# Calculus 1 : Spatial Calculus

## Example Questions

### Example Question #11 : Spatial Calculus

Find the velocity at  if the position function of a spring is:  .

Explanation:

To find the velocity function, take the derivative of the position function.

Substitute  into the velocity function.

### Example Question #1 : Derivatives

The position of an object is given by the following equation:

Determine the equation for the velocity of the object.

Explanation:

Velocity is the derivative of position, so in order to find the equation for the velocity of an object, all we must do is take the derivative of the equation for its position:

We will use the power rule to get the derivative.

Therefore we get,

### Example Question #12 : Spatial Calculus

The acceleration of an object is given by the following equation:

If , find the velocity of the object at  seconds.

Explanation:

Acceleration is the derivative of velocity, so in order to obtain in equation for the velocity we must integrate the equation for acceleration with respect to time:

Now we must use the given initial condition, v(0)=1, to solve for C:

Now we can simply plug in t=2 seconds to find the velocity of the object at that time:

### Example Question #13 : Spatial Calculus

The displacement of a spacecraft is defined by , what is the instantaneous velocity of the ship at  seconds?

Explanation:

To find the velocity of the of the spacecraft, we can differentiate the position equation:

Now we can use  in the velocity equation to find the velocity at 3 seconds.

### Example Question #2 : Derivatives

The position of a particle is represented by . What is the velocity at ?

Explanation:

Differentiate the position equation, to get the velocity equation

Now we plug 4 into the equation to find the velocity

is approximately equal to 2.72. Therefore

### Example Question #14 : Spatial Calculus

What is the instantaneous velocity at  of a particle whose positional equation is represented by ?

Explanation:

We find the velocity equation by differentiating the position equation. Since cos(t) is also a function we need to use the power rule along with the chain rule. This states to take the derivative of the outside function and multiply it by the derivative of the inside function. In math terms this is as follows:

Using  as the value of

Therefore the velocity at  is 0

### Example Question #15 : Spatial Calculus

A weight hanging from a spring is stretched down by  units from its rest position and released at time equals zero to bob up and down. Its position is given by . What is its velocity at time ?

Explanation:

To find the velocity of the weight, we can differentiate the position equation.

Doing so gives us:

where,

thus,

### Example Question #16 : Spatial Calculus

The position of a particle at time  is given by . What is the particle's velocity at time ?

Explanation:

To find the velocity, we must first find the velocity equation by differentiating the position equation of the particle.

We can now use 7 as the value for  to give

### Example Question #17 : Spatial Calculus

The position function of a particle is given as: . What is the velocity function?

None of these answers are correct.

Explanation:

To find the velocity function from the position function, we only need to differentiate the position function.

### Example Question #18 : Spatial Calculus

Find the velocity of a function if the acceleration is: