# Calculus 1 : Calculus

## Example Questions

### Example Question #1 : Calculus Review

Consider the position function , which describes the positon of an oxygen molecule.

Find the function which models the velocity of the oxygen molecule.

Explanation:

Recall that velocity is the first derivative of position and acceleration is the second derivative of position.

So given:

Apply the power rule to each term to find the velocity.

### Example Question #41 : Calculus

Consider the position function , which describes the positon of an oxygen molecule.

Find the velocity of the oxygen molecule after  seconds.

Explanation:

Recall that velocity is the first derivative of position and acceleration is the second derivative of position.

So given:

To find the velocity after 4 seconds, find v(4)

### Example Question #41 : How To Find Velocity

The displacement of an object at time  is defined by the function . What is the velocity at ?

Explanation:

The velocity is defined as the derivative of the displacement equation. With the displacement equation , we can use the power rule to differentiate.

Remember that .

In our case our

Thus, the velocity equation is defined as:

. Now  can be used in the velocity equation.

The velocity at  is 60.

### Example Question #41 : Velocity

The position of an object is given by the equation . What is the velocity of the object?

Explanation:

The velocity of the object can be found by differentiating the equation . We must use the chain rule and the power rule to differentiate this equation.

We can rewrite the equation as

using the power rule we get:

from here we need to use the chain rule and multiply this by the derivative of the inside piece which is the sine function. Since the derivative of sine is cosine we  get the following:

Thus, the velocity equation is:

### Example Question #45 : Calculus

The position of an object is given by the equation . What is the velocity of the object at ?

Explanation:

The velocity equation can be found by differentiating the equation for position .

This gives the equation

At t=0

### Example Question #41 : Calculus

The acceleration of an object is given by the equation . What is the velocity equation of the object if it has an initial velocity of ?

Explanation:

The velocity equation can be found by integrating the acceleration equation.

Differentiating the equation  gives

Using the initial velocity of the equation gives us

The complete equation for the velocity is

### Example Question #41 : How To Find Velocity

If the acceleration of an object is , and its acceleration function is , what is the velocity of the object?

Explanation:

To find the velocity from the acceleration function, integrate the acceleration function . To integrate, remember to add an exponent to each term and divide the whole term by the value of the exponent.

We will also need to know the time when the object at .

Substitute this value back into the acceleration function and solve for time.

Substitute this value to the velocity function to find the velocity.

### Example Question #42 : How To Find Velocity

If the position function of a ball is , what is the velocity of the ball when ?

Explanation:

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

Remember when integrating, increase the exponent of each term by one and then divide the term by the value of the new exponent.

Substitute the time into the velocity function.

### Example Question #43 : Calculus

The acceleration of an object is given by the equation . What is the velocity of the object at  if its initial velocity is ?

Explanation:

Using the acceleration equation,  we can find the velocity equation by taking the integral.

Now using the initial velocity of 50, we can find

Now we can solve for the velocity at t=10

### Example Question #41 : Calculus

The position equation for an object is . What is the velocity of the object at ?

Explanation:

The velocity of the object can be found by differentiating the position equation. To differentiate this equation, the power rule and the chain rule must be used.

The power rule says that

.

The chain rule says that

Differentiating the equation , gives

.

Using 0 as the value for t gives us