AP Calculus BC : AP Calculus BC

Study concepts, example questions & explanations for AP Calculus BC

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Example Questions

Example Question #1 : Points Of Inflection

Find all the points of infection of

.

Possible Answers:

There are no points of inflection.

Correct answer:

Explanation:

In order to find the points of inflection, we need to find  using the power rule .

Now lets factor .

Now to find the points of inflection, we need to set .

.

From this equation, we already know one of the point of inflection, .

To figure out the rest of the points of inflection 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=20, b=0, c=-18.

 

Thus the other 2 points of infection are

To verify that they are all inflection points we need to plug in values higher and lower than each value and see if the sign changes.

Lets plug in  

Since there is a sign change at each point, all are points of inflection.

Example Question #2 : Inflection Points

Which of the following is a point of inflection of  on the interval ?

Possible Answers:

Correct answer:

Explanation:

Which of the following is a point of inflection of f(x) on the interval ?

To find points of inflection, we need to know where the second derivative of the function is equal to zero. So, find the second derivative:

So, where on the given interval does ?

Well, we know from our unit circle that ,

So we would have a point of inflection at , but we still need to find the y-coordinate of our POI. find this by finding 

So our POI is:

 

Example Question #1 : Inflection Points

Which of the following functions has an inflection point where concavity changes?

Possible Answers:

Correct answer:

Explanation:

For a graph to have an inflection point, the second derivative must be equal to zero. We also want the concavity to change at that point. 

, for all real numbers, but this is a line and has no concavity associated with it, so not this one.

, there are no real values of  for which this equals zero, so no inflection points.

 

, same story here.

 

, so no inflection points here.

 

This leaves us with 

, whose derivatives are a bit more difficult to take.

 

, so by the chain rule we get

So,  when . So 

.  This is our correct answer.

Example Question #11 : Derivatives

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Example Question #11 : Derivatives

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Example Question #11 : Derivatives

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Example Question #14 : Second Derivatives

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Example Question #15 : Second Derivatives

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Example Question #16 : Second Derivatives

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Example Question #21 : Second Derivatives

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