Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Introduction to Integrals

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Fundamental Theorem Of Calculus With Definite Integrals

Evaluate f'(1) when $f(x)=\int_{0}^{x}(6t^{2}-3t+10)dt$ .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function $f(x)=\int_{0}^{x}(6t^{2}-3t+10)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(6t^{2}-3t+10)dt=6x^{2}-3x+10$ . Therefore, $f'(1)=6(1)^{2}-3(1)+10=6-3+10=3+10=13$ .

Report an Error

Example Question #151 : Introduction To Integrals

Evaluate f'(7) when $f(x)=\int_{0}^{x}(t^{2}-2t+5)dt$ .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function $f(x)=\int_{0}^{x}(t^{2}-2t+5)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(t^{2}-2t+5)dt=x^{2}-2x+5$ . Therefore, $f'(7)=(7)^{2}-2(7)+5=49-14+5=35+5=40$ .

Report an Error

Example Question #2 : Fundamental Theorem Of Calculus With Definite Integrals

Suppose we have the function

$\small g(x)=\int_{0}^{\cos x}(1+\sin t)dt$

What is the derivative, $\small g'(x)$ ?

Possible Answers:

$\small \small \small 1-\sin\left(\cos(x) \right )$

$\small \small \small 1+\sin\left(\cos(x) \right )$

$\small \small -\sin x\left[1+\sin\left(\cos(x) \right )\right]$

$\small \small \small \small \sin x\left[1+\sin\left(\cos x \right )\right]$

Correct answer:

$\small \small -\sin x\left[1+\sin\left(\cos(x) \right )\right]$

Explanation:

We can view the function $\small g(x)$ as a function of $\small \cos x$ , as so

$\small g(x)=F(\cos x)$

where $\small F(x)=\int_0^x (1+\sin t)dt$ .

We can find the derivative of $\small g(x)$ using the chain rule:

$\small g'(x)=F'(\cos x)\cdot (\cos x)'=F'(\cos x)\cdot (-\sin x)$

where $\small F'(z)$ can be found using the fundamental theorem of calculus:

$\small \small F'(x)=\frac{d}{dz}\int_0^z (1+\sin t)dt=1+\sin z$

So we get

$\small \small \small \small g'(x)=-\sin x\cdot F'(\cos x)=-\sin x\cdot[1+\sin(\cos x)]$

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus

If $f(x)=\int_{0}^{x}(4t^{2}-6t+9)dt$ , what is f'(3) ?

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function $f(x)=\int_{0}^{x}(4t^{2}-6t+9)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(4t^{2}-6t+9)dt=4x^{2}-6x+9$ .

Thus, $f'(3)=4(3)^{2}-6(3)+9=36-18+9=18+9=27$

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus

If $f(x)=\int_{0}^{x}(t^{2}+7t-12)dt$ , what is f'(2) ?

Possible Answers:

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function $f(x)=\int_{0}^{x}(t^{2}+7t-12)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(t^{2}+7t-12)dt=t^{2}+7t-12$ .

Thus, $f'(2)=(2)^{2}+7(2)-12=4+14-12=4+2=6$

Report an Error

Example Question #2 : Fundamental Theorem Of Calculus

If $f(x)=\int_{0}^{x}(8t^{2}-t+14)dt$ , what is f'(5) ?

Possible Answers:

189

209

219

199

229

Correct answer:

209

Explanation:

By the Fundamental Theorem of Calculus, we know that given a function $f(x)=\int_{0}^{x}(8t^{2}-t+14)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(8t^{2}-t+14)dt=8x^{2}-x+14$ .

Thus, $f'(5)=8(5)^{2}-(5)+14=200-5+14=200+9=209$

Report an Error

Example Question #2 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Given

$f(x)=\int_{0}^{x}\left(\frac{1}{t^{2}}-2t+3\right)dt$ , what is f'(6) ?

Possible Answers:

$-\frac{36}{323}$

$\frac{323}{36}$

$\frac{36}{323}$

$-\frac{323}{36}$

None of the above.

Correct answer:

$-\frac{323}{36}$

Explanation:

By the Fundamental Theorem of Calculus, for all functions that are continuously defined on the interval [a,b] with in [a,b] and for all functions defined by by $F(x)=\int_{a}^{x}f(t)dt$ , we know that F(x)=f'(x) .

Thus, for

$f(x)=\int_{0}^{x}\left(\frac{1}{t^{2}}-2t+3\right)dt$ ,

$f'(x)=\frac{d}{dx}\int_{0}^{x}\left(\frac{1}{t^{2}}-2t+3\right)dt=\frac{1}{x^{2}}-2x+3$ .

Therefore,

$f'(6)=\frac{1}{6^{2}}-2(6)+3=\frac{1}{36}-12+3=\frac{1}{36}-9=\frac{1}{36}-\frac{324}{36}=-\frac{323}{36}$

Report an Error

Example Question #3 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Given

$f(x)=\int_{0}^{x}(\frac{2}{t^{2}+4t-5})dt$ , what is f'(2) ?

Possible Answers:

None of the above.

$\frac{7}{2}$

$-\frac{2}{7}$

$\frac{2}{7}$

$-\frac{7}{2}$

Correct answer:

$\frac{2}{7}$

Explanation:

Given

$f(x)=\int_{0}^{x}\left(\frac{2}{t^{2}+4t-5}\right)dt$ , then

$f'(x)=\frac{d}{dx}\int_{0}^{x}\left(\frac{2}{t^{2}+4t-5}\right)dt=\frac{2}{x^{2}+4x-5}$ .

Therefore,

$f'(2)=\frac{2}{2^{2}+4(2)-5}=\frac{2}{4+8-5}=\frac{2}{4+3}=\frac{2}{7}$ .

Report an Error

Example Question #11 : Fundamental Theorem Of Calculus

Given

$f(x)=\int_{0}^{x}({3}t^{2}-7t-12})dt$ , what is f'(7) ?

Possible Answers:

Correct answer:

Explanation:

Thus, since

$f(x)=\int_{0}^{x}({3}t^{2}-7t-12})dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(3t^{2}-7t-12)dt=3x^{2}-7x-12$ .

Therefore,

$f'(7)=3(7)^{2}-7(7)-12=147-49-12=98-12=86$ .

Report an Error

Example Question #12 : Fundamental Theorem Of Calculus

Given $f(x)=\int_{0}^{x}(5t^{2}-8t+13)dt$ , what is f'(0) ?

Possible Answers:

Correct answer:

Explanation:

According to the Fundamental Theorem of Calculus, if is a continuous function on the interval [a,b] with as the function defined for all on [a,b] as $F(x)=\int_{a}^{x}f(t)dt$ , then F'(x)=f(x) . Therefore, if $f(x)=\int_{0}^{x}(5t^{2}-8t+13)dt$ , then $f'(x)=\frac{d}{dx}\int_{0}^{x}(5t^{2}-8t+13)dt=5t^{2}-8t+13$ . Thus, $f'(0)=5(0)^{2}-8(0)+13=0-0+13=13$ .

Report an Error

View Calculus 2 Tutors

Mikah
Certified Tutor

Ivy Tech Community College, Associate in Science, Biotechnology. Western Governors University, Bachelor in Business Administr...

View Calculus 2 Tutors

Sayantan
Certified Tutor

Western Michigan University, Bachelor of Science, Applied Mathematics.

View Calculus 2 Tutors

Michael
Certified Tutor

University of Richmond, Bachelor of Science, Mathematics. Central Connecticut State University, Master of Arts, Mathematics.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Phoenix, GMAT Tutors in Atlanta, Spanish Tutors in Dallas Fort Worth, LSAT Tutors in Dallas Fort Worth, Biology Tutors in Boston, ISEE Tutors in Miami, Biology Tutors in Philadelphia, ACT Tutors in Washington DC, SSAT Tutors in Dallas Fort Worth, Biology Tutors in Miami

Popular Courses & Classes

ACT Courses & Classes in Atlanta, ACT Courses & Classes in Miami, SAT Courses & Classes in Seattle, ISEE Courses & Classes in New York City, GRE Courses & Classes in Phoenix, ACT Courses & Classes in Chicago, LSAT Courses & Classes in Phoenix, SAT Courses & Classes in Phoenix, MCAT Courses & Classes in Philadelphia, ACT Courses & Classes in Seattle

Popular Test Prep

SSAT Test Prep in Philadelphia, ISEE Test Prep in Seattle, ACT Test Prep in Seattle, GRE Test Prep in San Diego, ACT Test Prep in Atlanta, MCAT Test Prep in Atlanta, ACT Test Prep in Denver, ACT Test Prep in Chicago, ISEE Test Prep in Washington DC, ACT Test Prep in Los Angeles

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 2 : Introduction to Integrals

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #1 : Fundamental Theorem Of Calculus With Definite Integrals

Example Question #151 : Introduction To Integrals

Example Question #2 : Fundamental Theorem Of Calculus With Definite Integrals

Example Question #1 : Fundamental Theorem Of Calculus

Example Question #1 : Fundamental Theorem Of Calculus

Example Question #2 : Fundamental Theorem Of Calculus

Example Question #2 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Example Question #3 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Example Question #11 : Fundamental Theorem Of Calculus

Example Question #12 : Fundamental Theorem Of Calculus

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: