Calculus 2 : Derivatives

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #41 : Derivatives

Use the definition of a derivative to find  when .

Possible Answers:

Correct answer:

Explanation:

In order to find , we must remember that we can define a derivative as . Given , we can set , calculate , and solve the limit as:

 

Example Question #42 : Derivatives

Use the definition of a derivative to find  when .

Possible Answers:

Correct answer:

Explanation:

In order to find , we must remember that we can define a derivative as .

Given , we can set , calculate , and solve the limit as:

 

 

Example Question #43 : Derivatives

Find the velocity of a car at  when the position of the car is given by the following function:

Possible Answers:

Correct answer:

Explanation:

The limit definition of a derivative is as follows:

where  represents a very small change in .

Now, use the function given for the above formula:

which simplified becomes

Finally, plug in the given point:

 

Example Question #44 : Derivatives

Find the derivative of the given function using the definition of derivative.

Possible Answers:

Correct answer:

Explanation:

The definition of the derivative is 

For this problem, the derivative is

As such the derivative is

Example Question #45 : Derivatives

Find dy/dx:

Possible Answers:

Correct answer:

Explanation:

To solve for the derivative of the given function, we must realize the following:

Given:

This simplifies to:

This is one of the answer choices.

Example Question #46 : Derivatives

Find dy/dx:

Possible Answers:

Correct answer:

Explanation:

Before attempting to take the derivative of the given function, the following identity must be realized:

                                                                            

Using this identity, the given function can be simplified to:

From this taking the derivative is a fairly straightforward process:

We can simplify this expression by placing all the terms under a common denominator like so:

This is one of the answer choices.

Example Question #47 : Derivatives

Use a definition of the derivative with the function  to evaluate the following limit:

Possible Answers:

Correct answer:

Explanation:

Using the definition 

And plugging in our function, we get that

.

if we factor out  inside the limit we get 

since the  term doesn't contain an h we can factor it out, and then divide by both sides, getting that

but we know that 

So we find that the limit is equal to .

Example Question #48 : Derivatives

If the definition of the derivative is used to find the derivative of . Which of these expressions must be evaluated?

Possible Answers:

None of the other answers

Correct answer:

Explanation:

The definition of the derivative of a function is given by

With , we substitute into our function, and get . Substituting these into the formula gives us

.

Note: this limit is not easy to evaluate by hand, it would be easier to find the derivative using the Product Rule.

Example Question #49 : Derivatives

1a

Possible Answers:

 

 

 

 

Correct answer:

Explanation:

1

Example Question #50 : Derivatives

Find the derivative of  using the definition of the derivative.

Possible Answers:

Correct answer:

Explanation:

The definition of a derivative is 

Substituting these expressions into the definition of a derivative gives us

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