### All Calculus 3 Resources

## Example Questions

### Example Question #1 : Calculus 3

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

**Possible Answers:**

**Correct answer:**

### 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:**

### Example Question #2 : Calculus 3

Let .

Find .

**Possible Answers:**

**Correct answer:**

### Example Question #4 : Calculus 3

Let .

Find its linear approximation at .

**Possible Answers:**

Hint: Use Taylor's formula.

**Correct answer:**

### Example Question #1 : Calculus 3

Let . Which of the following is equal to ?

**Possible Answers:**

**Correct answer:**

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 #4 : Numerical Approximations To Definite Integrals

Evaluate:

**Possible Answers:**

cannot be determined

**Correct answer:**

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 #2 : Calculus 3

Find if .

**Possible Answers:**

**Correct answer:**

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 #3 : Calculus 3

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

**Possible Answers:**

**Correct answer:**

First, mulitply both sides by r.

Then, use the identities and .

The answer is .

### Example Question #6 : Calculus 3

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**Correct answer:**

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 #4 : Calculus 3

Evaluate the following limit:

**Possible Answers:**

Does not exist.

**Correct answer:**

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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