# Calculus 3 : Calculus 3

## Example Questions

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### Example Question #1 : Calculus 3

Suppose the vectors and are orthogonal. Find all real values of .

Possible Answers:

Correct answer:

Explanation:

### Example Question #2 : Calculus 3

The vector-valued function paremeterizes a curve , where .

Find a tangent vector to at the point .

Possible Answers:

Correct answer:

Explanation:

### Example Question #2 : Calculus 3

Let .

Find .

Possible Answers:

Correct answer:

Explanation:

### Example Question #4 : Calculus 3

Let .

Find its linear approximation at .

Possible Answers:

Hint: Use Taylor's formula.

Correct answer:

Explanation:

### Example Question #1 : Calculus 3

Let . Which of the following is equal to ?

Possible Answers:

Correct answer:

Explanation:

Notice that  can be expressed as a composite function, i.e. a function within a function. If we let and  , then . In order to differentiate , we will need to apply the Chain Rule, as shown below:

First, we need to find , which equals .

Then, we need to find by applying the Product Rule.

The answer is .

### Example Question #11 : Integrals

Evaluate:

Possible Answers:

cannot be determined

Correct answer:

Explanation:

First, we can write out the first few terms of the sequence , where  ranges from 1 to 3.

Notice that each term , is found by multiplying the previous term by . Therefore, this sequence is a geometric sequence with a common ratio of . We can find the sum of the terms in an infinite geometric sequence, provided that , where  is the common ratio between the terms. Because  in this problem,  is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

The answer is .

### Example Question #171 : Functions, Graphs, And Limits

Find  if .

Possible Answers:

Correct answer:

Explanation:

We will have to find the first derivative of  with respect to  using implicit differentiation. Then, we can find , which is the second derivative of  with respect to .

We will apply the chain rule on the left side.

We now solve for the first derivative with respect to .

In order to get the second derivative, we will differentiate with respect to . This will require us to employ the Quotient Rule.

We will replace  with .

But, from the original equation, . Also, if we solve for , we obtain .

The answer is .

### Example Question #1 : Interpretations And Properties Of Definite Integrals

Which of the following represents the graph of the polar function  in Cartestian coordinates?

Possible Answers:

Correct answer:

Explanation:

First, mulitply both sides by r.

Then, use the identities  and .

The answer is .

### Example Question #1 : Calculus 3

Possible Answers:

Correct answer:

Explanation:

We can use the substitution technique to evaluate this integral.

Let .

We will differentiate  with respect to .

, which means that .

We can solve for  in terms of , which gives us .

We will also need to change the bounds of the integral. When , , and when , .

We will now substitute  in for the , and we will substitute  for .

The answer is .

### Example Question #1 : Calculus 3

Evaluate the following limit:

Possible Answers:

Does not exist.

Correct answer:

Explanation:

First, let's multiply the numerator and denominator of the fraction in the limit by .

As  becomes increasingly large the and terms will tend to zero. This leaves us with the limit of .

.

The answer is .

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