Calculus 1 : Spatial Calculus

Example Questions

Example Question #51 : Velocity

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

Explanation:

Given the equation , the velocity can be found by differentiating the position equation. To properly differentiate this equation, we must use the chain rule where if

.

Therefore, the derivative of the function is,

We now use  to find the velocity.

Example Question #52 : Velocity

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

Explanation:

The velocity of the object is the integral of the accelertion. We can use the power rule to integrate the equation. The power rule is that

Therefore

To find the velocity we must now solve for the value of , using the initial velocity of the object.

Therefore  and

We can now find the velocity at time

Example Question #53 : Velocity

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

Explanation:

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

and where

Therefore the velocity equation is

Example Question #2 : Calculus Review

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

Explanation:

The velocity of the object can be found by differentiating the position equation of the object. To differentiate the position equation of the object, we can use the power rule for the second term where if

Using this rule we find that

We can now use the value of  to solve for the velocity at

Example Question #54 : Velocity

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 is the derivative of the position equation. To differentiate the position equation we must use the chain rule and the power rule, where if

and if

Applying these two rules to the position equation gives us

Example Question #55 : Velocity

The position of a car is defined as . What is the car's velocity at ?

Explanation:

The velocity  is defined as the first derivative of its position , or .

Using the power rule to differentiate,

We find.

.

Swapping

Example Question #56 : Velocity

The position of a particle is defined as . What is the particle's velocity at ?

Explanation:

The velocity  is defined as the first derivative of its position , or .

Using the power rule when differentiating,

we find,

.

Swapping

Example Question #57 : Velocity

The position of a particle is defined as . What is the particle's velocity at ?

Explanation:

The velocity  is defined as the first derivative of its position , or .

Use the power rule to differentiate the position function.

Since .

Swapping

Example Question #58 : Velocity

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

Explanation:

The velocity of the object can be found by integrating the acceleration of the object. This can be done using the power rule where if

Integrating the acceleration equation gives us

We can solve for the value of  using the velocity of the object at

Therefore  and

We can now find the velocity at time

Example Question #59 : Velocity

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