Call Now to Set Up Tutoring:

(888) 888-0446

AP Calculus AB : Numerical approximations to definite integrals

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

Write the equation of a tangent line to the given function at the point.

y = ln(x²) at (e, 3)

Possible Answers:

y = (2/e)(x – e)

y = (2/e)

y – 3 = ln(e²)(x – e)

y – 3 = (x – e)

y – 3 = (2/e)(x – e)

Correct answer:

y – 3 = (2/e)(x – e)

Explanation:

To solve this, first find the derivative of the function (otherwise known as the slope).

y = ln(x²)

y' = (2x/(x²))

Then, to find the slope in respect to the given points (e, 3), plug in e.

y' = (2e)/(e²)

Simplify.

y'=(2/e)

The question asks to find the tangent line to the function at (e, 3), so use the point-slope formula and the points (e, 3).

y – 3 = (2/e)(x – e)

Report an Error

Example Question #1 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at (1,-1) when

$y=\frac{x^{3}-2x(x^{1/2})}{x}$

Possible Answers:

y=x+1

y=1

y=x-2

y=-x-2

y=x

Correct answer:

y=x-2

Explanation:

The answer is

$y=\frac{x^{3}-2x(x^{1/2})}{x}$ let's go ahead and cancel out the 's. This will simplify things.

$y=x^{2}-2(x^{1/2})$

$y'=2x-\frac{1}{(x^{1/2})}$

$y'(1)=2(1)-\frac{1}{(1^{1/2})} =1$ this is the slope so let's use the point slope formula.

y+1=1(x-1)

y+1=x-1

y=x-2

Report an Error

Example Question #11 : Integrals

Differentiate $y=\frac{t+2}{t^{2}-4t-12}$

Possible Answers:

$y'=\frac{1}{(t^{2}-4t-12)^{2}}$

$y'=\frac{1}{(t-6)}$

$y'=\frac{(t+2)-1}{(t^{2}-4t-12)^{2}}$

$y'=\frac{-1}{(t-6)^{2}}$

$y'=\frac{1}{2t-4}$

Correct answer:

$y'=\frac{-1}{(t-6)^{2}}$

Explanation:

We see the answer is $y'=\frac{-1}{(t-6)^{2}}$ after we simplify and use the quotient rule.

$y=\frac{t+2}{t^{2}-4t-12}$ we could use the quotient rule immediatly but it is easier if we simplify first.

$y=\frac{t+2}{(t+2)(t-6)}$

$y=\frac{1}{(t-6)}$

$y=(t-6)^{-1}$

$y'=-(t-6)^{-2}$

$y'=\frac{-1}{(t-6)^{2}}$

Report an Error

Example Question #11 : Integrals

Find $\lim_{x\rightarrow \infty }\frac{-2x^3+x^2-2}{x^2+10}$

Possible Answers:

$1$

$0$

$\infty$

$-\infty$

$-2$

Correct answer:

$-\infty$

Explanation:

When taking limits to infinity, we usually only consider the highest exponents. In this case, the numerator has $-2x^3$ and the denominator has $x^2$ . Therefore, by cancellation, it becomes $-2x$ as approaches infinity. So the answer is $-\infty$ .

Report an Error

Example Question #12 : Integrals

Evaluate:

$\sum_{n=1}^{\infty }4(\frac{-1}{2})^{n-1}$

Possible Answers:

cannot be determined

$\infty$

$\frac{8}{3}$

Correct answer:

$\frac{8}{3}$

Explanation:

First, we can write out the first few terms of the sequence $a_{n}=$ $4(\frac{-1}{2})^{n-1}$ , where ranges from 1 to 3.

$a_{1}=4(\frac{-1}{2})^{1-1}=4$

$a_{2}=4(\frac{-1}{2})^{2-1}=-2$

$a_{3}=4(\frac{-1}{2})^{3-1}=1$

Notice that each term $\left ( n>1 \right )$ , is found by multiplying the previous term by $-\frac{1}{2}$ . Therefore, this sequence is a geometric sequence with a common ratio of $-\frac{1}{2}$ . We can find the sum of the terms in an infinite geometric sequence, provided that |r|<1 , where is the common ratio between the terms. Because $r=-\frac{1}{2}$ in this problem, $\left | r \right |$ is indeed less than 1. Therefore, we can use the following formula to find the sum, , of an infinite geometric series.

$S=\frac{a_{1}}{1-r}=\frac{4}{1-(\frac{-1}{2})}=\frac{4}{3/2}=\frac{8}{3}$

The answer is $\frac{8}{3}$ .

Report an Error

Example Question #11 : Integrals

If $f(x)=4, f'(x)=3, g(x)=2, \ and\ g'(x)=1$ then find h'(x) .

$h(x)=\frac{2f}{g}$

Possible Answers:

$\frac{1}{2}$

Correct answer:

Explanation:

The answer is 1.

$h(x)=\frac{2f}{g}$

$h'(x)=\frac{2(f'(x)g(x)-f(x)g'(x))}{g^{2}}$

$h'(x)=\frac{2(3*2-4*1)}{4} = 1$

Report an Error

Example Question #2 : Numerical Approximations To Definite Integrals

Find the equation of the tangent line at (4,1) on graph f(x)=(x-2)(x+4)

Possible Answers:

y=3x-7

y=7x+8

y=x-11

y=x+40

y=10x-39

Correct answer:

y=10x-39

Explanation:

The answer is 10x-39

f(x)=(x-2)(x+4)

$f(x)=x^{2}+2x-8$

$f'(x)=2x+2$

f'(x) = 2(4)+2=10

(This is the slope. Now use the point-slope formula)

y-1=10(x-4)

y-1=10x-40

y=10x-39

Report an Error

Example Question #13 : Integrals

Find the equation of the tangent line at (1,1) in

$f(x)=ax^{2}+bx+c$

Possible Answers:

y=2ax-b+1

y=2ab+b-1

y=2ax+bx-2a-b+1

y=2ax-bx-2a+1

y=2a+b

Correct answer:

y=2ax+bx-2a-b+1

Explanation:

The answer is y=2ax+bx-2a-b+1

$f(x)=ax^{2}+bx+c$

$f'(x)=2ax+b$

$f'(1)=2a(1)+b = 2a+b$

(This is the slope. Now use the point-slope formula.)

y-1=(2a+b)(x-1)

y-1=2ax+bx-2a-b

y=2ax+bx-2a-b+1

Report an Error

Example Question #14 : Integrals

If $f(x)=\cos x$ then

$f^{'}\left ( \frac{\pi }{2} \right )=$

Possible Answers:

$\frac{2^{1/2}}{-2}$

$\frac{1}{2}$

Correct answer:

Explanation:

The answer is .

f(x)=cos(x)

f'(x)=-sin(x)

We know that

$\frac{\pi }{2}=90^{\circ}$

so,

$f'\left ( \frac{\pi }{2}\right )=-sin(90^{\circ})=-1$

Report an Error

Example Question #7 : Numerical Approximations To Definite Integrals

$\lim_{x\rightarrow 0}\frac{2\sin(x^2)}{x\tan(x)}=$

Possible Answers:

1/2

1/4

does not exist

Correct answer:

Explanation:

When we let x = 0 in our original limit, we obtain the 0/0 indeterminate form. Therefore, we can apply L'Hospital's Rule, which requires that we take the derivative of the numerator and denominator separately.

$\lim_{x\rightarrow 0}\frac{2\sin(x^2)}{x\tan(x)}=\lim_{x\rightarrow 0}\frac{\frac{\mathrm{d} }{\mathrm{d} x}2\sin(x^2)}{\frac{\mathrm{d} }{\mathrm{d} x}x\tan(x)}$

Apply the Chain Rule in the numerator and the Product Rule in the denominator.

$\lim_{x\rightarrow 0}\frac{4x\cos(x^2)}{\tan(x)+x\sec^2(x)}$

If we again substitute x = 0, we still obtain the 0/0 indeterminate form. Thus, we can apply L'Hospital's Rule one more time.

$\lim_{x\rightarrow 0}\frac{4x\cos(x^2)}{\tan(x)+x\sec^2(x)}=\lim_{x\rightarrow 0}\frac{\frac{\mathrm{d} }{\mathrm{d} x}4x\cos(x^2)}{\frac{\mathrm{d} }{\mathrm{d} x}(\tan(x)+x\sec^2(x))}$

$=\lim_{x\rightarrow 0}\frac{4\cos(x^2)+4x(2x)(-\sin(x^2))}{\sec^2(x)+(\sec^2(x)+x(2\sec^2x\tan x))}$

If we now let x = 0, we can evaluate the limit.

$=\lim_{x\rightarrow 0}\frac{4\cos(x^2)+4x(2x)(-\sin(x^2))}{\sec^2(x)+(\sec^2(x)+x(2\sec^2x\tan x))}=\frac{4\cos(0)+0}{\sec^2(0)+\sec^2(0)+0}$

$=\frac{4}{1+1}=2$

The answer is 2.

Report an Error

View AP Calculus AB Tutors

Andrew
Certified Tutor

University of Wisconsin-Stevens Point, Bachelor of Science, Mathematics. University of Wisconsin-Oshkosh, Master of Science, ...

View AP Calculus AB Tutors

Lyric
Certified Tutor

Spelman College, Bachelor of Science, Mathematics.

View AP Calculus AB Tutors

Jeremy
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Physics.

All AP Calculus AB Resources

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

Popular Subjects

GRE Tutors in Dallas Fort Worth, Physics Tutors in San Diego, Math Tutors in Boston, Physics Tutors in Denver, GMAT Tutors in Philadelphia, ACT Tutors in Houston, English Tutors in Phoenix, LSAT Tutors in Washington DC, SSAT Tutors in Boston, Biology Tutors in Los Angeles

Popular Courses & Classes

SSAT Courses & Classes in Seattle, GMAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Seattle, GMAT Courses & Classes in New York City, ACT Courses & Classes in New York City, GRE Courses & Classes in Phoenix, GRE Courses & Classes in Philadelphia, MCAT Courses & Classes in Seattle, MCAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Denver

Popular Test Prep

MCAT Test Prep in Seattle, MCAT Test Prep in New York City, SSAT Test Prep in San Francisco-Bay Area, GRE Test Prep in Boston, SAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Houston, SSAT Test Prep in Chicago, SSAT Test Prep in Phoenix, LSAT Test Prep in San Francisco-Bay Area, SSAT 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

AP Calculus AB : Numerical approximations to definite integrals

Study concepts, example questions & explanations for AP Calculus AB

All AP Calculus AB Resources

Example Questions

Example Question #1 : Integrals

Example Question #1 : Numerical Approximations To Definite Integrals

Example Question #11 : Integrals

Example Question #11 : Integrals

Example Question #12 : Integrals

Example Question #11 : Integrals

Example Question #2 : Numerical Approximations To Definite Integrals

Example Question #13 : Integrals

Example Question #14 : Integrals

Example Question #7 : Numerical Approximations To Definite Integrals

All AP Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: