### All Calculus 2 Resources

## Example Questions

### Example Question #1 : Derivatives Of Polar Form

For the polar equation , find when .

**Possible Answers:**

None of the other answers.

**Correct answer:**

When

.

Taking the derivative of our given equation with respect to , we get

To find , we use

Substituting our values of into this equation and simplifying carefully using algebra, we get the answer of .

### Example Question #1 : Derivatives Of Parametric, Polar, And Vector Functions

Find the derivative of the following polar equation:

**Possible Answers:**

**Correct answer:**

Our first step in finding the derivative dy/dx of the polar equation is to find the derivative of r with respect to . This gives us:

Now that we know dr/d, we can plug this value into the equation for the derivative of an expression in polar form:

Simplifying the equation, we get our final answer for the derivative of r:

### Example Question #2 : Derivatives Of Polar Form

Evaluate the area given the polar curve: from .

**Possible Answers:**

**Correct answer:**

Write the formula to find the area in between two polar equations.

The outer radius is .

The inner radius is .

Substitute the givens and evaluate the integral.

### Example Question #4 : Derivatives Of Parametric, Polar, And Vector Functions

Find the derivative of the polar function .

**Possible Answers:**

**Correct answer:**

The derivative of a polar function is found using the formula

The only unknown piece is . Recall that the derivative of a constant is zero, and that

, so

Substiting this into the derivative formula, we find

### Example Question #4 : Derivatives Of Polar Form

Find the first derivative of the polar function

.

**Possible Answers:**

**Correct answer:**

In general, the dervative of a function in polar coordinates can be written as

.

Therefore, we need to find , and then substitute into the derivative formula.

To find , the chain rule,

, is necessary.

We also need to know that

.

Therefore,

.

Substituting into the derivative formula yields

### Example Question #3 : Derivatives Of Polar Form

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

The formula for the derivative of a polar function is

First, we must find the derivative of the function given:

Now, we plug in the derivative, as well as the original function, into the above formula to get

### Example Question #4 : Derivatives Of Polar Form

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

The derivative of a polar function is given by the following:

First, we must find

The derivative was found using the following rule:

Finally, plug in the derivative we just found along with r, the function given, into the above formula:

### Example Question #5 : Derivatives Of Polar Form

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

The derivative of a polar function is given by the following:

First, we must find

The derivative was found using the following rules:

, ,

Finally, plug in the above derivative and our original function into the above formula:

### Example Question #6 : Derivatives Of Polar Form

Find the derivative of the following function:

**Possible Answers:**

**Correct answer:**

The derivative of a polar function is given by the following:

First, we must find

We found the derivative using the following rules:

,

Finally, we plug in the above derivative and the original function into the above formula:

### Example Question #7 : Derivatives Of Polar Form

Find the derivative of the function:

**Possible Answers:**

**Correct answer:**

The derivative of a polar function is given by

First, we must find the derivative of the function, r:

which was found using the following rules:

,

Now, using the derivative we just found and our original function in the above formula, we can write out the derivative of the function in terms of x and y: