AP Calculus AB : Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval

Example Questions

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Example Question #1 : Interpretations And Properties Of Definite Integrals

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

28

16

4

28

Explanation:

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 #1 : Interpretations And Properties Of Definite Integrals

Find the limit.

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 6)

0

4

nonexistent

–6

1

4

Explanation:

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 6)

Use L'Hopitals rule to find the limit.

lim as n approaches infiniti of ((4n3) – 6n)/((n3) – 2n+ 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?      Explanation:

The answer is seconds.  now set because that is what the question is asking for.  seconds

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

A particle's movement is represented by At what time is the velocity at it's greatest?      Explanation:

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 #5 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

What is the domain of ?      Explanation: because the denominator cannot be zero and square roots cannot be taken of negative numbers     Example Question #6 : 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?      Explanation:    Example Question #1 : Interpretations And Properties Of Definite Integrals

What is the average value of the function from to ?      Explanation:

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 .      Explanation:

We see the answer is 0 after we do the quotient rule.   Example Question #9 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval      Explanation: Example Question #10 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval        