All AP Calculus BC Resources
Example Questions
Example Question #21 : Finding Maximums
Compute the maximum value attained by the function
.
We use the quotient rule to differentiate and set the derivative to to find any extrema.
(where in the last step we multiplied both sides by , which is never )
So any extrema occur at
.
Evaluating the original function at these values of gives:
.
We choose the larger value and conclude that the maximum value attained by the function is
.
Graph: Both extrema can be seen on the graph of the function below.
Example Question #1 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
Since the interval is , satisfies the MVT.
Example Question #2 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
Multiple solutions will solve this function, but on the interval , only fits within, satisfying the MVT.
Example Question #2 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
which falls within the interval , satisfying the MVT.
Example Question #3 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
, which falls between , satisfying the MVT.
Example Question #3 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
which falls within the interval satisfying the mean value theorem.
Example Question #4 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
, which falls within the interval , satisfying the MVT.
Example Question #1 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
There are multiple solutions; within the interval , satisfies the mean value theorem.
Example Question #2 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
This solution falls within , validating the mean value theorem.
Example Question #3 : The Mean Value Theorem
Let on the interval . Find a value for the number(s) that satisfies the mean value theorem for this function and interval.
The mean value theorem states that for a planar arc passing through a starting and endpoint , there exists at a minimum one point, , within the interval for which a line tangent to the curve at this point is parallel to the secant passing through the starting and end points.
In other words, if one were to draw a straight line through these start and end points, one could find a point on the curve where the tangent would have the same slope as this line.
Note that the value of the derivative of a function at a point is the function's slope at that point; i.e. the slope of the tangent at said point.
First, find the two function values of on the interval
Then take the difference of the two and divide by the interval.
Now find the derivative of the function; this will be solved for the value(s) found above.
Derivative of an exponential:
Derivative of a natural log:
Product rule:
Using a calculator, we find the solution , which fits within the interval , satisfying the mean value theorem.