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Example Questions
Example Question #1 : Integrals
Evaluate .
Does not exist
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).
.
Example Question #2 : Integrals
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 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 .
Example Question #1 : Integrals
Evaluate
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
Example Question #1 : Fundamental Theorem Of Calculus
Find
Does not exist
Does not exist
The one side limits are not equal: left is 0 and right is 3
Example Question #2 : Fundamental Theorem Of Calculus
Which of the following is a vertical asymptote?
When approaches 3,
approaches
.
Vertical asymptotes occur at values. The horizontal asymptote occurs at
.
Example Question #3 : Fundamental Theorem Of Calculus
What are the horizontal asymptotes of ?
Compute the limits of as
approaches infinity.
Example Question #4 : Fundamental Theorem Of Calculus
Write the domain of the function.
The answer is
The denominator must not equal zero and anything under a radical must be a nonnegative number.
Example Question #5 : Fundamental Theorem Of Calculus
What is the value of the derivative of at x=1?
First, find the derivative of the function, which is:
Then, plug in 1 for x:
The result is .
Example Question #1 : Calculus 3
Evaluate the following limit:
Does not exist.
First, let’s multiply the numerator and denominator of the fraction in the limit by .
As becomes increasingly large the
and
terms will tend to zero. This leaves us with the limit of
.
.
The answer is .
Example Question #1 : Fundamental Theorem Of Calculus
Let and
be inverse functions, and let
.
What is the value of ?
Since and
are inverse functions,
. We can differentiate both sides of the equation
with respect to
to obtain the following:
We are asked to find , which means that we will need to find
such that
. The given information tells us that
, which means that
. Thus, we will substitute 3 into the equation.
The given information tells us that.
The equation then becomes .
We can now solve for .
.
The answer is .
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