# Calculus 1 : Calculus

## Example Questions

### Example Question #61 : Spatial Calculus

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

Explanation:

The velocity of the object can be found by differentiating the position equation. This can be done using the power rule where if

.

Appying this to the position equation gives us

.

We can now solve for the velocity to by inputting  into the velocity equation

.

### Example Question #1 : Derivatives

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

Explanation:

The velocity of the object can be found by differentiating the position equation. The position equation can be accurately differentiated using the power rule and the product rule where if

and where if

Using these two rules we find the velocity equation to be

### Example Question #62 : Spatial Calculus

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

None of these.

Explanation:

The velocity of the object can be found by integrating the acceleration equation. To integrate this equation we can use the power rule where if

.

Therefore the velocity of the object is

.

The value of  can be found  using the initial velocity of the object.

Therefore  and .

### Example Question #61 : Calculus

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

Explanation:

The velocity of the object can be found by differentiating the object's position equation. To accurately differentiate the position equation we can use the chain rule and the power rule where if

and where if

.

Therefore the velocity equation of the object is

.

We now find the velocity of ,

### Example Question #64 : Spatial Calculus

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

Explanation:

The velocity of the equation can be found by integrating the acceleration equation. To integrate the acceleration equation, we can use the power rule where if

.

Therefore the velocity equation is

.

We can solve for the value of  using the initial velocity of the object.

Therefore  and .

We can now solve velocity at time  by inputting this into the velocity equation.

### Example Question #66 : Calculus

The jerk of an object is given by the equation . What is the velocity of the object at time , if the object has an initial acceleration of  and a velocity of  at time ?

Explanation:

The velocity of the object can be found by integrating the jerk of the object twice. To do this we must use the power rule where if

.

Therefore the acceleration of the object is

.

We can find the value of  using the initial acceleration of the object.

Therefore  and .

Repeating the integration for the acceleration equation gives us

.

To find the value of  we use the velocity of the object at .

Therefore  and .

Now we can find the velocity at .

### Example Question #61 : Calculus

The jerk of an object is given by the equation . What is the velocity of the object, if the the acceleration at time  is  and the velocity at time  is

Explanation:

To find the velocity we can integrate the jerk of the equation twice. To integrate the equation for jerk we can use the power rule where if

.

Therefore the acceleration equation is

.

Using the acceleration at time , we can find the value of .

Therefore  and .

Repeating this integration for the velocity equation we obtain

.

We can solve for this value of  the using the same method as before, but with the velocity at time .

Therefore  and .

### Example Question #65 : Spatial Calculus

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 position equation. To differentiate this equation accurately we can use the power rule and the product rule where if

and if

.

Therefore the velocity of the object is

### Example Question #66 : Spatial Calculus

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

Explanation:

The velocity of this object can be found by differentiating the object's position. To accurately differentiate this equation we can use the quotient rule, the chain rule, and the power rule where if

.

Where the chain rule is as follows,

The power rule is,

.

Therefore the velocity of the object is

.

### Example Question #62 : Calculus

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 object's position. To accurately differentiate the object's position we can use the quotient rule, the chain rule, and power rule.

The quotient rule is,

.

The chain rule is,

.

The power rule is,

.

Also remember that when differentiating

.

Therefore the object's velocity equation is

.