Multivariable Calculus : Multivariable Calculus

Study concepts, example questions & explanations for Multivariable Calculus

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Example Questions

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Example Question #1 : Tangents & Normals

Find the equation of the tangent plane to  at .

Possible Answers:

Correct answer:

Explanation:

First, we need to find the partial derivatives in respect to , and , and plug in .

 

Remember that the general equation for a tangent plane is as follows:

Now lets apply this to our problem

Example Question #2 : Tangents & Normals

Find the equation of the tangent plane to  at .

Possible Answers:



Correct answer:

Explanation:

First, we need to find the partial derivatives in respect to , and , and plug in .

 

Remember that the general equation for a tangent plane is as follows:

Now lets apply this to our problem

Example Question #1 : Dot & Cross Products

Let , and .

Find .

Possible Answers:

Correct answer:

Explanation:

We are trying to find the cross product between  and .

Recall the formula for cross product.

If  , and , then

.

Now apply this to our situation.

Example Question #2 : Dot & Cross Products

Let , and .

Find .

Possible Answers:

Correct answer:

Explanation:

We are trying to find the cross product between  and .

Recall the formula for cross product.

If  , and , then

.

Now apply this to our situation.

Example Question #1 : Vector Equations

Write down the equation of the line in vector form that passes through the points , and .

Possible Answers:

Correct answer:

Explanation:

Remember the general equation of a line in vector form:

, where  is the starting point, and  is the difference between the start and ending points.

Lets apply this to our problem.

Distribute the 

Now we simply do vector addition to get

Example Question #2 : Vector Equations

Write down the equation of the line in vector form that passes through the points , and .

Possible Answers:

Correct answer:

Explanation:

Remember the general equation of a line in vector form:

, where  is the starting point, and  is the difference between the start and ending points.

Lets apply this to our problem.

Distribute the 

Now we simply do vector addition to get

Example Question #1 : Differentiation Rules

Find .

Possible Answers:

Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

 

Example Question #2 : Differentiation Rules

Find .

Possible Answers:


Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to  , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

Example Question #3 : Differentiation Rules

Find .

Possible Answers:

Correct answer:

Explanation:

In order to find , we need to take the derivative of  in respect to , and treat , and  as constants. We also need to remember what the derivatives of natural log, exponential functions and power functions are for single variables.

Natural Log:

Exponential Functions:

Power Functions:

 

Example Question #1 : Arc Length

Determine the length of the curve , on the interval .

Possible Answers:

Correct answer:

Explanation:

First we need to find the tangent vector, and find its magnitude.

 

Now we can set up our arc length integral

 

 

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