### All Calculus 2 Resources

## Example Questions

### Example Question #1 : Definition Of Derivative

Evaluate the limit using one of the definitions of a derivative.

**Possible Answers:**

Does not exist

**Correct answer:**

Evaluating the limit directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let and notice . It follows that .

Thus, the limit is

### Example Question #2 : Definition Of Derivative

Evaluate the limit using one of the definitions of a derivative.

**Possible Answers:**

Does not exist

**Correct answer:**

Evaluating the derivative directly will produce an indeterminant solution of .

The limit definition of a derivative is . However, the alternative form, , better suits the given limit.

Let and notice . It follows that . Thus, the limit is .

### Example Question #1 : Definition Of Derivative

Suppose and are differentiable functions, and . Calculate the derivative of , at

**Possible Answers:**

None of the other answers

**Correct answer:**

None of the other answers

The correct answer is 11.

Taking the derivative of involves the product rule, and the chain rule.

Substituting into both sides of the derivative we get

.

### Example Question #4 : Definition Of Derivative

Evaluate the limit

without using L'Hopital's rule.

**Possible Answers:**

**Correct answer:**

If we recall the definition of a derivative of a function at a point , one of the definitions is

.

If we compare this definition to the limit

we see that that this is the limit definition of a derivative, so we need to find the function and the point at which we are evaluating the derivative at. It is easy to see that the function is and the point is . So finding the limit above is equivalent to finding .

We know that the derivative is , so we have

.

### Example Question #2 : Definition Of Derivative

Approximate the derivative if where .

**Possible Answers:**

**Correct answer:**

Write the definition of the limit.

Substitute .

Since is approaching to zero, it would be best to evaluate when we assume that is progressively decreasing. Let's assume and check the pattern.

The best answer is:

### Example Question #2 : Derivative Review

## Given:

## Find f'(x):

**Possible Answers:**

**Correct answer:**

Computation of the derivative requires the use of the Product Rule and Chain Rule.

The Product Rule is used in a scenario when one has two differentiable functions multiplied by each other:

This can be easily stated in words as: "First times the derivative of the second, plus the second times the derivative of the first."

In the problem statement, we are given:

is the "First" function, and is the "Second" function.

The "Second" function requires use of the Chain Rule.

When:

Applying these formulas results in:

Simplifying the terms inside the brackets results in:

We notice that there is a common term that can be factored out in the sets of equations on either side of the "+" sign. Let's factor these out, and make the equation look "cleaner".

Inside the brackets, it is possible to clean up the terms into one expanded function. Let us do this:

Simplifying this results in one of the answer choices:

### Example Question #1 : Derivatives

What is the value of the limit below?

**Possible Answers:**

**Correct answer:**

Recall that one definition for the derivative of a function is .

This means that this question is asking us to find the value of the derivative of at .

Since

and , the value of the limit is .

### Example Question #8 : Definition Of Derivative

**Possible Answers:**

**Correct answer:**

Evaluation of this integral requires use of the Product Rule. One must also need to recall the form of the derivative of .

*Product Rule:*

* *

Applying these two rules results in:

This matches one of the answer choices.

### Example Question #9 : Definition Of Derivative

Use the definition of the derivative to solve for .

**Possible Answers:**

**Correct answer:**

In order to find , we need to remember how to find by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

### Example Question #10 : Definition Of Derivative

Use the definition of the derivative to solve for .

**Possible Answers:**

**Correct answer:**

In order to find , we need to remember how to find by using the definition of derivative.

Definition of Derivative:

Now lets apply this to our problem.

Now lets expand the numerator.

We can simplify this to

Now factor out an h to get

We can simplify and then evaluate the limit.

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