We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus BC : Fundamental Theorem of Calculus with Definite Integrals

Study concepts, example questions & explanations for AP Calculus BC

Create An Account

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

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

Find the result:

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} t \ln t \; \mathrm{d}t$

Possible Answers:

$xe^{2x}$

$xe^{x }+ e$

2xe^x + 1

2xe^x

$xe^{x+1}$

Correct answer:

$xe^{2x}$

Explanation:

Set u = e^x . Then $\frac{\mathrm{d} u}{\mathrm{d} x} = e^x$ , and by the chain rule,

$\frac{\mathrm{d} }{\mathrm{d} x} \int_{1}^{e^x} t \ln t \; \mathrm{d}t$

$=\frac{\mathrm{d} u}{\mathrm{d} x} \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \ln t \; \mathrm{d}t$

$= e^x \cdot \frac{\mathrm{d} }{\mathrm{d} u} \int_{1}^{u} t \ln t \; \mathrm{d}t$

By the fundamental theorem of Calculus, the above can be rewritten as

$e^x \cdot u \ln u$

$= e^x \cdot e^x \ln e^x$

$= e^{2x} \cdot x = xe^{2x}$

Report an Error

Example Question #151 : Ap Calculus Bc

Evaluate f'(0) :

$f(x)=\int_{0}^{x} \sin\left(\sqrt{t^2+\frac{\pi^2}{4}} \right )dt$

Possible Answers:

$\frac{\pi}{2}$

$-\frac{\pi}{2}$

Correct answer:

Explanation:

By the Fundamental Theorem of Calculus, we have that ${f'(x) = \frac{d}{dx} \int_0^x \sin \sqrt{t^2 + \frac{\pi ^2}{4}} dt = \sin \sqrt{x^2 + \frac{\pi ^2}{4}}$ . Thus, $f'(0) = \sin \sqrt{0^2 + \frac{\pi ^2}{4}} =1$ .

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Evaluate f'(4) when $f(x)=\int_{0}^{x}(2t^{2}+5t-4)dt$ .

Possible Answers:

Correct answer:

Explanation:

Via the Fundamental Theorem of Calculus, we know that, given a function $f(x)=\int_{0}^{x}(2t^{2}+5t-4)dt$ , $f'(x)=\frac{d}{dx}\int_{0}^{x}(2t^{2}+5t-4)dt=2x^{2}+5x-4$ .

Therefore $f'(4)=2(4)^{2}+5(4)-4=32+20-4=52-4=48$ .

Report an Error

Example Question #2 : Fundamental Theorem Of Calculus

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 #5 : 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:

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

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

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

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

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 #2 : Fundamental Theorem Of Calculus

Given

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

Possible Answers:

$-\frac{36}{323}$

$-\frac{323}{36}$

$\frac{36}{323}$

None of the above.

$\frac{323}{36}$

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

Given

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

Possible Answers:

$-\frac{7}{2}$

$\frac{7}{2}$

None of the above.

$\frac{2}{7}$

$-\frac{2}{7}$

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 #161 : Ap Calculus Bc

Evaluate $\int_{0}^{6} (x-3)^2 dx$

Possible Answers:

$\frac{1}{3}$

Correct answer:

Explanation:

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

Use the fundamental theorem of calculus to evaluate:

$\frac{1}{3} (6-3)^3 - [ \frac{1}{3}(0-3)^3] = \frac{1}{3} (3)^3 - \frac{1}{3} (-3)^3$

$9 + \frac{1}{3}(27) = 9+9 = 18$

Report an Error

Example Question #162 : Ap Calculus Bc

$\int _{\frac{\pi}{3}} ^ {\pi } \sin (x/2) dx$

Possible Answers:

$2 \sqrt3$

$\sqrt3$

$\frac{\sqrt3}{4}$

$\frac{\sqrt3}{2}$

Correct answer:

$\sqrt3$

Explanation:

$\int \sin (x/2) dx = -2 \cos (x/2 )$

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints:

$-2 \cos (\pi/2 ) - -2 \cos (\frac{\pi}{3} / 2 ) = -2 \cos (\frac{\pi}{2}) + 2 \cos (\frac{\pi}{6})$

$= -2(0) + 2(\frac{\sqrt3}{2} )= \sqrt3$

Report an Error

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

$\int_{-1}^{3} \left (4x^2 - x \right )dx$

Possible Answers:

$32.\overline{3}$

$0.\overline{3}$

$31.\overline{6}$

$33.\overline{3}$

Correct answer:

$33.\overline{3}$

Explanation:

$\int \left ( 4x^2 -x\right ) dx = \frac{4}{3} x^3 - \frac{1}{2} x + C$

Use the Fundamental Theorem of Calculus and evaluate the integral at both endpoints:

$\frac{4}{3} (3)^3 - \frac{1}{2} (3)^2 - [ \frac{4}{3} (-1)^3 - \frac{1}{2} (-1)^2]$

$= 36 - 4.5 - [ -\frac{4}{3} - \frac{1}{2} ] = 36 - 4.5 + 1. \overline{3} + .5 = 33. \overline{3}$

Report an Error

View AP Calculus BC Tutors

Lilian Natalia
Certified Tutor

University of Texas Rio Grande Valley, Bachelor of Science, Mechanical Engineering. University of Texas Rio Grande Valley, Ma...

View AP Calculus BC Tutors

Robert
Certified Tutor

Rensselaer Polytechnic Institute, Bachelors, Electrical Engineering. Brooklyn Polytechnic, Masters, Computer Science.

View AP Calculus BC Tutors

Alysse
Certified Tutor

All AP Calculus BC Resources

2 Diagnostic Tests 86 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Math Tutors in Boston, French Tutors in Dallas Fort Worth, English Tutors in Houston, French Tutors in Seattle, Physics Tutors in Denver, ACT Tutors in Los Angeles, GRE Tutors in San Francisco-Bay Area, Reading Tutors in Miami, MCAT Tutors in San Francisco-Bay Area, Chemistry Tutors in Philadelphia

Popular Courses & Classes

GRE Courses & Classes in New York City, Spanish Courses & Classes in Philadelphia, ISEE Courses & Classes in Houston, SSAT Courses & Classes in Boston, GRE Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in Denver, SAT Courses & Classes in Chicago, MCAT Courses & Classes in Chicago, MCAT Courses & Classes in Miami, ISEE Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Dallas Fort Worth, LSAT Test Prep in Miami, SSAT Test Prep in Washington DC, ISEE Test Prep in Dallas Fort Worth, MCAT Test Prep in Miami, LSAT Test Prep in Houston, GRE Test Prep in Miami, ISEE Test Prep in Phoenix, SSAT Test Prep in Denver, SSAT Test Prep in Boston

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 BC : Fundamental Theorem of Calculus with Definite Integrals

Study concepts, example questions & explanations for AP Calculus BC

All AP Calculus BC Resources

Example Questions

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

Example Question #151 : Ap Calculus Bc

Example Question #1 : Fundamental Theorem Of Calculus And Techniques Of Antidifferentiation

Example Question #2 : Fundamental Theorem Of Calculus

Example Question #5 : Fundamental Theorem Of Calculus With Definite Integrals

Example Question #2 : Fundamental Theorem Of Calculus

Example Question #3 : Fundamental Theorem Of Calculus

Example Question #161 : Ap Calculus Bc

Example Question #162 : Ap Calculus Bc

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

All AP Calculus BC Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: