All AP Calculus BC Resources
Example Questions
Example Question #1 : Derivatives
Determine the intervals on which the function is concave up:
on the interval
The function is never concave up
To determine the intervals on which the function is concave up, we must find the intervals on which the second derivative of the function is positive.
First, we must find the second derivative of the function:
The derivatives were found using the following rules:
, , ,
Next, we find the values on the given interval for which the second derivative is equal to zero:
We now use these values as bounds for the intervals on which we check the sign of the second derivative:
Note that at the bounds of the intervals, the second derivative is neither positive nor negative.
Evaluating the sign simply by plugging in any value on the given interval into the second derivative function, we find that on the first interval, the second derivative is negative, while on the second interval, the second derivative is positive. Thus, the function is concave up on the second interval, .
Example Question #1 : Derivatives
Tell whether g(w) is concave up or concave down when . How do you know?
Concave down, because
Concave down, because
Concave up, because
Concave up, because
Concave up, because
Tell whether g(w) is concave up or concave down when . How do you know?
To test for concavity, we need to know the sign of our function's second derivative at the given value.
Let's begin by finding the first derivative. To do so, we need to recall the following:
Those two rules tell us how to derive functions made up of polynomials and exponential terms.
Apply the rules to get the following.
Now, do derive again to get the 2nd derivative.
Now, we are almost there, but we need to find g"(-3)
So, our second derivative is positive when w=-3. This means our original function is concave up at this point.
Concave up, because
Example Question #1 : Derivatives
At which of the following values of is the function
concave down?
The function is concave down wherever , so we compute and see where it is negative. We have:
(a parabola, opening upwards)
To find where is negative, we first find its zeros by setting :
,
so when or , and we conclude that is negative ( is concave down) between them. That is, . The only answer choice completely inside this interval (not outside or at the endpoints) is
.
Example Question #1 : How To Find Local Maximum Graphing Functions Of Curves
What type of point is in ?
Asymptote
Inflection point
Local minimum
Hole
Local maximum
Inflection point
Even though the first derivative () is at , there is no max or min because the function is increasing on both sides (derivative is positive on both sides). However, the function is changing its concavity (from negative to positive) at . We can tell because the second derivative () is also at , and it's going from negative to positive. Hole and asymptote are irrelevant. When the second derivative changes signs around a specific point we call this an inflection point. An inflection point describes a point that changes the concavity of a function.
Example Question #1 : Inflection Points
Which of the following is true about the twice-differentiable function above?
Since the function is increasing at , , and since is below the x-axis, .
Furthermore, there exists an inflection point at , where the concavity of function changes.
Thus at , .
Therefore, the correct answer is .
Example Question #1 : How To Graph Functions Of Points Of Inflection
Find the inflection point(s) of .
Inflection points can only occur when the second derivative is zero or undefined. Here we have
.
Therefore possible inflection points occur at and . However, to have an inflection point we must check that the sign of the second derivative is different on each side of the point. Here we have
.
Hence, both are inflection points
Example Question #1 : Points Of Inflection
Below is the graph of . How many inflection points does have?
Not enough information
Possible inflection points occur when . This occurs at three values, . However, to be an inflection point the sign of must be different on either side of the critical value. Hence, only are critical points.
Example Question #1 : Points Of Inflection
Find the point(s) of inflection for the function .
and
and
There are no points of inflection.
A point of inflection is found where the graph (or image) of a function changes concavity. To find this algebraically, we want to find where the second derivative of the function changes sign, from negative to positive, or vice-versa. So, we find the second derivative of the given function
The first derivative using the power rule
is,
and the seconds derivative is
We then find where this second derivative equals . when .
We then look to see if the second derivative changes signs at this point. Both graphically and algebraically, we can see that the function does indeed change sign at, and only at, , so this is our inflection point.
Example Question #3 : Second Derivatives
What are the coordinates of the points of inflection for the graph
There are no points of inflection on this graph.
Infelction points are the points of a graph where the concavity of the graph changes. The inflection points of a graph are found by taking the double derivative of the graph equation, setting it equal to zero, then solving for .
To take the derivative of this equation, we must use the power rule,
.
We also must remember that the derivative of an constant is 0.
After taking the first derivative of the graph equation using the power rule, the equation becomes
.
In this problem the double derivative of the graph equation comes out to , factoring this equation out it becomes .
Solving for when the equation is set equal to zero, the inflection points are located at .
Example Question #1 : Points Of Inflection
Find all the points of inflection of
There are no points of inflection.
In order to find the points of inflection, we need to find using the power rule .
Now to find the points of inflection, we need to set .
.
Now we can use the quadratic equation.
Recall that the quadratic equation is
,
where a,b,c refer to the coefficients of the equation .
In this case, a=12, b=0, c=-4.
Thus the possible points of infection are
.
Now to check if or which are inflection points we need to plug in a value higher and lower than each point. If there is a sign change then the point is an inflection point.
To check lets plug in .
Therefore is an inflection point.
Now lets check with .
Therefore is also an inflection point.
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