# Calculus 2 : Parametric Calculations

## Example Questions

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### Example Question #1 : Parametric Calculations

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

Explanation:

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 #2 : Parametric Calculations

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

Explanation:

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?

Explanation:

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 .

Explanation:

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 .

None of the above

Explanation:

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 .

None of the above

Explanation:

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 .

None of the above

Explanation:

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 .

Explanation:

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 #9 : Parametric Calculations

Solve for  if  and .

None of the above

Explanation:

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 #10 : Parametric Calculations

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

Explanation:

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:

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