We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus AB : Limits and Continuity

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

← Previous 1 2 3 4 5 6 7 8 9 … 12 13 Next →

Example Question #11 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Find the slope of the tangent line to the function $h(x)=\frac{2ln(x)}{x^2}$ at x=1 .

Possible Answers:

ln(2)-1

Correct answer:

Explanation:

The slope of the tangent line to a function at a point is the value of the derivative of the function at that point. In this problem, h(x) is a quotient of two functions, $2ln(x) \ and\ x^2$ , so the quotient rule is needed.

In general, the quotient rule is

$\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right)=\frac{f'(x)g(x)-f(x)g'(x)}{g^2(x)}$ .

To apply the quotient rule in this example, you must also know that $\frac{d}{dx}ln(x)=\frac{1}{x}$ and that $\frac{d}{dx}x^n=nx^{n-1}$ .

Therefore, the derivative is

$h'(x)=\frac{x^2(\frac{2}{x})-2ln(x)(2x)}{(x^2)^2}=\frac{2x-4xln(x)}{x^4}$

The last step is to substitute for in the derivative, which will tell us the slope of the tangent line to h(x) at x = 1 .

$h'(1)=\frac{2(1)-4(1)ln(1)}{1^4}=\frac{2-0}{1}=2$

Report an Error

Example Question #12 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

$\frac{\mathrm{d} }{\mathrm{d} x} \enspace 2 e^x x^2 -xe^x$

Possible Answers:

e^x(3x^2 + 2x - 1 )

e^x (4x-1)

e^x (2x^2 +3x-1)

e^x (2x^2 -x) + 4x - 1

Correct answer:

e^x (2x^2 +3x-1)

Explanation:

First, factor out e^x : e^x (2x^2 -x ) . Now we can differentiate using the product rule, f(x)g(x) '= f'(x)g(x) + f(x)g'(x) .

Here, f(x) = e^x so f'(x) = e^x . g(x) = 2x^2 - x so g'(x) = 4x - 1 .

The answer is e^x (2x^2 -x) + e^x (4x - 1 ) = e^x [ 2x^2 - x + 4x - 1 ] = e^x (2x^2 + 3x -1)

Report an Error

Example Question #13 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

$\frac{\mathrm{d} }{\mathrm{d} x} 2 \sin x \cos x$

Possible Answers:

$4 \cos ^2 x$

$2 \cos (2x )$

Correct answer:

$2 \cos (2x )$

Explanation:

According to the product rule, f(x)g(x) ' = f'(x)g(x) + f(x)g'(x) . Here $f(x) = 2 \sin x$ so $f'(x) = 2 \cos x$ . $g(x) = \cos x$ so $g'(x) = -\sin x$ .

The derivative is $2 \cos x \cdot \cos x + 2 \sin x \cdot - \sin x = 2 \cos ^2 x - 2 \sin ^2 x$

Factoring out the 2 gives $2(\cos ^2 x - \sin ^2 x )$ . Remembering the double angle trigonometric identity finally gives $2 (\cos (2x ))$

Report an Error

Example Question #14 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

If $f(t) = \frac{t}{\ln(t)}$ , find f'(e^2)

Possible Answers:

$\frac{1}{4}$

$\frac{1}{2}$

$\frac{1}{e}$

Correct answer:

$\frac{1}{4}$

Explanation:

First, we need to find f'(t) . We can do that by using the quotient rule.

$f'(t) = \frac{(\ln(t))(1)-(t)(\frac{1}{t})}{(\ln(t))^2}$ .

Plugging e^2 in for and simplifying, we get

$f'(e^2) = \frac{(\ln(e^2))(1)-(e^2)(\frac{1}{e^2})}{(\ln(e^2))^2} = \frac{2-1}{2^2} = \frac{1}{4}$ .

Report an Error

Example Question #15 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Find the derivative of f:

$f=\frac{x^3}{x\cos(x)}$

Possible Answers:

$\frac{x^3\cos(x)+x^4\sin(x)}{x^2\cos^2(x)}$

$\frac{2x^3\cos(x)+x^4\sin(x)}{x\cos(x)}$

$\frac{3x^3\cos(x)-x^4\sin(x)}{x^2\cos^2(x)}$

$\frac{2x^3\cos(x)+x^4\sin(x)}{x^2\cos^2(x)}$

Correct answer:

$\frac{2x^3\cos(x)+x^4\sin(x)}{x^2\cos^2(x)}$

Explanation:

The derivative of the function is equal to

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

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \frac{f(x)}{g(x)}=\frac{g(x)f'(x)-f(x)g'(x)}{g(x)^2}$ , $\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$

Report an Error

Example Question #16 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Find the derivative of the function:

$f=x^3+x^2+\alpha^2+2\alpha^2x$

where $\alpha$ is a constant

Possible Answers:

$3x^2+2x+\alpha+2\alpha^2$

$3x^2+2x+2\alpha^2$

3x^2+2x

$x^2+2x+2\alpha^2$

Correct answer:

$3x^2+2x+2\alpha^2$

Explanation:

When taking the derivative of the sum, we simply take the derivative of each component.

The derivative of the function is

$f'=3x^2+2x+2\alpha^2$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ , $\frac{\mathrm{d} }{\mathrm{d} x} ax=a$

Report an Error

Example Question #17 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Compute the first derivative of the following function.

f(x)=sin(x)cox(x)+x^2sin(x)

Possible Answers:

cos^2(x)-sin^2(x)+2xsin(x)-x^2cos(x)

cos^2(x)-sin^2(x)+2xsin(x)+x^2cos(x)

1+2xsin(x)+x^2cos(x)

1+2sin(x)+x^2cos(x)

Correct answer:

cos^2(x)-sin^2(x)+2xsin(x)+x^2cos(x)

Explanation:

Compute the first derivative of the following function.

f(x)=sin(x)cox(x)+x^2sin(x)

To solve this problem, we need to apply the product rule:

(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)

So, we need to apply this rule to each of the terms in our function. Let's start with the first term

(sin(x)cos(x))'=cos(x)cos(x)+sin(x)(-sin(x))=cos^2(x)-sin^2(x)

Next, let's tackle the second part

(x^2sin(x))=2xsin(x)+x^2cos(x)

Now, combine the two to get:

cos^2(x)-sin^2(x)+2xsin(x)+x^2cos(x)

Report an Error

Example Question #18 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Suppose and are differentiable functions, and f(3)=1, f'(3)=4, g(-1)=3, g'(-1)=-1 .

Calculate the derivative of h(x) = f(x+2)g(-x) , at x = 1 .

Possible Answers:

None of the other answers

Correct answer:

None of the other answers

Explanation:

The correct answer is 11.

Taking the derivative of h(x) involves the product rule, and the chain rule.

h'(x)=[f(x+2)][-g'(-x)]+[g(-x)][f'(x+2)]

= -g'(-x)f(x+2)+g(-x)f'(x+2)

Substituting x=1 into both sides of the derivative we get

$h'(1)= -g'(-1)f(3)+g(-1)f'(3) = -1\cdot1 + 3 \cdot 4 = 11$

Report an Error

Example Question #19 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Find the second derivative of g(x)

g(x)=3x^4ln(x)

Possible Answers:

36x^2ln(x)+21x^2

12x^3ln(x)+3x^3

36x^2ln(x)+21x

36xln(x)+21x^2

Correct answer:

36x^2ln(x)+21x^2

Explanation:

Find the second derivative of g(x)

g(x)=3x^4ln(x)

To find this derivative, we need to use the product rule:

(f(x)g(x))'=f'(x)g(x)+f(x)g'(x)

So, let's begin:

g(x)=3x^4ln(x)

$g'(x)=12x^3ln(x)+3x^4(\frac{1}{x})=12x^3ln(x)+3x^3$

So, we are closer, but we need to derive again to get the 2nd derivative

g'(x)=12x^3ln(x)+3x^3

$g''(x)=36x^2ln(x)+12x^3(\frac{1}{x})+9x^2=36x^2ln(x)+21x^2$

So, our answer is:

36x^2ln(x)+21x^2

Report an Error

Example Question #20 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Evaluate the derivative of the function $y = x^2\cos(3x)$ .

Possible Answers:

$2x\cos(3x)-3x^2\sin(3x)$

$2x\cos(3x)+3x^2\sin(3x)$

$2x\cos(3x)-x^2\sin(3x)$

$x^2\cos(3x)-2x\sin(3x)$

$2x\cos(3x)+x^2\sin(3x)$

Correct answer:

$2x\cos(3x)-3x^2\sin(3x)$

Explanation:

Use the product rule: (fg)' = f'g + fg'

where f(x) = x^2 and $g(x) = \cos(3x)$ .

By the power rule, f'(x)=2x .

By the chain rule, $g'(x) = -\sin(3x)\cdot3$ .

Therefore, the derivative of the entire function is:

$y'=(2x)(\cos(3x))+(x^2)(-\sin(3x)\cdot 3)$

$y'=2x\cos(3x)-3x^2\sin(3x)$ .

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 … 12 13 Next →

View Tutors

Zhi Xuan
Certified Tutor

Queen's University, Bachelor, Engineering .

View Tutors

Kaye
Certified Tutor

Loyola University-Chicago, Bachelor in Arts, English. Loyola University-Chicago, Masters in Education, Reading Teacher Educat...

View Tutors

Fahad
Certified Tutor

De Montfort University Leicester, Master's/Graduate, Computing .

All Calculus AB Resources

45 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Denver, Math Tutors in Denver, ISEE Tutors in Atlanta, Math Tutors in Los Angeles, ACT Tutors in Miami, Computer Science Tutors in Denver, Calculus Tutors in Miami, Math Tutors in Miami, ISEE Tutors in Chicago, LSAT Tutors in Dallas Fort Worth

Popular Courses & Classes

LSAT Courses & Classes in New York City, GRE Courses & Classes in Miami, Spanish Courses & Classes in Miami, ISEE Courses & Classes in Miami, SSAT Courses & Classes in Miami, ACT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Los Angeles, ISEE Courses & Classes in Denver, ACT Courses & Classes in Washington DC, SSAT Courses & Classes in Atlanta

Popular Test Prep

GMAT Test Prep in Seattle, SSAT Test Prep in Boston, GRE Test Prep in New York City, LSAT Test Prep in San Diego, LSAT Test Prep in Los Angeles, MCAT Test Prep in Chicago, GMAT Test Prep in Denver, GRE Test Prep in San Diego, ACT Test Prep in Los Angeles, SSAT Test Prep in Houston

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 : Limits and Continuity

Study concepts, example questions & explanations for Calculus AB

All Calculus AB Resources

Example Questions

Example Question #11 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #12 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #13 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #14 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #15 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #16 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #17 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #18 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #19 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

Example Question #20 : Determine Limits Using Algebraic Properties And The Squeeze Theorem

All Calculus AB Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: