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 #74 : Derivatives Of Functions

Find the limit of the function below using L'Hopital's rule

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}$

Possible Answers:

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=\frac{6}{\pi}$

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=\frac{\pi}{6}$

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=-6\pi$

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=6\pi$

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=\frac{0}{0}$

Correct answer:

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=\frac{6}{\pi}$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$\binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}$

Plug in 4 for q to test to use L'Hopital's rule

$\binom{lim}{q\rightarrow 4}=\frac{4^2-2(4)-8}{sin(4\pi)}$

$\binom{lim}{q\rightarrow 4}=\frac{0}{0}$

From this indeterminate form, realize we can now use L'Hopital's rule, deriving the expressions in the numerator and denominator independently of one another

$\binom{lim}{q\rightarrow 4}\frac{d(q^2-2q-8)}{d(sin(q\pi))}$

$\binom{lim}{q\rightarrow 4}\frac{2q-2}{\pi(cos(q\pi))}$ (pi comes from chain rule with qpi)

$\binom{lim}{q\rightarrow 4}=\frac{2(4)-2}{\pi(cos((4)\pi))}$

$\binom{lim}{q\rightarrow 4}=\frac{6}{\pi*1}$

Thus, we arrive at our correct answer:

*****************************************************************

CORRECT ANSWER

*****************************************************************

${\color{Red} \binom{lim}{q\rightarrow 4}\frac{q^2-2q-8}{sin(q\pi)}=\frac{6}{\pi}}$

Report an Error

Example Question #75 : Derivatives Of Functions

Find the derivative of the function:

$s(r)=7^{2r}$

Possible Answers:

s'(r)=2r*2ln(7)

s'(r)=rln(7)

s'(r)=-2r*2ln(7)

$s'(r)=7^{2r}*2ln(7)$

$s'(r)=7^{2r}*7ln(2)$

Correct answer:

$s'(r)=7^{2r}*2ln(7)$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$s(r)=7^{2r}$

Understand the derivative of a^x:

d(a^x)=a^xln(a)

Thus:

$s'(r)=7^{2r}*ln(7)*(chain)$

We find the chain by deriving the compound operation "2r"

d(2r)=2

Plug 2 in for the chain:

$s'(r)=7^{2r}*ln(7)*2$

Thus, we arrive at the correct answer:

*****************************************************************

CORRECT ANSWER

*****************************************************************

$s'(r)=7^{2r}*2ln(7)$

Report an Error

Example Question #76 : Derivatives Of Functions

Find the derivative of the function:

$y=cos^{-1}(x^2)$

Possible Answers:

$y'=-\frac{\sqrt{1-x^4}}{2x}$

$y'=\frac{\sqrt{1-x^4}}{2x}$

$y'=\frac{-2x}{\sqrt{1-x^4}}$

$y'=-\frac{\sqrt{1-x^2}}{x}$

$y'=\frac{2x}{\sqrt{1-x^4}}$

Correct answer:

$y'=\frac{-2x}{\sqrt{1-x^4}}$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$y=cos^{-1}(x^2)$

$y'=d(cos^{-1}(x^2))*(chain)$

$d(cos^-1(x^2))=\frac{-1}{\sqrt{1-(x^2)^2}}$

$d(cos^-1(x^2))=\frac{-1}{\sqrt{1-x^4}}$

Now, take the derivative of x^2, and plug in for the "chain"

$d(cos^-1(x^2))=\frac{-1}{\sqrt{1-x^4}}*(chain)$

$d(cos^-1(x^2))=\frac{-1}{\sqrt{1-x^4}}*(d(x^2))$

$d(cos^-1(x^2))=\frac{-1}{\sqrt{1-x^4}}*2x$

Multiplying, we arrive at the correct answer

*****************************************************************

CORRECT ANSWER

*****************************************************************

$d(cos^-1(x^2))={\color{Red} y'=\frac{-2x}{\sqrt{1-x^4}}}$

Report an Error

Example Question #77 : Derivatives Of Functions

Find the limit of the function below using L'Hopital's Rule

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}$

Possible Answers:

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{-5}{6}$

The limit does not exist

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{5}{6}$

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{6}{5}$

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{-6}{5}$

Correct answer:

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{-6}{5}$

Explanation:

*****************************************************************

STEPS

*****************************************************************

Given:

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}$

Try the limit. Plug in two for y and check the result:

$\binom{lim}{y\rightarrow 2}=\frac{(2)^3-7(2)^2+10(2)}{(2)^2+(2)-6}$

$\binom{lim}{y\rightarrow 2}=\frac{8-28+20}{4+2-6}$

$\binom{lim}{y\rightarrow 2}=\frac{0}{0}$

Thus, we realize me must use L'Hopital's Rule on the original quotient, deriving the expressions in the numerator and denominator independently

$\binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}$

$\binom{lim}{y\rightarrow 2}\frac{d(y^3-7y^2+10y)}{d(y^2+y-6)}$

$\binom{lim}{y\rightarrow 2}\frac{3y^2-14y+10}{2y+1}$

Try the limit once more:

$\binom{lim}{y\rightarrow 2}=\frac{3(2)^2-14(2)+10}{2(2)+1}$

$\binom{lim}{y\rightarrow 2}=\frac{12-28+10}{4+1}$

$\binom{lim}{y\rightarrow 2}=\frac{-16+10}{5}$

Simplifying the numerator, we arrive at the correct answer:

*****************************************************************

CORRECT ANSWER

*****************************************************************

${\color{Red} \binom{lim}{y\rightarrow 2}\frac{y^3-7y^2+10y}{y^2+y-6}=\frac{-6}{5}}$

Report an Error

Example Question #78 : Derivatives Of Functions

Find the derivative of the function

$g(x)=e^{\lambda x^3}+\cos(\lambda x)$ ,

where $\lambda$ is a constant.

Possible Answers:

$-\lambda\sin(\lambda x)$

$3 \lambda x^2 e^{\lambda x^3}-\lambda\sin(\lambda x)$

$3 \lambda x^2 e^{\lambda x^3}-\sin(\lambda x)$

$3 \lambda x^2 e^{\lambda x^3}+\lambda\sin(\lambda x)$

Correct answer:

$3 \lambda x^2 e^{\lambda x^3}-\lambda\sin(\lambda x)$

Explanation:

The derivative of the function is equal to

$g'(x)=3 \lambda x^2 e^{\lambda x^3}-\lambda\sin(\lambda x)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u \cdot \frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x}ax=a$

Report an Error

Example Question #96 : Derivatives

Find the derivative of the function:

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

Possible Answers:

$2x^3+2xe^{\sin(x)}-x^2\cos(x)e^{\sin(x)}$

$4x^3+2xe^{\sin(x)}+x^2\cos(x)e^{\sin(x)}$

$2x^3+2xe^{\sin(x)}$

$2x^3+2xe^{\sin(x)}+x^2\cos(x)e^{\sin(x)}$

Correct answer:

$2x^3+2xe^{\sin(x)}+x^2\cos(x)e^{\sin(x)}$

Explanation:

The derivative of the function is equal to

$f'(x)=2x^3+2xe^{\sin(x)}+x^2\cos(x)e^{\sin(x)}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sin(x)=\cos(x)$

Report an Error

Example Question #81 : Computation Of The Derivative

Find the derivative of the function:

$f=\ln(6x+x^2)$

Possible Answers:

$\frac{2x+6}{x^2+6x}$

$\frac{8}{x+6}$

$-\frac{2x+6}{x^2+6x}$

$\frac{1}{x^2+6x}$

Correct answer:

$\frac{2x+6}{x^2+6x}$

Explanation:

The derivative of the function is equal to

$f'=\frac{2x+6}{x^2+6x}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \ln(u)=\frac{1}{u}\frac{\mathrm{d} u}{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Report an Error

Example Question #82 : Computation Of The Derivative

Find the derivative of the following function:

$f(x)=\frac{x^2+3x+1}{x^4}$

Possible Answers:

$\frac{-2x^2-9x-4}{x^5}$

$\frac{6x^2+15x+4}{x^5}$

$\frac{2x^2+9x+4}{x^5}$

Correct answer:

$\frac{-2x^2-9x-4}{x^5}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{-2x^2-9x-4}{x^5}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

The derivative was simplified from

$\frac{-2x^5-9x^4-4x^3}{x^8}$

to its most simple form.

Report an Error

Example Question #83 : Computation Of The Derivative

Find the derivative of the function:

$f(x)=\frac{x\cos(x)}{x+3}$

Possible Answers:

$\frac{(x+3)(\cos(x)+x\sin(x))-x\cos(x)}{(x+3)^2}$

$\frac{\cos(x)-x\sin(x)-x\cos(x)}{x+3}$

$\frac{(x+3)(\cos(x)-x\sin(x))-x\cos(x)}{(x+3)^2}$

$\frac{(x+3)(\cos(x)-x\sin(x))+x\cos(x)}{(x+3)^2}$

Correct answer:

$\frac{(x+3)(\cos(x)-x\sin(x))-x\cos(x)}{(x+3)^2}$

Explanation:

The derivative of the function is equal to

$f'(x)=\frac{(x+3)(\cos(x)-x\sin(x))-x\cos(x)}{(x+3)^2}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} \sin(x)=\cos(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$

Report an Error

Example Question #84 : Computation Of The Derivative

Find the derivative of the function:

$f=2\sec(x)+2\csc(x)$

Possible Answers:

$2\ln\left |\sec(x)+\tan(x) \right |-2\ln\left |\csc(x)+\cot(x) \right |$

$2\csc(x)\cot(x)-2\sec(x)\tan(x)$

$-2\sec(x)\tan(x)-2\csc(x)\cot(x)$

$2\sec(x)\tan(x)-2\csc(x)\cot(x)$

Correct answer:

$2\sec(x)\tan(x)-2\csc(x)\cot(x)$

Explanation:

The derivative of the function is equal to

$f'=2\sec(x)\tan(x)-2\csc(x)\cot(x)$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \sec(x)=\sec(x)\tan(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x }\csc(x)=-\csc(x)\cot(x)$

Report an Error

View AP Calculus AB Tutors

Gary
Certified Tutor

SUNY at Albany, Bachelor of Science, Economics. Pace University-New York, Masters in Business Administration, Analysis and Fu...

View AP Calculus AB Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

View AP Calculus AB Tutors

Lyric
Certified Tutor

Spelman College, Bachelor of Science, Mathematics.

All AP Calculus AB Resources

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

Popular Subjects

Physics Tutors in Boston, Algebra Tutors in Phoenix, Statistics Tutors in Seattle, French Tutors in Phoenix, Spanish Tutors in Dallas Fort Worth, MCAT Tutors in Philadelphia, Computer Science Tutors in Washington DC, Computer Science Tutors in Boston, Physics Tutors in Atlanta, Algebra Tutors in Miami

Popular Courses & Classes

GMAT Courses & Classes in Boston, ACT Courses & Classes in San Diego, GRE Courses & Classes in Miami, ACT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Los Angeles, GMAT Courses & Classes in Washington DC, MCAT Courses & Classes in Houston, ACT Courses & Classes in Atlanta, ACT Courses & Classes in Boston, GRE Courses & Classes in Denver

Popular Test Prep

ACT Test Prep in Seattle, MCAT Test Prep in Washington DC, MCAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Phoenix, SSAT Test Prep in Boston, SAT Test Prep in San Diego, GMAT Test Prep in Boston, LSAT Test Prep in Miami, ACT Test Prep in San Francisco-Bay Area, GRE Test Prep in Denver

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 #74 : Derivatives Of Functions

Example Question #75 : Derivatives Of Functions

Example Question #76 : Derivatives Of Functions

Example Question #77 : Derivatives Of Functions

Example Question #78 : Derivatives Of Functions

Example Question #96 : Derivatives

Example Question #81 : Computation Of The Derivative

Example Question #82 : Computation Of The Derivative

Example Question #83 : Computation Of The Derivative

Example Question #84 : Computation Of The Derivative

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: