All AP Calculus AB Resources
Example Questions
Example Question #1 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval
If f(1) = 12, f' is continuous, and the integral from 1 to 4 of f'(x)dx = 16, what is the value of f(4)?
27
12
16
4
28
28
You are provided f(1) and are told to find the value of f(4). By the FTC, the following follows:
(integral from 1 to 4 of f'(x)dx) + f(1) = f(4)
16 + 12 = 28
Example Question #503 : Ap Calculus Ab
Find the limit.
lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n2 + 6)
0
1
nonexistent
–6
4
4
lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n2 + 6)
Use L'Hopitals rule to find the limit.
lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n2 + 6)
lim as n approaches infiniti of ((12n2) – 6)/((3n2) – 4n + 6)
lim as n approaches infiniti of 24n/(6n – 4)
lim as n approaches infiniti of 24/6
The limit approaches 4.
Example Question #1 : Interpretations And Properties Of Definite Integrals
If a particle's movement is represented by
, then when is the velocity equal to zero?
The answer is
seconds.
now set
because that is what the question is asking for.
seconds
Example Question #44 : Integrals
A particle's movement is represented by
At what time is the velocity at it's greatest?
The answer is at 6 seconds.
We can see that this equation will look like a upside down parabola so we know there will be only one maximum.
Now we set
to find the local maximum.
seconds
Example Question #1 : Interpretations And Properties Of Definite Integrals
What is the domain of
?
because the denominator cannot be zero and square roots cannot be taken of negative numbers
Example Question #2 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval
If
,then at
, what is 's instantaneous rate of change?
The answer is 8.
Example Question #5 : Calculus 3
Which of the following represents the graph of the polar function
in Cartestian coordinates?
First, mulitply both sides by r.
Then, use the identities
and .
The answer is
.Example Question #3 : Interpretations And Properties Of Definite Integrals
What is the average value of the function
from to ?
The average function value is given by the following formula:
, evaluated from to .
Example Question #2 : Interpretations And Properties Of Definite Integrals
If
then find
.
We see the answer is 0 after we do the quotient rule.
Example Question #2 : Interpretations And Properties Of Definite Integrals
If
, then which of the following is equal to ?
According to the Fundamental Theorem of Calculus, if we take the derivative of the integral of a function, the result is the original function. This is because differentiation and integration are inverse operations.
For example, if
, where is a constant, then .We will apply the same principle to this problem. Because the integral is evaluated from 0 to
, we must apply the chain rule.
The answer is
.