Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined - AP Calculus AB
Card 1 of 33
Evaluate 
.
Evaluate
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Even though an antideritvative of 
does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of 
, 
, and then substituting 
in the integrand. Lastly the Theorem states you must multiply your result by 
(similar to the directions in using the chain rule).
.
Even though an antideritvative of
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The graph of a function 
is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function 
?
The graph of a function
Tap to reveal answer
The function 
represents the area under the curve 
from 
to some value of 
.
Do not be confused by the use of 
in the integrand. The reason we use 
is because are writing the area as a function of 
, which requires that we treat the upper limit of integration as a variable 
. So we replace the independent variable of 
with a dummy index 
when we write down the integral. It does not change the fundamental behavior of the function 
or 
.
The graph of the derivative of 
is the same as the graph for 
. This follows directly from the Second Fundamental Theorem of Calculus.
If the function 
is continuous on an interval 
containing 
, then the function defined by:
has for its' derivative 
.
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →
Evaluate
Evaluate
Tap to reveal answer
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.
Instead, we make a clever observation of the graph of
Namely, that
This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that
← Didn't Know|Knew It →
Evaluate .
Evaluate
Tap to reveal answer
Even though an antideritvative of does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.
. Start
. You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , , and then substituting in the integrand. Lastly the Theorem states you must multiply your result by (similar to the directions in using the chain rule).
.
Even though an antideritvative of
← Didn't Know|Knew It →
The graph of a function is drawn below. Select the best answers to the following:
What is the best interpretation of the function?
Which plot shows the derivative of the function ?
The graph of a function
Tap to reveal answer
The function represents the area under the curve from to some value of .
Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of with a dummy index when we write down the integral. It does not change the fundamental behavior of the function or .
The graph of the derivative of is the same as the graph for . This follows directly from the Second Fundamental Theorem of Calculus.
If the function is continuous on an interval containing , then the function defined by:
has for its' derivative .
The function
Do not be confused by the use of
If the function
has for its' derivative
← Didn't Know|Knew It →