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

Study concepts, example questions & explanations for AP Calculus AB

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Example Questions

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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)?

Possible Answers:

12

16

4

28

27

Correct answer:

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

Find the limit. 

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

Possible Answers:

1

–6

nonexistent

0

4

Correct answer:

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

If a particle's movement is represented by p=3t^{2}-t+16, then when is the velocity equal to zero?

Possible Answers:

Correct answer:

Explanation:

The answer is  seconds.

 

p=3t^{2}-t+16

v=p'=6t-1

now set  because that is what the question is asking for. 

v=0=6t-1

t=\frac{1}{6} seconds

Example Question #4 : 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 p=-t^{2}+12t+2

 At what time is the velocity at it's greatest?

Possible Answers:

Correct answer:

Explanation:

The answer is at 6 seconds. 

 

p=-t^{2}+12t+2 

We can see that this equation will look like a upside down parabola so we know there will be only one maximum.

v=p'=-2t+12

Now we set  to find the local maximum. 

v=0=-2t+12

t=6 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 f(x)=\frac{x+5}{\sqrt{x^2-9}}?

Possible Answers:

(-\infty,-3)\cup (3,+\infty)

(-\infty,-3]\cup [3,+\infty)

(-5,+\infty)

(3,+\infty)

Correct answer:

(-\infty,-3)\cup (3,+\infty)

Explanation:

{\sqrt{x^2-9}}>0 because the denominator cannot be zero and square roots cannot be taken of negative numbers

x^2-9>0

x^2>9

\sqrt{x^2}>\sqrt{9}

\left | x \right |>3

x>3\: or\, x<-3

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 y-6x-x^{2}=4,

then at , what is 's instantaneous rate of change?

Possible Answers:

Correct answer:

Explanation:

The answer is 8.

 

y-6x-x^{2}=4

y=x^{2}+6x+4

y'=2x+6

Example Question #7 : 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 average value of the function  from  to ?

Possible Answers:

Correct answer:

Explanation:

The average function value is given by the following formula:

, evaluated from  to .

Example Question #8 : Definite Integral Of The Rate Of Change Of A Quantity Over An Interval Interpreted As The Change Of The Quantity Over The Interval

h(x)=\frac{g(x)}{(1+f(x))}

If 

then find .

Possible Answers:

\frac{5}{2}

\frac{-1}{2}

Correct answer:

Explanation:

We see the answer is 0 after we do the quotient rule. 

 

h(x)=\frac{g(x)}{(1+f(x))}

h'(x)=\frac{g'(x)(1+f(x))-g(x)(f'(x))}{(1+f(x))^{2}}

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

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

Explanation:

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