Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

Create An Account

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #21 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=7 - 8x\text{ as h approaches 0.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=7 - 8x\text{ and we can find that }\\&f(x+h)=7 - 8x - 8\cdot h\\&f'(x)=\frac{7 - 8x - 8\cdot h-(7 - 8x)}{h}\\&f'(x)=\frac{-8\cdot h}{h}\\&f'(x)=-8\\&\text{Then, if we take the limit as h goes to zero, we find:}\\&-8\end{align*}$

Report an Error

Example Question #31 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=- 5x - 1\text{ as h approaches 0.}\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=- 5x - 1\text{ and we can find that }\\&f(x+h)=- 5\cdot h - 5x - 1\\&f'(x)=\frac{- 5\cdot h - 5x - 1-(- 5x - 1)}{h}\\&f'(x)=\frac{-5\cdot h}{h}\\&f'(x)=-5\\&\text{Then, if we take the limit as h goes to zero, we find:}\\&-5\end{align*}$

Report an Error

Example Question #32 : Concept Of The Derivative

Evaluate the following difference quotient:

$\lim_{x\rightarrow \Delta x}\frac{cot(x+\Delta x)-cot(x)}{\Delta x}$

Possible Answers:

-csc(x)cot(x)

-csc^2(x)

sec(x)

-csc^2x

sec(x)tan(x)

Correct answer:

-csc^2(x)

Explanation:

One should recognize the difference quotient as being equivalent to the derivative or instantaneous rate of change.

That is, for some continuous function, f(x) .

$f'(x)=\lim_{x\rightarrow \Delta x}\frac{f(x+\Delta x)-f(x)}{\Delta x}$

The function in our question is cot(x)

We note from this that because the difference quotient is the derivative:

$\lim_{x\rightarrow \Delta x}\frac{cot(x+\Delta x)-cot(x)}{\Delta x}=-csc^2(x)$

Report an Error

Example Question #32 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=- x - 10\text{ as h approaches 0 for }x=-17\end{align*}$

Possible Answers:

$\text{Nonexistent.}$

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=- x - 10\text{ and we can find that }\\&f(x+h)=- h - x - 10\\&f'(x)=\frac{- h - x - 10-(- x - 10)}{h}\\&f'(x)=\frac{-h}{h}\\&f'(x)=-1\\&\text{Then, if we take the limit as h goes to zero, we find at }x=-17:\\&-1\end{align*}$

Report an Error

Example Question #33 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=4x - 8x^{2} + 8\text{ as h approaches 0 for }x=19\end{align*}$

Possible Answers:

$\frac{75}{2}$

$\text{Nonexistent.}$

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=4x - 8x^{2} + 8\text{ and we can find that }\\&f(x+h)=4\cdot h + 4x - 8\cdot (h + x)^{2} + 8\\&f'(x)=\frac{4\cdot h + 4x - 8\cdot (h + x)^{2} + 8-(4x - 8x^{2} + 8)}{h}\\&f'(x)=\frac{4\cdot h - 16\cdot hx - 8\cdot h^{2}}{h}\\&f'(x)=4 - 16x - 8\cdot h\\&\text{Then, if we take the limit as h goes to zero, we find at }x=19:\\&-300\end{align*}$

Report an Error

Example Question #34 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=9 - 8x^{2} - 7x\text{ as h approaches 0 for }x=19\end{align*}$

Possible Answers:

$\text{Nonexistent.}$

$\frac{311}{8}$

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=9 - 8x^{2} - 7x\text{ and we can find that }\\&f(x+h)=9 - 7x - 8\cdot (h + x)^{2} - 7\cdot h\\&f'(x)=\frac{9 - 7x - 8\cdot (h + x)^{2} - 7\cdot h-(9 - 8x^{2} - 7x)}{h}\\&f'(x)=\frac{- 7\cdot h - 16\cdot hx - 8\cdot h^{2}}{h}\\&f'(x)=- 8\cdot h - 16x - 7\\&\text{Then, if we take the limit as h goes to zero, we find at }x=19:\\&-311\end{align*}$

Report an Error

Example Question #35 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=3x + x^{2} + 7\text{ as h approaches 0 for }x=-4\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=3x + x^{2} + 7\text{ and we can find that }\\&f(x+h)=3\cdot h + 3x + (h + x)^{2} + 7\\&f'(x)=\frac{3\cdot h + 3x + (h + x)^{2} + 7-(3x + x^{2} + 7)}{h}\\&f'(x)=\frac{3\cdot h + 2\cdot hx + h^{2}}{h}\\&f'(x)=h + 2x + 3\\&\text{Then, if we take the limit as h goes to zero, we find at }x=-4:\\&-5\end{align*}$

Report an Error

Example Question #36 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=4x + 8x^{2} + 10\text{ as h approaches 0 for }x=13\end{align*}$

Possible Answers:

212

$\frac{53}{2}$

1414

108

Correct answer:

212

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=4x + 8x^{2} + 10\text{ and we can find that }\\&f(x+h)=4\cdot h + 4x + 8\cdot (h + x)^{2} + 10\\&f'(x)=\frac{4\cdot h + 4x + 8\cdot (h + x)^{2} + 10-(4x + 8x^{2} + 10)}{h}\\&f'(x)=\frac{4\cdot h + 16\cdot hx + 8\cdot h^{2}}{h}\\&f'(x)=8\cdot h + 16x + 4\\&\text{Then, if we take the limit as h goes to zero, we find at }x=13:\\&212\end{align*}$

Report an Error

Example Question #41 : Concept Of The Derivative

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=4 - 7x\text{ as h approaches 0 for }x=12\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=4 - 7x\text{ and we can find that }\\&f(x+h)=4 - 7x - 7\cdot h\\&f'(x)=\frac{4 - 7x - 7\cdot h-(4 - 7x)}{h}\\&f'(x)=\frac{-7\cdot h}{h}\\&f'(x)=-7\\&\text{Then, if we take the limit as h goes to zero, we find at }x=12:\\&-7\end{align*}$

Report an Error

Example Question #31 : Derivative Defined As The Limit Of The Difference Quotient

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=x - 7\text{ as h approaches 0 for }x=10\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=x - 7\text{ and we can find that }\\&f(x+h)=h + x - 7\\&f'(x)=\frac{h + x - 7-(x - 7)}{h}\\&f'(x)=\frac{h}{h}\\&f'(x)=1\\&\text{Then, if we take the limit as h goes to zero, we find at }x=10:\\&1\end{align*}$

Report an Error

View AP Calculus AB Tutors

Bindu
Certified Tutor

View AP Calculus AB Tutors

Justin
Certified Tutor

University of Washington-Seattle Campus, Current Undergrad Student, Biomedical Engineering.

View AP Calculus AB Tutors

Samantha
Certified Tutor

Cornell University, Bachelor of Science, Computer Science.

All AP Calculus AB Resources

3 Diagnostic Tests 164 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Chemistry Tutors in Denver, Calculus Tutors in Chicago, Physics Tutors in Seattle, Physics Tutors in Miami, Reading Tutors in Seattle, SAT Tutors in Boston, ACT Tutors in Atlanta, Statistics Tutors in Atlanta, SSAT Tutors in New York City, English Tutors in Atlanta

Popular Courses & Classes

Spanish Courses & Classes in Atlanta, GMAT Courses & Classes in Atlanta, SAT Courses & Classes in Philadelphia, GMAT Courses & Classes in Houston, MCAT Courses & Classes in Seattle, MCAT Courses & Classes in Chicago, ACT Courses & Classes in Philadelphia, GRE Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Boston

Popular Test Prep

MCAT Test Prep in Washington DC, GMAT Test Prep in Boston, LSAT Test Prep in Phoenix, ACT Test Prep in San Francisco-Bay Area, SSAT Test Prep in Chicago, LSAT Test Prep in Atlanta, ISEE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Dallas Fort Worth, SAT Test Prep in Seattle, GRE Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

AP Calculus AB : AP Calculus AB

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #21 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #31 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #32 : Concept Of The Derivative

Example Question #32 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #33 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #34 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #35 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #36 : Derivative Defined As The Limit Of The Difference Quotient

Example Question #41 : Concept Of The Derivative

Example Question #31 : Derivative Defined As The Limit Of The Difference Quotient

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: