Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Derivatives

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 #1203 : Ap Calculus Ab

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=2x + 6x^{2} + 10\text{ as h approaches 0 for }x=-10\end{align*}$

Possible Answers:

580

$-\frac{59}{3}$

590

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=2x + 6x^{2} + 10\text{ and we can find that }\\&f(x+h)=2\cdot h + 2x + 6\cdot (h + x)^{2} + 10\\&f'(x)=\frac{2\cdot h + 2x + 6\cdot (h + x)^{2} + 10-(2x + 6x^{2} + 10)}{h}\\&f'(x)=\frac{2\cdot h + 12\cdot hx + 6\cdot h^{2}}{h}\\&f'(x)=6\cdot h + 12x + 2\\&\text{Then, if we take the limit as h goes to zero, we find at }x=-10:\\&-118\end{align*}$

Report an Error

Example Question #1204 : Ap Calculus Ab

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=- 10x - 6\text{ as h approaches 0 for }x=-1\end{align*}$

Possible Answers:

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=- 10x - 6\text{ and we can find that }\\&f(x+h)=- 10\cdot h - 10x - 6\\&f'(x)=\frac{- 10\cdot h - 10x - 6-(- 10x - 6)}{h}\\&f'(x)=\frac{-10\cdot h}{h}\\&f'(x)=-10\\&\text{Then, if we take the limit as h goes to zero, we find at }x=-1:\\&-10\end{align*}$

Report an Error

Example Question #1205 : Ap Calculus Ab

$\begin{align*}&\text{Calculate the limit of the difference quotient }f'(x)=\frac{f(x+h)-f(x)}{h}\\&\text{For }f(x)=5x^{2} - 4\text{ as h approaches 0 for }x=-9\end{align*}$

Possible Answers:

401

$\text{Nonexistent.}$

Correct answer:

Explanation:

$\begin{align*}&\text{The difference quotient is a means of approximating a rate of}\\&\text{change over a given increment, h. As the increment, h, grows}\\&\text{smaller and smaller, we get increasingly refine this approximation,}\\&\text{gaining an increasingly accurate representation of an instantaneous}\\&\text{rate of change, rather than a broader average. This is what}\\&\text{is also known as the derivative. For our problem:}\\&f(x)=5x^{2} - 4\text{ and we can find that }\\&f(x+h)=5\cdot (h + x)^{2} - 4\\&f'(x)=\frac{5\cdot (h + x)^{2} - 4-(5x^{2} - 4)}{h}\\&f'(x)=\frac{10\cdot hx + 5\cdot h^{2}}{h}\\&f'(x)=5\cdot h + 10x\\&\text{Then, if we take the limit as h goes to zero, we find at }x=-9:\\&-90\end{align*}$

Report an Error

Example Question #1206 : Ap Calculus Ab

Find the derivative of the function using the limit of the difference quotient:

f(x)=12x+3

Possible Answers:

The limit does not exist

None of the other answers

Correct answer:

Explanation:

The derivative of a function, f(x) , as defined by the limit of the difference quotient is

$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Taking the limit of our function - and remembering the limit at each step! - we get

$\lim_{h\rightarrow 0}\frac{12(x+h)+3-(12x+3)}{h}=\lim_{h\rightarrow 0}\frac{12x+12h+3-12x-3}{h}=\lim_{h\rightarrow 0}\frac{12h}{h}=12$

Report an Error

Example Question #1207 : Ap Calculus Ab

Find the derivative of the function using the limit of the difference quotient:

f(x)=3x^2+x

Possible Answers:

6x+3h+1

6x+1

Correct answer:

6x+1

Explanation:

The derivative of a function, f(x) , as defined by the limit of the difference quotient is

$\lim_{h\rightarrow 0}\frac{f(x+h)-f(x)}{h}$

Taking the limit of our function (and remembering to write the limit at each step) we get

$\lim_{h\rightarrow 0}\frac{3(x+h)^2+(x+h)-(3x^2+x)}{h}=\lim_{h\rightarrow 0}\frac{3(x^2+2xh+h^2)+x+h-3x^2-x}{h}=\lim_{h\rightarrow 0}\frac{3x^2+6xh+3h^2+x+h-3x^2-x}{h}=\lim_{h\rightarrow 0}\frac{h(6x+3h+1)}{h}=6x+1$

Report an Error

Example Question #51 : Concept Of The Derivative

Find the derivative of the function using the limit of the difference quotient:

f(x)=5x^2+1

Possible Answers:

10x+1

10x

15xh

10x+5h

Correct answer:

10x

Explanation:

The derivative of a function f(x) is defined by the limit of the difference quotient, as follows:

$\lim_{h\rightarrow 0}\frac{ f(x+h)-f(x)}{h}$

Using this limit for our function, and remembering to write the limit at every step, we get

$\lim_{h\rightarrow 0} \frac{5(x+h^2)+1-(5x^2+1)}{h}=\lim_{h\rightarrow 0} \frac{5(x^2+2xh+h^2)+1-5x^2-1}{h}=\lim_{h\rightarrow 0}\frac{10xh+5h^2}{h}=\lim_{h\rightarrow 0} (10x+5h) = 10x$

Report an Error

Example Question #1211 : Ap Calculus Ab

Let f(x)=sin(x)-cos(x) .

Find the second derivative of f(x) .

Possible Answers:

f''(x)=sin^2(x)-cos^2(x)

f''(x)=sin(x)-cos(x)

f''(x)=cos(x)-sin(x)

f''(x)=sin^2(x)+cos^2(x)

f''(x)=sin(x)+cos(x)

Correct answer:

f''(x)=cos(x)-sin(x)

Explanation:

The second derivative is just the derivative of the first derivative. So first we find the first derivative of f(x) . Remember the derivative of $\sin x$ is $\cos x$ , and the derivative for $\cos x$ is $-\sin x$ .

f'(x)= cos(x)+sin(x)

Then to get the second derivative, we just derive this function again. So

f''(x)=-sin(x)+cos(x)=cos(x)-sin(x)

Report an Error

Example Question #51 : Calculus I — Derivatives

Define $f (x) = 6x^{3} - 12x^{2} + 4x -8$ .

What is f ' ' (x) ?

Possible Answers:

f ' '(x) = 18

f ' '(x) = 36x

f ' '(x) = 6x - 1 2

f ''(x) = 18x - 24

f ' '(x) = 36x - 24

Correct answer:

f ' '(x) = 36x - 24

Explanation:

Take the derivative of , then take the derivative of .

$f (x) = 6x^{3} - 12x^{2} + 4x -8$

$f '(x) = 3 \cdot 6x^{3-1} - 2 \cdot 12x^{2-1} + 4 -0$

$f '(x) = 18x^{2} - 24x+ 4$

$f ' '(x) = 2 \cdot 18x^{2-1} - 24+ 0$

f ' '(x) = 36x - 24

Report an Error

Example Question #12 : Finding Second Derivative Of A Function

Define $g(x) = \cos \left (x^{2} \right )$ .

What is g''(x) ?

Possible Answers:

$g''(x) = - 2 \sin(x^2)-4x^2 \cos(x^2) \right ]$

$g''(x) = - 2 \cos(x^2)-4x^2 \sin(x^2) \right ]$

$g''(x) =-4x^2 \cos(x^2)$

$g''(x) = -4x^2 \sin(x^2)$

$g''(x) =4x^2 \cos(x^2)$

Correct answer:

$g''(x) = - 2 \sin(x^2)-4x^2 \cos(x^2) \right ]$

Explanation:

Take the derivative of , then take the derivative of .

$g(x) = \cos \left (x^{2} \right )$

$g'(x) = 2x \cdot [-\sin (x^2)]$

$g'(x) = - 2x \sin (x^2)$

$g''(x) = - 2\left [ 1 \cdot \sin(x^2)+ x \cdot 2x \cos(x^2) \right ]$

$g''(x) = - 2\left [ \sin(x^2)+ 2x^2 \cos(x^2) \right ]$

$g''(x) = - 2 \sin(x^2)-4x^2 \cos(x^2) \right ]$

Report an Error

Example Question #11 : Finding Second Derivative Of A Function

Define $f(x) = \frac{1}{e^{4x}}$ .

What is f ''(x) ?

Possible Answers:

$f(x) = \frac{1}{16e^{4x}}$

$f''(x) = \frac{1 }{e^{4x-2}}$

$f''(x) = \frac{16 }{e^{4x}}$

$f''(x) =- \frac{16 }{e^{4x}}$

$f''(x) = \frac{1 }{e^{4x+2}}$

Correct answer:

$f''(x) = \frac{16 }{e^{4x}}$

Explanation:

Rewrite:

$f(x) = \frac{1}{e^{4x}}$

$f(x) = e^{-4x}$

Take the derivative of , then take the derivative of .

$f'(x) = -4 e^{-4x}$

$f''(x) = -4 \cdot \left (-4 \right )e^{-4x} = 16e^{-4x} = \frac{16 }{e^{4x}}$

Report an Error

View AP Calculus AB Tutors

Bertha
Certified Tutor

California Institute of Technology, Bachelor's (in progress), Chemical Engineering.

View AP Calculus AB Tutors

Sayantan
Certified Tutor

Western Michigan University, Bachelor of Science, Applied Mathematics.

View AP Calculus AB Tutors

Adam
Certified Tutor

Michigan Technological University, Bachelor of Science, Business, General. Ramkhamhaeng, Master of Science, Business Administ...

All AP Calculus AB Resources

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

Popular Subjects

English Tutors in Los Angeles, Chemistry Tutors in Dallas Fort Worth, Math Tutors in San Francisco-Bay Area, English Tutors in Boston, Chemistry Tutors in Boston, English Tutors in Miami, ISEE Tutors in New York City, MCAT Tutors in Philadelphia, Reading Tutors in Los Angeles, Spanish Tutors in Miami

Popular Courses & Classes

SAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Los Angeles, GMAT Courses & Classes in Boston, ISEE Courses & Classes in New York City, ISEE Courses & Classes in Miami, ISEE Courses & Classes in Houston, ACT Courses & Classes in Boston, Spanish Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Chicago

Popular Test Prep

GMAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Miami, SSAT Test Prep in Denver, GMAT Test Prep in Denver, SSAT Test Prep in Los Angeles, SAT Test Prep in Dallas Fort Worth, GMAT Test Prep in Washington DC, SAT Test Prep in Miami, MCAT Test Prep in Boston

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 : Derivatives

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1203 : Ap Calculus Ab

Example Question #1204 : Ap Calculus Ab

Example Question #1205 : Ap Calculus Ab

Example Question #1206 : Ap Calculus Ab

Example Question #1207 : Ap Calculus Ab

Example Question #51 : Concept Of The Derivative

Example Question #1211 : Ap Calculus Ab

Example Question #51 : Calculus I — Derivatives

Example Question #12 : Finding Second Derivative Of A Function

Example Question #11 : Finding Second Derivative Of A Function

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: