We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Fundamental Theorem of Calculus

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 #1 : Integrals

Evaluate $\frac{d}{dx}\int_4^{x^2} e^{t^2}dt$ .

Possible Answers:

$\frac{e^{t^2}}{t^2}$

Does not exist

$e^{x^2}$

$x^2e^{x^2}$

$2xe^{x^4}$

Correct answer:

$2xe^{x^4}$

Explanation:

Even though an antideritvative of $e^{t^2}$ does not exist, we can still use the Fundamental Theorem of Calculus to "cancel out" the integral sign in this expression.

$\frac{d}{dx}\int_4^{x^2} e^{t^2}dt$ . Start

$= e^{(x^2)^2} \times (x^2)'$ . You can "cancel out" the integral sign with the derivative by making sure the lower bound of the integral is a constant, the upper bound is a differentiable function of , f(x) , and then substituting t = f(x) in the integrand. Lastly the Theorem states you must multiply your result by f'(x) (similar to the directions in using the chain rule).

$=2xe^{x^4}$ .

Report an Error

Example Question #1 : Integrals

The graph of a function f(x) is drawn below. Select the best answers to the following:

Pbstm

What is the best interpretation of the function?

$F(x)=\int_a^xf(t)dt$

Which plot shows the derivative of the function F(x) ?

Possible Answers:

Wrn4

Question 10 correct answer

Wrngan2

Wrong3q10

Correct answer:

Question 10 correct answer

Explanation:

$F(x)=\int_a^xf(t)dt$

The function F(x) represents the area under the curve f(x) from x=a to some value of x>a .

Do not be confused by the use of in the integrand. The reason we use is because are writing the area as a function of , which requires that we treat the upper limit of integration as a variable . So we replace the independent variable of f(x) with a dummy index when we write down the integral. It does not change the fundamental behavior of the function F(x) or f(t) .

The graph of the derivative of F(x) is the same as the graph for f(x) . This follows directly from the Second Fundamental Theorem of Calculus.

If the function f(x) is continuous on an interval containing , then the function defined by:

$F(x)=\int_a^xf(t)dt$

has for its' derivative F'(x)=f(x) .

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus

Evaluate

$\int_{-(4\pi+e^4)}^{4\pi+e^4}3\sin(2x)dx$

Possible Answers:

$\frac{-3\cos(8\pi+2e^4)}{2}+\frac{3\cos(-8\pi-2e^4)}{2}$

$\frac{-3\cos(\pi+e^4)}{2}$

$\frac{-3\cos(8\pi+2e^4)}{2}$

Correct answer:

Explanation:

Here we could use the Fundamental Theorem of Calculus to evaluate the definite integral; however, that might be difficult and messy.

Instead, we make a clever observation of the graph of

$3\sin(2x)$

Namely, that

$3\sin(2x)=-3\sin(-2x)$

This means that the values of the graph when comparing x and -x are equal but opposite. Then we can conclude that

$\int_{-(4\pi+e^4)}^{4\pi+e^4}3\sin(2x)dx=\int_{-(4\pi+e^4)}^{0}3\sin(2x)dx+\int_{0}^{4\pi+e^4}3\sin(2x)dx$

=-(SOMETHING)+(SOMETHING)=0

Report an Error

Example Question #1 : Fundamental Theorem Of Calculus

Wolframalpha--plot_piecewisexxlt03-x0ltxlt213-xxgt2--2013-09-10_1008

Find $\lim _{x\rightarrow 0}f(x)$

Possible Answers:

$+\infty$

Does not exist

Correct answer:

Does not exist

Explanation:

The one side limits are not equal: left is 0 and right is 3

Report an Error

Example Question #2 : Fundamental Theorem Of Calculus

Wolframalpha--plot_piecewisexxlt03-x0ltxlt213-xxgt2--2013-09-10_1008

Which of the following is a vertical asymptote?

Possible Answers:

y=0

x=3

x=0

y=3

Correct answer:

x=3

Explanation:

When approaches 3, approaches $\pm \infty$ .

Vertical asymptotes occur at values. The horizontal asymptote occurs at

y=0 .

Report an Error

Example Question #3 : Fundamental Theorem Of Calculus

$f(x) = \frac{x+1}{\sqrt{x^2 -4}}$

What are the horizontal asymptotes of f(x) ?

Possible Answers:

y=1

x=0

x=-1

y=0

x=1

Correct answer:

y=1

Explanation:

Compute the limits of f(x) as approaches infinity.

Report an Error

Example Question #4 : Fundamental Theorem Of Calculus

Write the domain of the function.

$f(x)=\frac{(x^{4}-81)^{1/2}}{x-4}$

Possible Answers:

$x\geq 4\ and\ x\neq 3$

$x\neq 4\ and\ \left | x \right |\geq 3$

$x\neq -9\ and\ x\neq 9$

${x\neq 4}$

$x\neq 4\ and\ x\neq 3$

Correct answer:

$x\neq 4\ and\ \left | x \right |\geq 3$

Explanation:

The answer is ${x\mid x\neq 4\ and\ \left | x \right |\geq 3}$

The denominator must not equal zero and anything under a radical must be a nonnegative number.

Report an Error

Example Question #4 : Fundamental Theorem Of Calculus

What is the value of the derivative of $f(x)=x^3+6x^2+\frac{3}{4}x$ at x=1?

Possible Answers:

$\frac{63}{4}$

$\frac{61}{4}$

$\frac{15}{4}$

Correct answer:

$\frac{63}{4}$

Explanation:

First, find the derivative of the function, which is:

$f{}'(x)=3x^2+12x+\frac{3}{4}$

Then, plug in 1 for x:

$3(1^2)+12(1)+\frac{3}{4}$

The result is $\frac{63}{4}$ .

Report an Error

Example Question #3 : Calculus 3

Evaluate the following limit:

$\lim_{x\rightarrow \infty }\frac{1-x^4}{x^2-4x^4}$

Possible Answers:

$\infty$

$\frac{1}{4}$

Does not exist.

$\frac{1}{2}$

Correct answer:

$\frac{1}{4}$

Explanation:

First, let's multiply the numerator and denominator of the fraction in the limit by $\frac{1}{x^{4}}$ .

$\lim_{x\rightarrow \infty }\frac{1-x^4}{x^2-4x^4}$ $=\lim_{x\rightarrow \infty }\frac{(\frac{1}{x^4})(1-x^4)}{\frac{1}{x^{4}}(x^2-4x^4)}$

$=\lim_{x\rightarrow \infty }\frac{\frac{1}{x^4}-1}{\frac{1}{x^2}-4}$

As becomes increasingly large the $\frac{1}{x^{4}}$ and $\frac{1}{x^{2}} ^{}$ terms will tend to zero. This leaves us with the limit of $\frac{1}{4}$ .

$\lim_{x\rightarrow \infty }\frac{-1}{-4}=\frac{1}{4}$ .

The answer is $\frac{1}{4}$ .

Report an Error

Example Question #6 : Fundamental Theorem Of Calculus

Let and be inverse functions, and let

$\\f(3)=4 \\g(3)=-5 \\g(4)=3 \\f'(3)=-2 \\f'(4)=4 \\g'(3)=-1$ .

What is the value of g'(4) ?

Possible Answers:

$\frac{1}{3}$

$-\frac{1}{2}$

Correct answer:

$-\frac{1}{2}$

Explanation:

Since and are inverse functions, g(f(x))=f(g(x))=x . We can differentiate both sides of the equation g(f(x)) with respect to to obtain the following:

$g'(f(x))\cdot f'(x)=1$

We are asked to find g'(4) , which means that we will need to find such that f(x)=4 . The given information tells us that f(3)=4 , which means that x=3 . Thus, we will substitute 3 into the equation.

$g'(f(3))\cdot f'(3)=1$

The given information tells us that f'(3)=-2 .

The equation then becomes $g'(4)\cdot (-2)=1$ .

We can now solve for g'(4) .

$g'(4)=-\frac{1}{2}$ .

The answer is $-\frac{1}{2}$ .

Report an Error

View AP Calculus AB Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

View AP Calculus AB Tutors

Mike Bahaa
Certified Tutor

Concordia University-Ann Arbor, Master of Arts Teaching, Mathematics Teacher Education.

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

Algebra Tutors in San Francisco-Bay Area, English Tutors in Seattle, Physics Tutors in Atlanta, Math Tutors in Washington DC, ISEE Tutors in Chicago, Calculus Tutors in Los Angeles, Computer Science Tutors in San Diego, LSAT Tutors in Atlanta, Biology Tutors in Los Angeles, Biology Tutors in Atlanta

Popular Courses & Classes

LSAT Courses & Classes in Phoenix, GRE Courses & Classes in Washington DC, SSAT Courses & Classes in Phoenix, MCAT Courses & Classes in San Diego, ISEE Courses & Classes in Los Angeles, GMAT Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Atlanta, GRE Courses & Classes in Phoenix, GMAT Courses & Classes in Seattle, GMAT Courses & Classes in Miami

Popular Test Prep

LSAT Test Prep in Washington DC, SAT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Miami, GMAT Test Prep in Seattle, ISEE Test Prep in Dallas Fort Worth, ISEE Test Prep in Los Angeles, GRE Test Prep in New York City, GRE Test Prep in Houston, GMAT Test Prep in Philadelphia, ACT Test Prep in Dallas Fort Worth

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 : Fundamental Theorem of Calculus

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Integrals

Example Question #1 : Integrals

Example Question #1 : Fundamental Theorem Of Calculus

Example Question #1 : Fundamental Theorem Of Calculus

Example Question #2 : Fundamental Theorem Of Calculus

Example Question #3 : Fundamental Theorem Of Calculus

Example Question #4 : Fundamental Theorem Of Calculus

Example Question #4 : Fundamental Theorem Of Calculus

Example Question #3 : Calculus 3

Example Question #6 : Fundamental Theorem Of Calculus

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: