Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

Create An Account

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find Average Value

Find the average value of the function f(x)=2x^5-6x^3+7 over the interval [1,2] .

Possible Answers:

7.5

5.5

Correct answer:

5.5

Explanation:

Because the objective of this problem is to find the average value of the function, the formula f $f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx$ will be useful. Since the interval and function are provided, this problem consists of recognizing the base components and making the appropriate substitutions:

$f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx=\frac{1}{2-1}\int_{1}^{2}2x^5-6x^3+7 dx =[\frac{1}{3}x^6-\frac{3}{2}x^4+7x] |_1^2$ $=\frac{64}{3}-24+14-\frac{1}{3}+\frac{3}{2}-7=5.5$

Report an Error

Example Question #4 : Find Average Value

Identify the correct integral expression for the average value of the function f(x)=1x+2x over the interval [1,4] .

Possible Answers:

$\frac{1}{3}\int_{1}^{4}\frac{1}{t}+\sqrt{2t} dx$

$\frac{1}{4}\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx$

$\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx$

$\frac{1}{3}\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx$

Correct answer:

$\frac{1}{3}\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx$

Explanation:

When finding the average value of a function, it is useful to keep the following formula in mind: $f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx$ . This equation allows the substitution of the function and interval to solve for the average value. Because the function indicated in the problem is in terms of , the definite integral expression should also be in terms of .

$f_{avg}=\frac{1}{4-1}\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx =\frac{1}{3}\int_{1}^{4}\frac{1}{x}+\sqrt{2x} dx$

Report an Error

Example Question #5 : Find Average Value

Find the average value of the function f(x)=20x+x^3 over the interval [-1,1] .

Possible Answers:

Correct answer:

Explanation:

$f_{avg}=\frac{1}{1-(-1)}\int_{-1}^{1}20x+x^3 dx$ $=\frac{1}{2}[10x^2+(\frac{1}{4})(x)^4] |_-1^1=\frac{1}{2}(10+\frac{1}{4}-10-\frac{1}{4})=0$

Report an Error

Example Question #2 : Find Average Value

Let f(x)=4x+11 . What value of c allows the average value of f(x) over the interval [0,c] to be ?

Possible Answers:

Correct answer:

Explanation:

$27=\frac{1}{c-0}\int_{c}^{0}4x +11 dx$

Next, the definite integral can be taken to continue solving for .

$27=\frac{1}{c}[2x^2+11x] |_0^c$

$27=(\frac{1}{c})(2c^2+11c)$

27=2c+11

c=8

Report an Error

Example Question #2 : Find Average Value

Let f(x)=3x2+4x-3 . What value of allows the average value of f(x) over the interval [0,c] to be ?

Possible Answers:

Correct answer:

Explanation:

$12=\frac{1}{c-0}\int_{0}^{c}3x^2+4x-3 dx$

Next, the definite integral can be taken to continue solving for .

$12=\frac{1}{c}[x^3+2x^2-3x] |_0^c$

$12=(\frac{1}{c})(c^3+2c^2-3c)$

12=c^2+2c-3

0=c^2+2c-15

0=(c+5)(c-3)

c={-5,3}

Because the problem states that c>0 , the answer c=-5 can be eliminated. Therefore, the correct answer is c=3 .

Report an Error

Example Question #1 : Find Average Value

Which of the following theorems is related to finding the Average Value of a Function?

Possible Answers:

Mean Value Theorem for Integrals

Extreme Value Theorem

Intermediate Value Theorem

Fundamental Theorem of Calculus

Correct answer:

Mean Value Theorem for Integrals

Explanation:

The following equation is used for finding the Average Value of a Function: $f_{avg}=\frac{1}{b-a} \int_{a}^{b} f(x) dx$ . A rearrangement of this equation could be multiplying (b-a) to both sides. Making this rearrangement, and substituting $f_{avg}=f(c)$ with a<c<b , results in the following: $\int_{a}^{b}f(x) dx=f(c)(b-a)$ . Assuming f(x) is continuous, this is the correct equation for the Mean Value Theorem for Integrals.

Report an Error

Example Question #2 : Find Average Value

Find the average value of the function f(x)=2x^7+5x^2+4 over the interval [1,3] . Round to the nearest hundredth.

Possible Answers:

1691.33

103.45

846.50

845.67

Correct answer:

845.67

Explanation:

When finding the average value of a function, it is useful to keep the following formula in mind: $f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx$ . This equation is very helpful because it provides a simple way to determine the average value by substituting in values of the bounds and the function itself:

$f_{avg}=\frac{1}{3-1}\int_{1}^{3}2x^7+5x^2+4 dx=(12)[\frac{1}{4}x^8+\frac{5}{3}x^3+4x] |_1^3$ $=(\frac{1}{2})(\frac{6561}{4}+45+12-1\frac{}{4}-\frac{5}{3}-4)=845.67$

Report an Error

Example Question #1 : Find Average Value

Find the average value of the function $f(x)=4e^{2x}$ over the interval [0,5]

Possible Answers:

$\frac{2}{5}(e^5-1)$

e^5

$\frac{2}{5}(e^{10}-1)$

2e^10

Correct answer:

$\frac{2}{5}(e^{10}-1)$

Explanation:

$f_{avg}=\frac{1}{5-0}\int_{0}^{5}4e^{2x} dx=(\frac{1}{5})[4e^{2x} * \frac{1}{2}] |_0^5 =\frac{2}{5}e^{2(5)}-\frac{2}{5}e^{2(0)}=\frac{2}{5}(e^{10}-1)$

Report an Error

Example Question #1 : Find Average Value

Find the average value of the function f(t)=4+2cos(2t) over the interval $[0,\frac{\pi}{4}]$ .

Possible Answers:

$\frac{\pi}{4} + 2$

$4+\frac{4}{\pi}$

$4-\frac{4}{\pi}$

$1+\frac{4}{\pi}$

Correct answer:

$4+\frac{4}{\pi}$

Explanation:

When finding the average value of a function, it is useful to keep the following formula in mind: $f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx$ . This equation allows the substitution of the function and interval to solve for the average value. While the function in this problem contains a trigonometric function, the same approach can be applied. Remember that the function is in terms of t, so the definite integral expression should likewise be in terms of .

$f_{avg}=\frac{1}{\frac{\pi}{4}-0}\int_{0}^{\frac{\pi}{4}}4+2cos(2t) dt=(\frac{4}{\pi})[4t+sin(2t)] |_0^{\frac{\pi}{4}}$ $=(\frac{4}{\pi})[\pi +1-0-0]=4+\frac{4}{\pi}$

Report an Error

Example Question #3 : Find Average Value

Identify the correct integral expression for the average value of the function f(t)=6-3tsin(t^2) over the interval $[0,\pi]$ .

Possible Answers:

$\frac{1}{\pi}\int_{0}^{\pi}6-3xsin(x^2) dt$

$\frac{1}{\pi}\int_{0}^{\pi}6-3tsin(t^2) dt$

$\int_{0}^{\pi}6-3tsin(t^2) dt$

$\int_{0}^{\pi}6-3tsin(t^2) dx$

Correct answer:

$\frac{1}{\pi}\int_{0}^{\pi}6-3tsin(t^2) dt$

Explanation:

When finding the average value of a function, it is useful to keep the following formula in mind: $f_{avg}=\frac{1}{b-a}\int_{a}^{b}f(x) dx$ . This equation allows the substitution of the function and interval to solve for the average value. While the function in this problem contains a trigonometric function, the same approach can be applied. Remember that the function is in terms of , so the definite integral expression should likewise be in terms of .

$f_{avg}=\frac{1}{\pi-0}\int_{0}^{\pi}6-(3t)sin(t^2) dt =\frac{1}{\pi}\int_{0}^{\pi}6-3tsin(t^2) dt$

Report an Error

View Tutors

Jolan
Certified Tutor

San Francisco State University, Bachelor in Arts, Biology, General. Western University of Health Sciences, Master of Science,...

View Tutors

Mark
Certified Tutor

Murray State University, Bachelor of Science, Mathematics.

View Tutors

Makayla
Certified Tutor

Highline Community College, Associate in Arts, Arts, General. Washington State University, Bachelor of Science, International...

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

MCAT Tutors in Washington DC, Calculus Tutors in Boston, French Tutors in New York City, MCAT Tutors in Boston, Statistics Tutors in Los Angeles, Physics Tutors in Houston, Physics Tutors in Denver, ISEE Tutors in Houston, Physics Tutors in New York City, ACT Tutors in Philadelphia

Popular Courses & Classes

LSAT Courses & Classes in San Diego, SSAT Courses & Classes in Denver, GRE Courses & Classes in Seattle, ISEE Courses & Classes in Phoenix, SAT Courses & Classes in Miami, MCAT Courses & Classes in Denver, LSAT Courses & Classes in Houston, LSAT Courses & Classes in Miami, MCAT Courses & Classes in Los Angeles, GRE Courses & Classes in Washington DC

Popular Test Prep

MCAT Test Prep in Washington DC, MCAT Test Prep in Chicago, SAT Test Prep in New York City, GRE Test Prep in Washington DC, GRE Test Prep in San Francisco-Bay Area, SSAT Test Prep in New York City, GRE Test Prep in New York City, ACT Test Prep in Philadelphia, SAT Test Prep in San Francisco-Bay Area, ISEE 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

Calculus AB : Calculus AB

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #1 : Find Average Value

Example Question #4 : Find Average Value

Example Question #5 : Find Average Value

Example Question #2 : Find Average Value

Example Question #2 : Find Average Value

Example Question #1 : Find Average Value

Example Question #2 : Find Average Value

Example Question #1 : Find Average Value

Example Question #1 : Find Average Value

Example Question #3 : Find Average Value

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: