### All Calculus 2 Resources

## Example Questions

### Example Question #1 : Parametric Calculations

Calculate the length of the curve drawn out by the vector function from .

**Possible Answers:**

None of the other answers.

**Correct answer:**

The formula for arc length of a parametric curve in space is for .

Taking derivatives of each of the vector function components and substituting the values into this formula gives

We need to recognize that underneath the square root we have a perfect square, and we can write it as

Solving this we get

### Example Question #1 : Parametric Calculations

Calculate at the point on the curve defined by the parametric equations

,

**Possible Answers:**

None of the other answers

**Correct answer:**

None of the other answers

The correct answer is .

We use the equation

But we need a value for to substitute into our derivative. We can obtain such a by setting as our given point suggests.

Since our values of match, is our correcct value. Substituting this into the derivative and simplifying gives us our answer of

### Example Question #3 : Parametric Calculations

Which of the answers below is the equation obtained by eliminating the parametric from the following set of parametric equations?

**Possible Answers:**

**Correct answer:**

When the problem asks us to eliminate the parametric, that means we want to somehow get rid of our variable t and be left with an equation that is only in terms of x and y. While the equation for x is a polynomial, making it more difficult to solve for t, we can see that the equation for y can easily be solved for t:

Now that we have an expression for t that is only in terms of y, we can plug this into our equation for x and simplify, and we will be left with an equation that is only in terms of x and y:

### Example Question #4 : Parametric Calculations

Suppose and . Find the arc length from .

**Possible Answers:**

**Correct answer:**

Write the arc length formula for parametric curves.

Find the derivatives. The bounds are given in the problem statement.

### Example Question #5 : Parametric Calculations

Solve for if and .

**Possible Answers:**

None of the above

**Correct answer:**

Given equations for and in terms of , we can find the derivative of parametric equations as follows:

, as the terms will cancel out.

Using the Power Rule

for all and given and :

.

### Example Question #6 : Parametric Calculations

Solve for if and .

**Possible Answers:**

None of the above

**Correct answer:**

Given equations for and in terms of , we can find the derivative of parametric equations as follows:

, as the terms will cancel out.

Using the Power Rule

for all and given and :

### Example Question #7 : Parametric Calculations

Solve for if and .

**Possible Answers:**

None of the above

**Correct answer:**

Since we have two equations and , we can find by dividing the derivatives of the two equations - thus:

since the terms cancel out by standard rules of division of fractions.

In order to find the derivatives of and , let's use the Power Rule

for all :

Therefore,

.

### Example Question #8 : Parametric Calculations

Solve for if and .

**Possible Answers:**

**Correct answer:**

Since we have two equations and , we can find by dividing the derivatives of the two equations - thus:

since the terms cancel out by standard rules of division of fractions.

In order to find the derivatives of and , let's use the Power Rule

for all :

Therefore,

.

### Example Question #1 : Parametric Calculations

Solve for if and .

**Possible Answers:**

None of the above

**Correct answer:**

Since we have two equations and , we can find by dividing the derivatives of the two equations - thus:

(since the terms cancel out by standard rules of division of fractions).

In order to find the derivatives of and , let's use the Power Rule

for all :

Therefore, .

### Example Question #1 : Parametric Calculations

Given and , what is the length of the arc from ?

**Possible Answers:**

**Correct answer:**

In order to find the arc length, we must use the arc length formula for parametric curves:

.

Given and , we can use using the Power Rule

for all , to derive

and .

Plugging these values and our boundary values for into the arc length equation, we get:

Now, using the Power Rule for Integrals for all , we can determine that: