Equations - AP Calculus AB
Card 0 of 5558
Find
for the equation:

Find for the equation:
Note that:
![d[sin(u)]=du(cos(u))](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482151/gif.latex)
Product Rule: ![d[uv]=udv+vdu](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482152/gif.latex)
Take the derivative of each term in the equation twice: with respect to
and then with respect to
. When taking the derivative with respect to one variable, treat the other variable as a constant.
For the function

The derivative is then found using the product rule to be:

Notice how the chain rule needs to be utilized an additional time when taking the derivative of the
term with respect to
.
Now bring
and
terms to opposite sides of the equation:

Now rearraging variables gives
:

Note that:
Product Rule:
Take the derivative of each term in the equation twice: with respect to and then with respect to
. When taking the derivative with respect to one variable, treat the other variable as a constant.
For the function
The derivative is then found using the product rule to be:
Notice how the chain rule needs to be utilized an additional time when taking the derivative of the term with respect to
.
Now bring and
terms to opposite sides of the equation:
Now rearraging variables gives :
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Find
if 
Find if
For this problem, note that:
![d[e^u]=du(e^u)](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482153/gif.latex)
![d[sin(u)]=du(cos(u))](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482154/gif.latex)
![d[cos(u)]=-du(sin(u))](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482155/gif.latex)
Product Rule ![d[uv]=udv+vdu](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/482156/gif.latex)
To solve this problem, differentiate the expression one variable at a time, treating other variables as constants:
If we're looking for
for the function
then we'll begin by differentiating with respect to
first:

Next, differentiate with respect to
:

Now finally we'll differentiate with respect to
; remember to use the product rule:



For this problem, note that:
Product Rule
To solve this problem, differentiate the expression one variable at a time, treating other variables as constants:
If we're looking for for the function
then we'll begin by differentiating with respect to
first:
Next, differentiate with respect to :
Now finally we'll differentiate with respect to ; remember to use the product rule:
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Find
given:

Find given:
To solve, simply find the first derivative and let
. Thus,

To solve, simply find the first derivative and let . Thus,
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Write an integral expression which will solve the area under the curve of
, bounded by
, from
.
Write an integral expression which will solve the area under the curve of , bounded by
, from
.
Recall that area under the curve requires integration from either top minus bottom curve, or right minus left curve.
The top curve is: 
The bottom curve is: 
The
interval indicates that the lower bound is 2 and the upper bound is 5.
Write the integral.

Recall that area under the curve requires integration from either top minus bottom curve, or right minus left curve.
The top curve is:
The bottom curve is:
The interval indicates that the lower bound is 2 and the upper bound is 5.
Write the integral.
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What is the area under the curve
for
?
What is the area under the curve for
?
By normal exponent rules
, and so we set up the definite integral
,
which integrates to:
![\left[ \frac{e^{x \ln 2}}{\ln 2}\right]_{-\infty}^0 = \frac{2^0 - 2^{-\infty}}{\ln 2} = \frac{1 - 0}{\ln 2}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/126268/gif.latex)
By normal exponent rules , and so we set up the definite integral
,
which integrates to:
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The Gaussian integral formula states that
.
What is the integral of
?
The Gaussian integral formula states that
.
What is the integral of
?
Integrating by parts with
, 
,
,
we get:
![\int_{-\infty}^{\infty} x^2 e^{-x^2} dx = \int_A^B u ~ dv = \left[ u v\right ]_A^B - \int_A^B v ~ du](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/129650/gif.latex)
![= \left[ \frac{-x}{2} e^{-x^2} \right ]_{-\infty}^{\infty} - \int_{-\infty}^{\infty} \frac{-1}{2} e^{-x^2}~ dx = [0 - 0] - \frac{-1}{2} \sqrt{\pi} = \frac{\sqrt{\pi}}{2}](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/105514/gif.latex)
Integrating by parts with
,
,
,
we get:
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Determine
given that
.
Determine given that
.
This problem requires you to evaluate an indefinite integral of the given function f’(x). Integrating each term of the function with respect to x, we simply divide each coefficient by (n+1), where n is the value of the exponent on x for that particular term, and then add 1 to the value of the exponent on each x. This gives us:


A correct answer must include a constant C, as the original function may have had a constant that is not reflected in the equation for its derivative.
This problem requires you to evaluate an indefinite integral of the given function f’(x). Integrating each term of the function with respect to x, we simply divide each coefficient by (n+1), where n is the value of the exponent on x for that particular term, and then add 1 to the value of the exponent on each x. This gives us:
A correct answer must include a constant C, as the original function may have had a constant that is not reflected in the equation for its derivative.
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What is the value of the following definite integral?

What is the value of the following definite integral?
We start by integrating the function in parentheses with respect to x, and we then subtract the evaluation of the lower limit from the evaluation of the upper limit:



We start by integrating the function in parentheses with respect to x, and we then subtract the evaluation of the lower limit from the evaluation of the upper limit:
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Determine the amount of work required to push a box from
meters to
meters, given the function for force below:

Determine the amount of work required to push a box from meters to
meters, given the function for force below:
This problem intends to demonstrate one possible application of calculus to the field of physics. An equation for work as given in physics is as follows:

Using this equation, we simply set up a definite integral with our bounds given by the interval of interest in the problem:



Joules
After evaluating our integral, we can see that the work required to push the box from x=2 meters to x=5 meters is 39.8 Joules.
This problem intends to demonstrate one possible application of calculus to the field of physics. An equation for work as given in physics is as follows:
Using this equation, we simply set up a definite integral with our bounds given by the interval of interest in the problem:
Joules
After evaluating our integral, we can see that the work required to push the box from x=2 meters to x=5 meters is 39.8 Joules.
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Write out the expression of the area under the following curve from
to
.

Write out the expression of the area under the following curve from to
.
Simply use an integral expression with the given x values as your bounds.
Don't forget to add the dx!

Simply use an integral expression with the given x values as your bounds.
Don't forget to add the dx!
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Evaluate the definite integral within the interval
.

Evaluate the definite integral within the interval .
In order to solve this problem we must know that

In this case we have:

Our first step is to integrate:

We then arrive to our solution by plugging in our values 
![\left [ (1)^{5}+\frac{1}3{(1)^{3}-\frac{1}{2}(1)^{2}} \right ]-\left [ (0)^{5}+\frac{1}{3}(0)^{3}-\frac{1}{2}(0)^{2} \right ]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/277086/gif.latex)

In order to solve this problem we must know that
In this case we have:
Our first step is to integrate:
We then arrive to our solution by plugging in our values
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Evaluate the definite integral within the interval
.

Evaluate the definite integral within the interval .
In order to solve this problem we must remember that:

In this case we have:

Our first step is to integrate:

We then arrive at our solution by plugging in our values 
![\left[\frac{1}{2}(2)^4+\frac{1}{6}(2)^3+7(2)\right]-\left[\frac{1}{2}(1)^4+\frac{1}{6}(1)^3+7(1)\right]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/279458/gif.latex)

In order to solve this problem we must remember that:
In this case we have:
Our first step is to integrate:
We then arrive at our solution by plugging in our values
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Evaluate the definite integral within the interval 

Evaluate the definite integral within the interval
In order to solve this problem we must remember that:

In this case we have:

Our first step is to integrate:

We then arrive to our solution by plugging in our values 
![\left [ 2\sqrt{9}+(9)^{3}-4(9) \right ]-\left [ 2\sqrt{4}+(4)^{3}-4(4) \right ]](https://vt-vtwa-assets.varsitytutors.com/vt-vtwa/uploads/formula_image/image/279464/gif.latex)


In order to solve this problem we must remember that:
In this case we have:
Our first step is to integrate:
We then arrive to our solution by plugging in our values
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Evaluate the indefinite integral.

Evaluate the indefinite integral.
We are being asked to integrate the function.
To do this we need to remember the power rule of integrals,

Using this rule, we can evaluate the following integral:

We are being asked to integrate the function.
To do this we need to remember the power rule of integrals,
Using this rule, we can evaluate the following integral:
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Evaluate the indefinite integral.

Evaluate the indefinite integral.
In this problem we are being asked to integrate the function.
In order to do this we need to remember,
.
Since

then the integral,
.
In this problem we are being asked to integrate the function.
In order to do this we need to remember,
.
Since
then the integral,
.
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Evaluate the indefinite integral.

Evaluate the indefinite integral.
This problem is asking us to integrate the function.
In order to do this we need to remember,
.
Therefore,

This problem is asking us to integrate the function.
In order to do this we need to remember,
.
Therefore,
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Which of the following integrals could be evaluated to find the area under the curve of the following function from 3 to 13?

Which of the following integrals could be evaluated to find the area under the curve of the following function from 3 to 13?
We are asked to set up an integral without solving it. We want to evaluate the function from 3 to 13, so 3 goes at the bottom of the integral and 13 goes at the top. No other options have this correct.
We are asked to set up an integral without solving it. We want to evaluate the function from 3 to 13, so 3 goes at the bottom of the integral and 13 goes at the top. No other options have this correct.
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Write the correct expression to find the area of
from
.
Write the correct expression to find the area of from
.
This is a tricky question.
If we chose
, the region in the third quadrant of
is negative area, since
is no longer the top curve. Evaluating this integral will cancel out the negative area in the third quadrant with the positive area on the first quadrant. This integral will give zero area, which is incorrect.
The correct method is to split this interval into 2 separate integrations: One from
and the other from
.
From interval
, the top curve is
minus the bottom curve,
, is
.
From interval
, the top curve is
minus the bottom curve,
, is
.
Set up the integral.

This is a tricky question.
If we chose , the region in the third quadrant of
is negative area, since
is no longer the top curve. Evaluating this integral will cancel out the negative area in the third quadrant with the positive area on the first quadrant. This integral will give zero area, which is incorrect.
The correct method is to split this interval into 2 separate integrations: One from and the other from
.
From interval , the top curve is
minus the bottom curve,
, is
.
From interval , the top curve is
minus the bottom curve,
, is
.
Set up the integral.
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Which of the following expressions could be evaluated to find the area under
on the interval
?

Which of the following expressions could be evaluated to find the area under on the interval
?
To find the area under a curve, we usually want to use an integral. All of our answer choices are integrals though, so this doesn't narrow things down. Then next thing to look for are our limits of integration. In this case, we are on the interval \[360,540\], so look for an option that has the lower limit of integration as 360 and the higher limit as 540. Only one option has this:

For further clarification, to set up an integral, we want our function within the integral and our limits of integration on the integral sign itself. Once we have the correct integral set up, we can evaluate it to find the area under the curve.
To find the area under a curve, we usually want to use an integral. All of our answer choices are integrals though, so this doesn't narrow things down. Then next thing to look for are our limits of integration. In this case, we are on the interval \[360,540\], so look for an option that has the lower limit of integration as 360 and the higher limit as 540. Only one option has this:
For further clarification, to set up an integral, we want our function within the integral and our limits of integration on the integral sign itself. Once we have the correct integral set up, we can evaluate it to find the area under the curve.
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Suppose a student wants to find the area under the curve between functions
,
, and
. Find the correct integral that will determine the area.
Suppose a student wants to find the area under the curve between functions ,
, and
. Find the correct integral that will determine the area.
The region bounded by the three functions represents a triangle. It will be easier to determine the bounded region by subtracting the left curve from the right curve, and using
to integrate. In order to integrate in terms of
, any equation in terms of
must be rewritten in terms of
.
Isolate
.


Find the intersection of the lines
and
.


The correct bounds are from
. Write the correct integral.

The region bounded by the three functions represents a triangle. It will be easier to determine the bounded region by subtracting the left curve from the right curve, and using to integrate. In order to integrate in terms of
, any equation in terms of
must be rewritten in terms of
.
Isolate .
Find the intersection of the lines and
.
The correct bounds are from . Write the correct integral.
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