# Calculus 1 : Spatial Calculus

## Example Questions

### Example Question #21 : Spatial Calculus

What is the velocity function when the position function is given by

.

Explanation:

To find the velocity function, we need to find the derivative of the position function.

So lets take the derivative of  with respect to .

The derivative of  is  because of Power Rule:

The derivative of  is  due to Power Rule

So...

### Example Question #22 : Spatial Calculus

What is the velocity when the velocity function is  at time ?

Explanation:

To find the velocity at  we just plug  into the veloctiy function

### Example Question #23 : Spatial Calculus

Find the velocity function when the position function is

.

Explanation:

To find the veloctiy function, we need to take the derivative of the position function

So  would turn into

because of the Power Rule as well as the rule that the derivative of a constant is .

### Example Question #21 : Calculus

What is the velocity when the position function is

.

Explanation:

To find the velocity function, we need to take the derivative of the position function in relation to

So

changes into

using the Power Rule

.

### Example Question #25 : Spatial Calculus

What is the velocity when  when the velocity function is

?

Explanation:

To find the velocity at time  we plug in for

So

becomes

.

### Example Question #26 : Spatial Calculus

What is the velocity function when the position function is

?

Explanation:

To find the velocity function we need to take the derivative of the position function

So  turns into  because of power rule.

### Example Question #27 : Spatial Calculus

If models the distance of a projectile as a function of time, find the velocity of the projectile at .

Explanation:

We are given a function dealing with distance and asked to find a velocity. recall that velocity is the first derivative of position. Find the first derivative of h(t) and evaluate at t=15.

### Example Question #28 : Spatial Calculus

Consider the following position function:

Find the velocity of a paper airplane after  seconds whose position is modeled by .

Explanation:

Recall that velocity is the first derivative of position.

To find the velocity after 3 seconds, we need p'(3).

### Example Question #29 : Spatial Calculus

The velocity of a driver is given by the following vector:

What velocity will she or he reach in the long run?

Explanation:

To find the velocity for the long run, we need to find the limit as t gets bigger and bigger.

Note that to find this limit, we need to take the limit componentwise.

We know the following:

.

Therefore as t becomes bigger we will reach the velocity given by:

### Example Question #30 : Spatial Calculus

Two drivers have the following position vectors:

Wich driver is driving faster?

The second driver.

They have the same velocity.

They follow the same path and have the same velocity.

We can't tell from the given information.

The first driver.

The second driver.

Explanation:

To find who is driving faster, we need to compare the norm of their velocities:

We will call   the velocities of the first and the second drivers respectively.

Using the Chain Rule on the position function we have the following velocities,

.

.

Now we find the speed of both drivers.

Use the trigonometric identity to find velocity two.

and we see that the second driver is driving faster.