# AP Calculus BC : Functions, Graphs, and Limits

## Example Questions

### Example Question #13 : Derivatives Of Polar Form

What is the derivative of ?

Explanation:

In order to find the derivative   of a polar equation , we must first find the derivative of  with respect to  as follows:

We can then swap the given values of  and  into the equation of the derivative of an expression into polar form:

Using the trigonometric identity , we can deduce that . Swapping this into the denominator, we get:

### Example Question #31 : Functions, Graphs, And Limits

What is the derivative of ?

Explanation:

In order to find the derivative   of a polar equation , we must first find the derivative of with respect to  as follows:

We can then swap the given values of  and  into the equation of the derivative of an expression into polar form:

Using the trigonometric identity , we can deduce that . Swapping this into the denominator, we get:

### Example Question #3 : Vector Form

Find the vector form of  to .

Explanation:

When we are trying to find the vector form we need to remember the formula which states to take the difference between the ending and starting point.

Thus we would get:

Given  and

In our case we have ending point at  and our starting point at .

Therefore we would set up the following and simplify.

### Example Question #1 : Vector Form

Given points  and , what is the vector form of the distance between the points?

Explanation:

In order to derive the vector form of the distance between two points, we must find the difference between the , and  elements of the points.

That is, for any point

and ,

the distance is the vector

.

Subbing in our original points  and , we get:

### Example Question #2 : Vector Form

Given points  and , what is the vector form of the distance between the points?

Explanation:

In order to derive the vector form of the distance between two points, we must find the difference between the , , and elements of the points.

That is, for any point and , the distance is the vector .

Subbing in our original points  and ,  we get:

### Example Question #3 : Vector Form

The graph of the vector function can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #4 : Vector Form

The graph of the vector function can also be represented by the graph of which of the following functions in rectangular form?

Explanation:

We can find the graph of  in rectangular form by mapping the parametric coordinates to Cartesian coordinates :

We can now use this value to solve for :

### Example Question #28 : Derivatives Of Vectors

Explanation:

In general:

If ,

then

Derivative rules that will be needed here:

• Taking a derivative on a term, or using the power rule, can be done by doing the following:
• Special rule when differentiating an exponential function: , where k is a constant.

In this problem,

Put it all together to get

### Example Question #5 : Vector Form

Calculate

Explanation:

Calculate the sum of vectors.

In general,

Solution:

### Example Question #1 : Limits

Evaluate the following limit: