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Find the domain of the function: f(x) = ln(3x + 1).
Find the domain of the function: f(x) = 3 + ln(x - 1).
(1, )
[A] f(x) = ln x
[B] f(x) = ex-1
[C] f(x) = ln(x - 1)
[D] f(x) = ex
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
Sketch the graph: f(x) = ln|x|.
Solve for x: ln(5x + 1) + ln x = ln 4
Solve for x: ln(5x - 1) - ln x = 3.
dy/dx for y = ln(5 - x)6
Find the derivative: f(x) = ln(x3 + 3x)3
Find the derivative:
x2  1
f ( x )  ln
.
3
2
x (2 x  1)
x2 4x  1
Find the derivative: f ( x)  ln 3 3 .
( x  5)
Find the derivative: f ( x )  ln
x( x 2  5)
x3  5
.
Differentiate: y = ln(ln tan x)
Find y’ y = ln|2x2 - 5|
Find y’ if ln xy = x + y
2
dy
(
x

2
)
1

x
Use logarithmic differentiation to find
: y
.
3
dx
4x
Find the slope of the tangent line to the graph of
y = ln x2 at the point where x = e2
1
X
dx.
Y
Zx
4e
Evaluate the integral:
e
ln 4
Evaluate the integral:
1
X
dx.
Y
Zax  b
ln|ax + b| + C
 4x
X
dx.
Y
Zx
e
Evaluate the integral:
2
1
-2
 7x
X
Evaluate the integral: Y
dx.
Zx
e
2
1
x+
1
2
ln(x2 + 1) + C
Evaluate the integral:
X
Y
Z
x2  x  1
dx.
2
x 1
x+
ln(x2 + 1) + C
8x  9 x  8
X
dx.
Evaluate the integral: Y
Z x 1
2
2
8x +
ln(x2 + 1) + C
9x  9x  9
X
dx.
Evaluate the integral: Y
Z x 1
2
2
9x -
ln(x2 + 1) + C
ln x
X
dx.
Evaluate the integral: Y
Zx
+C
1
3
ln|sec 3x| + C
Evaluate the integral:
ztan 3x dx.
ln|sec 3x| + C
X
Y
Z
sin 2 x  cos2 x
Evaluate the integral:
dx.
sin x
-2 cos x + ln|csc x + cot x| + C
csc x
X
dx.
Evaluate the integral: Y
Zcot x
2
ln|tan x| + C
Evaluate the integral:
1 sin 
X
d .
Y
Z cos
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
Differentiate:
1
f ( x) 
.
2x 4
(4  e )
Differentiate:
f ( x)  4  e .
2x
Differentiate:
y  e sin
x
.
Find:
dy if xey + 1 = xy
dx
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
e
Evaluate the integral: X
dx.
Y
Zx
x
+C
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
dy
Find
if y = 3xx3
dx
3xx2[3 + (ln 3)x]
Differentiate: y = x1-x
x1-x
Differentiate y = xx
xx[1 + ln x]
Evaluate the integral:
zx3
x2
dx.
+C
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
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
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%
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%
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.
Determine whether the function y = 2cos x is a solution
to the differential equation y   y   0.
No
that
2
y 4  2e x  Ce2 x
2
is a solution to the differential equation
 x2
verify that y  2e  Ce
equation y   xy  xe  x y 3 .
4
2 x2
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
Find the particular solution to the differential equation
4
dy
 y 2 (1  x 3 ) 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
+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=
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
x2 + sec2y = C
dx
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=
Find the general solution
to the first-order differential
2
x
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
Find the solution to the initial value problem
(e y  cos y) y   x, y(-1) = 0
ey + sin y =
(x2 + 1)
Find the solution to the initial value problem
(1  x 2 )(1  y 2 )  xyy ,
y(1) = 0
ln(1 + y2) = 2 ln x + x2 - 1