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

← Previous 1 2 … 9 10 11 12 13 14 15 16 17 … 20 21 Next →

Example Question #113 : Asymptotic And Unbounded Behavior

Integrate the following expression: $\int(2x^4-6x^3+9x)dx$ .

Possible Answers:

x^5-x^4+x^2+C

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

2x^5-3x^4+9x^2

Correct answer:

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

Explanation:

First, divide up into three different expressions so you can integrate each x term separately:

$\int2x^{4}dx-\int6x^{3}dx+\int9xdx$

Then, integrate and simplify:

$2(\frac{x^{5}}{5})-6(\frac{x^4}{4})+9(\frac{x^2}{2})$

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

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

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

Report an Error

Example Question #114 : Asymptotic And Unbounded Behavior

Find the general solution of $\int\frac{1}{x^3}dx$ to find the particular solution that satisfies the intitial condition F(1)=0

Possible Answers:

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

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

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

Correct answer:

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

Explanation:

To start the problem, it's easier if you bring up the denominator and make it a negative exponent:

$\int x^{-3}dx$

Then, integrate:

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

Simplify and add the "C" for an indefinite integral:

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

Plug in the initial conditions [F(1)=0] to find C and generate the particular solution:

$-\frac{1}{2x^2}+C=0$

$-\frac{1}{2(1^2)}+C=0$

$C=\frac{1}{2}$

Thus, your final equation is:

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

Report an Error

Example Question #115 : Asymptotic And Unbounded Behavior

Integrate:

$\int(2x-3x^2)dx$

Possible Answers:

$\frac{4}{3}x^3+\frac{9}{2}x^2+C$

$4x^2-\frac{9}{2}x^3+C$

-x^3+x^2 +C

x^3-x^2+C

Correct answer:

-x^3+x^2 +C

Explanation:

First, split up into 2 integrals:

$\int2xdx-\int3x^2dx$

Then integrate and simplify:

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

x^2-x^3

Don't forget to add C because it's an indefinite integral:

-x^3+x^2 +C

Report an Error

Example Question #116 : Asymptotic And Unbounded Behavior

Integrate:

$\int(x^3+2)^2dx$

Possible Answers:

$\frac{x^7}{7}+x^4+4x+C$

x^7+x^4+4x+C

7x^7+x^4+4x+C

$\frac{x^7}{7}-x^4+4x+C$

Correct answer:

$\frac{x^7}{7}+x^4+4x+C$

Explanation:

First, FOIL the binomial:

(x^3+2)(x^3+2)= x^6+4x^3+4

Once that's expanded, integrate each piece separately:

$\int x^6dx+\int4x^3dx+\int4dx$

$\frac{x^7}{7}+4(\frac{x^4}{4})+4x$

Then simplify and add C because it's an indefinite integral:

$\frac{x^7}{7}+x^4+4x+C$

Report an Error

Example Question #33 : Finding Definite Integrals

$\int^6_3 x^2+5x\ dx =$

Possible Answers:

Undefined

130.5

31.5

193.5

Correct answer:

130.5

Explanation:

Remember the Rundamental Theorem of Calculus: If g'(x)=f(x) , then $\int^b_af(x){\mathrm{d} x}=g(b)-g(a)$ .

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

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

Remember, when taking an integral, definite or indefinite, we always add , as there could be a constant involved.

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

Now we can plug that back into the problem.

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

Notice that the 's cancel out. Plug in the values given in the problem:

$\int^6_3 x^2+5x{\mathrm{d} x}=(\frac{1}{3}(6)^3+\frac{5}{2}(6)^2)-(\frac{1}{3}(3)^3+\frac{5}{2}(3)^2)$ $\int^6_3 x^2+5x{\mathrm{d} x}=(\frac{1}{3}*216+\frac{5}{2}*36)-(\frac{1}{3}*27+\frac{5}{2}*9)$

$\int^6_3 x^2+5x{\mathrm{d} x}=(72+90)-(9+22.5)$

$\int^6_3 x^2+5x{\mathrm{d} x}=(162)-(31.5)$

$\int^6_3 x^2+5x{\mathrm{d} x}=130.5$

Report an Error

Example Question #1 : Calculus 3

$\int_{-1}^{0}e^{1-t}dt =$

Possible Answers:

$1-e^{2}$

$e^{2}-1$

$undefined$

$e+1$

$e^{2}-e$

Correct answer:

$e^{2}-e$

Explanation:

We can use the substitution technique to evaluate this integral.

Let u=1-t .

We will differentiate with respect to .

$\frac{\mathrm{d}u }{\mathrm{d} t}=-1$ , which means that ${\mathrm{d} u}=-{\mathrm{d} t}$ .

We can solve for ${\mathrm{d}t }$ in terms of ${\mathrm{d} u}$ , which gives us ${\mathrm{d}t }=-{\mathrm{d} u}$ .

We will also need to change the bounds of the integral. When t=-1 , u=1-(-1) , and when t=0 , u=1-0=1 .

We will now substitute in for the 1-t , and we will substitute $-{\mathrm{d}u }$ for ${\mathrm{d} t}$ .

$\int_{2}^{1}-e^{u}du$

$\int_{2}^{1}-e^{u}du = -e^{u}|_{2}^{1}=-e^{1}-(-e^{2})=e^{2}-e^{1}$

The answer is $e^{2}-e$ .

Report an Error

Example Question #5 : Finding Integrals By Substitution

Evaluate:

$\dpi{120} \int_{0}^{\frac{\pi }{ 2}}\sin x \sqrt{\cos x }\; \; dx$

Possible Answers:

$\frac{4}{3}$

$\frac{ 2} {3}$

Correct answer:

$\frac{ 2} {3}$

Explanation:

Set $u = \cos x$ .

Then $\frac{\mathrm{du} }{\mathrm{d} x}= \frac{\mathrm{d} }{\mathrm{d} x} \cos x= -\sin x$ and $\sin x \; dx = - du$ .

Also, since $u = \cos x$ , the limits of integration change to $\cos \frac{\pi }{2} = 0$ and $\cos 0 = 1$ .

Substitute:

$\dpi{120} \int_{0}^{\frac{\pi }{ 2}}\sin x \sqrt{\cos x }\; \; dx$

$= \int_{1}^{0} \left (- \sqrt{u} \right ) \; du$

$= \int_{1}^{0} \left (-u ^{\frac{1}{2}} \right ) \; du$

$= \int_{0}^{1} u ^{\frac{1}{2}} \; du$

$\dpi{150} = \left.\begin{matrix} \frac{u^{\frac{3}{2}}}{{\frac{3}{2}}} & \\ & \end{matrix}\right|$ $\begin{matrix} 1\\ \\ 0 \end{matrix}$

$\dpi{150} = \left.\begin{matrix} \\ \frac{ 2u \sqrt{u}} {3} & \\ & \end{matrix}\right|$ $\begin{matrix} 1\\ \\ 0 \end{matrix}$

$=\frac{ 2\cdot 1\cdot \sqrt{1}} {3} - \frac{ 2\cdot 0\cdot \sqrt{0}} {3} = \frac{ 2} {3}$

Report an Error

Example Question #121 : Functions, Graphs, And Limits

Evaluate the following integral:

$\int \frac{dx}{x^2+36}$

Possible Answers:

$\frac{x^3}3{+ 36x}$

$\frac{1}{6}\tan^{-1}\bigg(\frac{x}{6}\bigg)+C$

$\ln(x^2+36x)$

$\sin^{-1}(x+36)$

Correct answer:

$\frac{1}{6}\tan^{-1}\bigg(\frac{x}{6}\bigg)+C$

Explanation:

First you must know that:

$tan^{-1}(x)= \frac{1}{x^2+1}$ and $\int \frac{1}{a^2+x^2}=\bigg(\frac{1}{a}\bigg)tan^{-1}\bigg(\frac{x}{a}\bigg)+C$

Therefore we can rewrite our problem in this form:

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

where a=6 .

Thus the integral becomes,

$\int \frac{1}{6^2+x^2}=\bigg(\frac{1}{a}\bigg)tan^{-1}\bigg(\frac{x}{a}\bigg)+C=\bigg(\frac{1}{6}\bigg)tan^{-1}\bigg(\frac{x}{6}\bigg)+C$

Report an Error

Example Question #121 : Functions, Graphs, And Limits

Evaluate:

$\int_{0}^{2} 2x+3$ .

Possible Answers:

Correct answer:

Explanation:

$\int2x+3 = x^2+3x$

Setting the limits from zero to two we can find that,

$\int2x+3 = x^2+3x$

${}=[(2)^2+3(2)]-[(0^2+3(0)]$

=4+6-0

=10

Report an Error

Example Question #122 : Functions, Graphs, And Limits

Evaluate:

$\int_{0}^{3} \left | x-1\right |dx$ .

Possible Answers:

2.5

Correct answer:

2.5

Explanation:

Seeing that the equation contains an absolute value you should know that the graph must always remain positive therefore resulting in a V-shaped graph.

Since the equation is y=x-1 , when x=1, y=0 then the vertex of the graph is at (1,0) .

The graph contains a triangle ranging from 0 to 1 and a triangle from 1 to 3. Remebering that taking the interal of a function is the same as finding the area under the curve we can use these triangles to solve our problem.

The area of the triangle from 0 to 1 is,

$\frac{1}{2}bh=\frac{1}{2}(1)(1)=0.5$ .

The area of the triangle from 1 to 3 is,

$\frac{1}{2}bh=\frac{1}{2}(2)(2)=\frac{4}{2}=2$ .

Thus the evaluated integral must be these areas added together,

0.5+2=2.5 .

Report an Error

← Previous 1 2 … 9 10 11 12 13 14 15 16 17 … 20 21 Next →

View AP Calculus AB Tutors

Alena
Certified Tutor

Mogilevsky State Pedagogical Institute, Bachelor in Arts, Mathematics. Mogilevsky State Pedagogical Institute, Masters, Mathe...

View AP Calculus AB Tutors

Vince
Certified Tutor

Salt Lake Community College, Associate in Science, Mathematics.

View AP Calculus AB Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

All AP Calculus AB Resources

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

Popular Subjects

GRE Tutors in Chicago, Computer Science Tutors in San Francisco-Bay Area, ACT Tutors in Phoenix, GRE Tutors in Washington DC, ACT Tutors in Houston, Biology Tutors in New York City, Biology Tutors in Los Angeles, Reading Tutors in Atlanta, GRE Tutors in Phoenix, Spanish Tutors in Miami

Popular Courses & Classes

MCAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Houston, GRE Courses & Classes in Seattle, GMAT Courses & Classes in Phoenix, GMAT Courses & Classes in Atlanta, GRE Courses & Classes in Washington DC, MCAT Courses & Classes in Denver, GRE Courses & Classes in Boston, GMAT Courses & Classes in Chicago, LSAT Courses & Classes in Seattle

Popular Test Prep

GMAT Test Prep in Phoenix, SSAT Test Prep in Boston, GRE Test Prep in New York City, SSAT Test Prep in Los Angeles, LSAT Test Prep in Washington DC, MCAT Test Prep in Boston, ISEE Test Prep in Atlanta, ISEE Test Prep in Washington DC, ACT Test Prep in Philadelphia, SAT Test Prep in Miami

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 #113 : Asymptotic And Unbounded Behavior

Example Question #114 : Asymptotic And Unbounded Behavior

Example Question #115 : Asymptotic And Unbounded Behavior

Example Question #116 : Asymptotic And Unbounded Behavior

Example Question #33 : Finding Definite Integrals

Example Question #1 : Calculus 3

Example Question #5 : Finding Integrals By Substitution

Example Question #121 : Functions, Graphs, And Limits

Example Question #121 : Functions, Graphs, And Limits

Example Question #122 : 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: