POPPER QUESTIONS
dy
x3
Find
if y = x .
dx
3
1.
A.
x
3
x −2
B.
3x
2
3 (l n 3)
x
C.
x
2
(
9−x
3
x +1
2
) D.
x
2
(3 − x (l n 3))
3
x
POPPER QUESTIONS
ex
2. Differentiate: y = x .
x
A. e x
B. e
C.
x
e
x −1
x
e
x
x
⎛ ex
x ⎞
+ ( ln x ) e ⎟
⎜
⎝ x
⎠
D. xe + e
( )
x
POPPER QUESTIONS
3. Evaluate the integral:
⎛ x2
A. ⎜
⎝ 2
⎞ ⎛⎜⎝ x3
⎟3
⎠
3 ⎞⎟
⎠
+C
⎛ ln 3 ⎞ x2
B. ⎜
⎟3 + C
⎝ 2 ⎠
C.
3
x
2
2 ln 3
1 ⎛ x2
D.
⎜3
2⎝
+C
⎞+C
⎟
⎠
∫ x3
x2
dx .
2
x
∫ ln x dx =
∫ 2x
2x − 3 dx =
∫ x cos 2 x dx =
Find the solution to this differential equation subject to the given initial
condition.
y' = y − 5
d
ln 6 x 2 =
dx
(
)
d
ln 3x 2 + cos 2 x =
dx
(
)
y( 0 ) = 3
d
cosh 7 x 2 + 3 =
dx
(
)
d
log 5 x =
dx
d
x sinh ( 5x ) − 4 cos x =
dx
(
)
( )
(
)
Give an equation for the tangent line to the graph of g x = ln 4 x + 1 − 2e − x
at the point where x = 0.
∫ 2x ln (10 x ) dx =
Is the function f (x) = x3 + 3x invertible? If so find the derivative of the inverse
of f at 14. Note that f (2) = 14.
∫
sin ( 9x )
(
4 + cos ( 9x )
)
2
dx =
Given an exponential growth or decay situation, if the tripling time is 30 years,
what is the doubling time?
Given an exponential growth or decay situation, if the doubling time is 2 months,
what is the tripling time?
∫
∫
(
sin 4e −3 x
e3x
) dx =
dx
=
2
x + 6 x + 10
A mold culture doubles its mass every three days. Find the growth model for a
plate seeded with 1.2 grams of mold.
Solve the differential equation if y = 3 at time = 0: 2y ' = y .
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?
Find the function y = f ( x ) passing through the point (0, 6) that has the first
derivative
dy
= y − 2.
dx
3
cos
2 θ dθ
∫
3
2
cos
θ
sin
θdθ
∫
Trigonometric Substitution
Section 8.4
1) Given an integral involving
a 2 − x 2 use x = a sin θ .
2) Given an integral involving
a 2 + x 2 use x = a tan θ .
3) Given an integral involving
x 2 − a2
use x = a sec θ .
For the following, what substitution should we use?
9 − x2
16 + x 2
x2 − 4
Examples:
1.
∫
9 − x 2 dx
2.
∫
2 − x 2 + 4 x dx
3.
∫
x2
x2 −1
dx
4.
∫
x2
4 + x2
dx