Calculus 1 : Calculus

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #51 : Spatial Calculus

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

Possible Answers:

Correct answer:

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 #51 : How To Find Velocity

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

Possible Answers:

Correct answer:

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 #52 : How To Find Velocity

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

Possible Answers:

Correct answer:

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 #54 : Velocity

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

Possible Answers:

Correct answer:

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 #53 : How To Find Velocity

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

Possible Answers:

Correct answer:

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 #54 : How To Find Velocity

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

Possible Answers:

Correct answer:

Explanation:

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

Using the power rule to differentiate,

We find.

.

Swapping 

Example Question #54 : How To Find Velocity

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

Possible Answers:

Correct answer:

Explanation:

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

Using the power rule when differentiating,

we find,

.

Swapping 

Example Question #55 : How To Find Velocity

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

Possible Answers:

Correct answer:

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 #56 : How To Find 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 ?

Possible Answers:

Correct answer:

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 #57 : How To Find Velocity

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

Possible Answers:

Correct answer:

Explanation:

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

and where if 

Therefore the velocity of the object is

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