctan
3
3
1
4x  1
2
dx.
1
arcsec|2x| + C
2
45
Review Problems
Evaluate the integral:
x3
X
dx.
Y
Zx  9
2
x
1
2
ln(x + 9) + arctan + C
2
3
Evaluate the integral:
x2
X
Y
Z 4 x
2
dx.
x
 4  x  2 arcsin + C
2
2
46
Review Problems
Evaluate the integral:
2 x
X
Y
Z 4 x
2
dx.
x
2 arcsin  4  x 2 + C
2
x
X
Evaluate the integral: Y
Z16  x
4
dx.
2
x
1
arctan
+C
4
8
47
Review Problems
Evaluate the integral:
x
X
Y
Z36  x
4
dx.
1
x2
arctan
+C
12
6
Evaluate the integral:
X
Y
Z
(arctan x ) 3
dx.
2
1 x
1
4
(arctan x ) + C
4
48
Review Problems
1
X
Evaluate the integral: Y
Z  3  4x  x
2
dx.
arcsin(x - 2) + C
Evaluate the integral: X
Y
Zx
1
dx.
2
 4x  7
F
I
+C
G
J
H K
1
x2
arctan
3
3
49
Review Problems
Evaluate the integral:
Evaluate the integral:
X
Y
Z( x  1)
X
Y
Z( x  3)
5
x 2  2 x  24
dx.
x 1
arcsec
+C
5
1
x  6 x  13
2
dx.
1
x3
arcsec
+C
2
2
50
Review Problems
Solve the differential equation:
9  x y   3.
2
x
y = 3 arcsin + C
3
Solve the differential equation:
y
x  1y   .
x
2
y = Cearcsec x
51
Review Problems
1.
1 I
F

, J
G
H3 K
(1, )
2. C
9.
4
5
3. Graph
4.
1
5  e3
5.
9( x 2  1)
x( x 2  3)
6.
7.
8.
10.
6
x 5
x
1 12 x 2
  3
2
x  1 x 2x  1
2
2
9x2


x 4x  1 x3  51
2x
3x 2
 2

x
x

5
2( x 3  5)
2
sec x
tan x[ln(tan x)]
4x
2x2  5
xy  y
2
(
x

2
)
1

x
1
x
3
x  xy


4x3
x  2 1 x2 x
2
e2
L
M
N
ln 4
1
ln|ax + b| + C
a
11. -2
12.
O
P
Q
8x +
x+
1
ln(x2 + 1) + C
2
9
ln(x2 + 1) + C
2
9x -
9
ln(x2 + 1) + C
2
13.
1
(ln x ) 2 +
4
14.
-2 cos x + ln|csc x + cot x| + C
Answers
C
1
ln|sec
3
ln|tan x| + C
3x| + C
15. ln1 sin   C
 8e2 x
(4  e 2 x ) 5
16. D
e2 x
17.
4e
cos x sin
e
2 x
2x
y  ey
18. y
xe  x
1I
F
G
H 2J
K
2e
x +C
x
22.
xx[1 + ln x]
23.
7
2 ln 2
24. 4.64
19.
-ecosx + C 26. 30.78%
20.
-95 e-t/5 + C
1  3/ x
e
+C
3
3xx2[3 + (ln 3)x]
x1-x
1 x
L
O

ln
x
M
P
Nx
Q
3
+C
2 ln 3
2828
1.6e0.23105t
25. 5.23%
e 2 ln 2 
21.
x2
5.49%
43.53%
27. y = 10eln(12/5)t/6
2
2
No
2
4 y 3 y   4 xe x  4 xCe2 x  y   xy  xe x y 
28. y = 1 - cos x
Answers
29.
30.
y
4
4
4x  x 
5

4
 20
20 x  5x 4  4
y = (x + 1)3/2
y = 3 ln|1 + x| + C
2I
F
x J + C
G
H 3K 37.
1
y=
31.
32.
y = C(4 -
34.
x2
y ln|y| + x = Cy
x2
y  e C
ey + sin y =
1
x4
C
y= ln
2
2
39.
3 13
13
y = 500 - 493e-x
40.
ln(1 +
1 2
(x + 1)
2
y2 )
= 2 ln x +
35. 13.86 years
22.24 years
x2
-1
41.
42.
sin y + cos x = C
2
3
x2
y  e C
F
I
G
H J
K
y  Ce x/2
y = 1 - cos x
38.
x)2
x2 + sec2y = C
2
33.
.
36. k  13 ln 53  01703
2
3 5
7
3 10
10
10 109
109
7
74
x 1 x2
1 x2
Answers
1
10

2x
x 1 4x2
43. 
44.
45.
46.
47.
8x
x 1  64 x 2
ex
1  e2 x
 2t
2
1 t4
x x 4
F
IJ
G
H K
2
1
sin x
arctan
+C
3
3
1
arcsec|2x| + C
2
1
2 + 9) + arctan
ln(x
2
2
2 arcsin
+C
x
 4  x  2 arcsin + C
2
x
2
 4 x +C
x2
1
arctan
+C
4
1
x2
arctan
+ C 18
6
(arctan x ) 4+ C
48. 12
2
4
49.
arcsin(x - 2) + C
F
I
G
H J
K
1
x2
arctan
+C
3
3
50.
51.
arcsec
x 1
+C
5
x3
1
arcsec
+C
2
2
x
y = 3 arcsin
+C
3
52.
53.
54.
55.
56.
Answers
y = Cearcsec x