We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Functions, Graphs, and Limits

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 #2203 : High School Math

$\int_0^4-8x^2+15=?$

Possible Answers:

$-230\tfrac{2}{3}$

$-110\tfrac{2}{3}$

$230\tfrac{2}{3}$

569

Correct answer:

$-110\tfrac{2}{3}$

Explanation:

Use the Fundamental Theorem of Calculus. If g'(x)=f(x) , then $\int_a^bf(x)=g(b)-g(a)$ .

Therefore we need to find the indefinite integral.

To find the indefinite integral, we use the reverse power rule. That means we raise the exponent on the variables by one and then divide by the new exponent.

$\int -8x^2+15{\mathrm{d} x}=\frac{-8x^{2+1}}{2+1}+\frac{15x^{0+1}}{0+1}+c$

Remember to include a when computing integrals. This is a place holder for any constant that might be in the new expression.

$\int -8x^2+15{\mathrm{d} x}=\frac{-8x^{3}}{3}+\frac{15x^{1}}{1}+c$

$\int -8x^2+15{\mathrm{d} x}=-\frac{8}{3}x^3+15x+c$

Plug that back into FTOC:

$\int_a^b -8x^2+15{\mathrm{d} x}=(-\frac{8}{3}b^3+15b+c)-(-\frac{8}{3}a^3+15a+c)$

Notice that the 's cancel out.

Plug in our given numbers.

$\int_0^4 -8x^2+15{\mathrm{d} x}=(-\frac{8}{3}(4)^3+15(4))-(-\frac{8}{3}(0)^3+15(0))$ $\int_0^4 -8x^2+15{\mathrm{d} x}=(-\frac{8}{3}(64)+(60))-(0+0)$

$\int_0^4 -8x^2+15{\mathrm{d} x}=(-170\tfrac{2}{3}+(60))$

$\int_0^4 -8x^2+15{\mathrm{d} x}=-110\tfrac{2}{3}$

Report an Error

Example Question #71 : Calculus Ii — Integrals

$\int 4x^2+5x-3\: dx=?$

Possible Answers:

$\frac{4}{3}x^3+\frac{5}{2}x^2-3x+c$

$\frac{3}{4}x^2+\frac{5}{2}x-3+c$

8x+5

$\frac{4}{3}x^3+\frac{5}{x^2}+3x+c$

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

Correct answer:

$\frac{4}{3}x^3+\frac{5}{2}x^2-3x+c$

Explanation:

To find the indefinite integral of our given equation, we can use the reverse power rule: we raise the exponent by one and then divide by that new exponent.

$\int 4x^2+5x-3\: dx=\frac{4x^{2+1}}{2+1}+\frac{5x^{1+1}}{1+1}-\frac{3x^{0+1}}{0+1}+c$

Don't forget to include a to compensate for any constant!

$\int 4x^2+5x-3\: dx=\frac{4x^{3}}{3}+\frac{5x^{2}}{2}-\frac{3x^{1}}{1}+c$

$\int 4x^2+5x-3\: dx=\frac{4}{3}x^3+\frac{5}{2}x^2-3x+c$

Report an Error

Example Question #104 : Asymptotic And Unbounded Behavior

What is the indefinite integral of 4x+3 with respect to ?

Possible Answers:

2x^2+3x+c

4+c

2x^3-c

2x^2+3x

Correct answer:

2x^2+3x+c

Explanation:

To find the indefinite integral, we're going to use the reverse power rule: raise the exponent of the variable by one and then divide by that new exponent.

$\int 4x+3 dx=(\frac{4x^{1+1}}{1+1})+(\frac{3x^{0+1}}{0+1})+c$

Be sure to include + c to compensate for any constant!

$\int 4x+3dx=(\frac{4x^{1+1}}{1+1})+(\frac{3x^{0+1}}{0+1})+c$

$\int 4x+3dx=(\frac{4x^{2}}{2})+(\frac{3x^{1}}{1})+c$

$\int 4x+3dx=2x^2+3x+c$

Report an Error

Example Question #2204 : High School Math

$\int_3^4x^3+2x^2+5=?$

Possible Answers:

$53\tfrac{1}{4}$

$179\tfrac{7}{12}$

$73\tfrac{5}{12}$

$126\tfrac{2}{3}$

Correct answer:

$73\tfrac{5}{12}$

Explanation:

The Fundamental Theorem of Calculus states that if g'(x)=f(x) , then $\int_a^bf(x)=g(b)-g(a)$ . Therefore, we need to find the indefinite integral of our given equation.

To find the indefinite integral, we can use the reverse power rule. Raise the exponent of the variable by one and then divide by that new exponent.

We're going to treat as 5x^0 .

$\int x^3+2x^2+5{\mathrm{d} x}=\frac{x^{3+1}}{3+1}+\frac{2x^{2+1}}{2+1}+\frac{5x^{0+1}}{0+1}+c$

Remember to include the when taking the integral to compensate for any constant.

$\int x^3+2x^2+5{\mathrm{d} x}=\frac{x^{4}}{4}+\frac{2x^{3}}{3}+\frac{5x^{1}}{1}+c$

Simplify.

$\int x^3+2x^2+5{\mathrm{d} x}=\frac{1}{4}x^4+\frac{2}{3}x^3+5x+c$

Plug that into FTOC:

$\int_a^b x^3+2x^2+5{\mathrm{d} x}=(\frac{1}{4}(b)^4+\frac{2}{3}(b)^3+5(b)+c)-(\frac{1}{4}(a)^4+\frac{2}{3}(a)^3+5(a)+c)$

Notice that the 's cancel out.

Plug in our given numbers.

$\int_3^4 x^3+2x^2+5{\mathrm{d} x}=(\frac{1}{4}(4)^4+\frac{2}{3}(4)^3+5(4))-(\frac{1}{4}(3)^4+\frac{2}{3}(3)^3+5(3))$ $\int_3^4 x^3+2x^2+5{\mathrm{d} x}=(\frac{1}{4}(256)+\frac{2}{3}(64)+20)-(\frac{1}{4}(81)+\frac{2}{3}(27)+15)$

$\int_3^4 x^3+2x^2+5{\mathrm{d} x}=(64+42\tfrac{2}{3}+20)-(20\tfrac{1}{4}+18+15)$

$\int_3^4 x^3+2x^2+5{\mathrm{d} x}=(126\tfrac{2}{3})-(53\tfrac{1}{4})$

$\int_3^4 x^3+2x^2+5{\mathrm{d} x}=73\tfrac{5}{12}$

Report an Error

Example Question #2205 : High School Math

$\int_2^44x^2+\frac{2}{3}x{\mathrm{d} x}=?$

Possible Answers:

$78\tfrac{1}{3}$

$102\tfrac{2}{3}$

308

$78\tfrac{2}{3}$

708

Correct answer:

$78\tfrac{2}{3}$

Explanation:

To find the definite integral, we can use the Fundamental Theorem of Calculus that states that if g'(x)=f(x) , then $\int_a^bf(x)=g(b)-g(a)$ .

Therefore, we need to find the indefinite integral of our equation to start.

To find the indefinite integral, we can use the reverse power rule. We raise the exponent of the variable by one and divide by our new exponent. For this problem that would look like this:

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4x^{2+1}}{2+1}+\frac{\frac{2}{3}x^{1+1}}{1+1}+c$

Remember to include a to cover any potential constant that might be in our new equation.

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4x^{3}}{3}+\frac{\frac{2}{3}x^{2}}{2}+c$

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4}{3}x^3+\frac{1}{3}x^2+c$

Plug that into FTOC:

$\int_a^b 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}b^3+\frac{1}{3}b^2+c)-(\frac{4}{3}a^3+\frac{1}{3}a^2+c)$

Notice that the 's cancel out.

Plug in our given values.

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}(4)^3+\frac{1}{3}(4)^2)-(\frac{4}{3}(2)^3+\frac{1}{3}(2)^2)$

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}(64)+\frac{1}{3}(16))-(\frac{4}{3}(8)+\frac{1}{3}(4))$

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{256}{3}+\frac{16}{3})-(\frac{32}{3}+\frac{4}{3})$

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{272}{3})-(\frac{36}{3})$

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{236}{3}$

$\int_2^4 4x^2+\frac{2}{3}x{\mathrm{d} x}=78\tfrac{2}{3}$

Report an Error

Example Question #2206 : High School Math

$\int_3^64x^2+\frac{2}{3}x{\mathrm{d} x}=?$

Possible Answers:

7.69

339

300

261

Correct answer:

261

Explanation:

To find the definite integral, we can use the Fundamental Theorem of Calculus which states that if g'(x)=f(x) , then $\int_a^bf(x)=g(b)-g(a)$ .

Therefore, we need to find the indefinite integral of our equation to start.

To find the indefinite integral, we can use the reverse power rule. We raise the exponent of the variable by one and divide by our new exponent.

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4x^{2+1}}{2+1}+\frac{\frac{2}{3}x^{1+1}}{1+1}+c$

Remember to include a to cover any potential constant that might be in our new equation.

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4x^{3}}{3}+\frac{\frac{2}{3}x^{2}}{2}+c$

$\int 4x^2+\frac{2}{3}x{\mathrm{d} x}=\frac{4}{3}x^3+\frac{1}{3}x^2+c$

Plug that into FTOC:

$\int_a^b 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}b^3+\frac{1}{3}b^2+c)-(\frac{4}{3}a^3+\frac{1}{3}a^2+c)$

Notice that the 's cancel out.

Plug in our given values.

$\int_3^6 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}(6)^3+\frac{1}{3}(6)^2)-(\frac{4}{3}(3)^3+\frac{1}{3}(3)^2)$

$\int_3^6 4x^2+\frac{2}{3}x{\mathrm{d} x}=(\frac{4}{3}(216)+\frac{1}{3}(36))-(\frac{4}{3}(27)+\frac{1}{3}(9))$

$\int_3^6 4x^2+\frac{2}{3}x{\mathrm{d} x}=(288+12)-(36+3)$

$\int_3^6 4x^2+\frac{2}{3}x{\mathrm{d} x}=(300)-(39)$

$\int_3^6 4x^2+\frac{2}{3}x{\mathrm{d} x}=261$

Report an Error

Example Question #21 : Finding Definite Integrals

$\int_{-\pi}^{\pi}\sin(x){\mathrm{d} x}=?$

Possible Answers:

$2\pi$

Correct answer:

Explanation:

The fundamental theorem of calculus states that if f'(x)=g(x) , then $\int_a^bg(x){\mathrm{d} x}=f(b)-f(a)$ .

First, we need to find the indefinite integral of our given equation. Just like with the derivatives, the indefinite integrals or anti-derivatives of trig functions must be memorized.

$\int \sin(x){\mathrm{d} x}=-\cos(x)+c$

Don't forget the to compensate for any potential constant!

Plug this in to our FTOC:

$\int_b^a \sin(x){\mathrm{d} x}=(-\cos(b)+c)-(-\cos(a)+c)$ .

Notice that the 's cancel out.

$\int_b^a \sin(x){\mathrm{d} x}=-\cos(b)+\cos(a)$ .

Now plug in the given values.

$\int_{-\pi}^{\pi} \sin(x){\mathrm{d} x}=-\cos(\pi)+\cos(-\pi)$

$\int_{-\pi}^{\pi} \sin(x){\mathrm{d} x}=0+0$

$\int_{-\pi}^{\pi} \sin(x){\mathrm{d} x}=0$

Report an Error

Example Question #29 : Finding Integrals

$\int_2^8 4x^2+5x-3\: dx=?$

Possible Answers:

216

804

810

$818\tfrac{2}{3}$

270

Correct answer:

804

Explanation:

To solve for the definite integral, use the fundamental theorem of calculus. If g'(x)=f(x) , then $\int_a^bf(x)=g(b)-g(a)$ .

First we need to find the indefinite integral.

To find the indefinite integral of our given equation, we can use the reverse power rule: we raise the exponent by one and then divide by that new exponent.

$\int 4x^2+5x-3\: dx=\frac{4x^{2+1}}{2+1}+\frac{5x^{1+1}}{1+1}-\frac{3x^{0+1}}{0+1}+c$

Don't forget to include a to compensate for any constant!

$\int 4x^2+5x-3\: dx=\frac{4x^{3}}{3}+\frac{5x^{2}}{2}-\frac{3x^{1}}{1}+c$

$\int 4x^2+5x-3\: dx=\frac{4}{3}x^3+\frac{5}{2}x^2-3x+c$

Plug this into our first FTOC equation:

$\int_a^b 4x^2+5x-3\: dx=(\frac{4}{3}b^3+\frac{5}{2}b^2-3b+c)-(\frac{4}{3}a^3+\frac{5}{2}a^2-3a+c)$

Notice that the 's cancel out.

$\int_a^b 4x^2+5x-3\: dx=(\frac{4}{3}b^3+\frac{5}{2}b^2-3b)-(\frac{4}{3}a^3+\frac{5}{2}a^2-3a)$

Plug in our given values.

$\int_2^8 4x^2+5x-3\: dx=(\frac{4}{3}(8)^3+\frac{5}{2}(8)^2-3(8))-(\frac{4}{3}(2)^3+\frac{5}{2}(2)^2-3(2))$

$\int_2^8 4x^2+5x-3\: dx=(682\tfrac{2}{3}+160-24)-(10\tfrac{2}{3}+10-6)$

$\int_2^8 4x^2+5x-3\: dx=(818\tfrac{2}{3})-(14\tfrac{2}{3})$

$\int_2^8 4x^2+5x-3\: dx=804$

Report an Error

Example Question #321 : Ap Calculus Ab

Find the indefinite ingtegral for $\int\frac{x+1}{\sqrt{x}} dx$ .

Possible Answers:

$x^{}\frac{3}{2}+ x^{\frac{1}{2}}+C$

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

$\frac{2}{3}x^{\frac{3}{2}}+2x^{\frac{1}{2}}+C$

Correct answer:

$\frac{2}{3}x^{\frac{3}{2}}+2x^{\frac{1}{2}}+C$

Explanation:

First, bring up the radical into the numerator and distribute to the (x+1) term.

$(x+1)x^{\frac{-1}{2}}$

$\int x^{\frac{1}{2}} + x^{\frac{-1}{2}}dx$

Then integrate.

$\frac{2}{3}x^{\frac{3}{2}}+ 2x^{\frac{1}{2}}$

Since it's indefinite, don't forget to add the C:

$\frac{2}{3}x^{\frac{3}{2}}+ 2x^{\frac{1}{2}}+C$

Report an Error

Example Question #111 : Functions, Graphs, And Limits

Integrate this function: $\int(x+6)dx$ .

Possible Answers:

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

x^2+6x

x^2 + 6x+C

Correct answer:

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

Explanation:

First, divide up into two different integral expressions:

$\int xdx + \int6dx$

Then, integrate each:

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

Don't forget "C" because it is an indefinite integral:

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

Report an Error

View AP Calculus AB Tutors

Rohan
Certified Tutor

Purdue University-Main Campus, Bachelor of Engineering, Aerospace Engineering.

View AP Calculus AB Tutors

Angela
Certified Tutor

Saint Josephs College, Bachelor in Arts, Mathematics.

View AP Calculus AB Tutors

James
Certified Tutor

University of Georgia, Bachelor in Arts, Political Science and Government.

All AP Calculus AB Resources

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

Popular Subjects

GMAT Tutors in Houston, Physics Tutors in Los Angeles, Spanish Tutors in Dallas Fort Worth, ISEE Tutors in Houston, Computer Science Tutors in Dallas Fort Worth, French Tutors in Denver, SSAT Tutors in Houston, English Tutors in Chicago, ACT Tutors in Washington DC, GMAT Tutors in New York City

Popular Courses & Classes

LSAT Courses & Classes in Philadelphia, SSAT Courses & Classes in Los Angeles, GMAT Courses & Classes in New York City, SAT Courses & Classes in New York City, GMAT Courses & Classes in San Diego, ACT Courses & Classes in Los Angeles, SSAT Courses & Classes in Boston, ISEE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Houston, Spanish Courses & Classes in New York City

Popular Test Prep

SSAT Test Prep in Seattle, ISEE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Seattle, MCAT Test Prep in Los Angeles, SAT Test Prep in Denver, MCAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Houston, GRE Test Prep in Seattle, ACT Test Prep in Washington DC, GRE Test Prep in Phoenix

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 : Functions, Graphs, and Limits

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #2203 : High School Math

Example Question #71 : Calculus Ii — Integrals

Example Question #104 : Asymptotic And Unbounded Behavior

Example Question #2204 : High School Math

Example Question #2205 : High School Math

Example Question #2206 : High School Math

Example Question #21 : Finding Definite Integrals

Example Question #29 : Finding Integrals

Example Question #321 : Ap Calculus Ab

Example Question #111 : Functions, Graphs, And Limits

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: