Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus Review

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #228 : Calculus 3

Calculate f' (1) of $f(x)= \ln \left(\frac{6x^2}{\left(x+4 \right )} \right )$ .

Possible Answers:

$-\frac{2}{15}$

$\frac{9}{5}$

$\ln 6$

Correct answer:

$\frac{9}{5}$

Explanation:

We begin by rewriting our function in a more convenient way,

$f(x) = \ln \left(\frac{6x^2}{\left(x+4 \right )} \right )= \ln \left(6x^2 \right )- \ln \left(x+4 \right )$ .

This makes taking the derivative a little easier. We can therefore write

$f' (x) = \frac{2}{x}-\frac{1}{x+4}$ , or $f' (1) = 2 - \frac{1}{5} = \frac{9}{5}$ .

Report an Error

Example Question #41 : Calculus Review

Calculate f'(5) of $f(x)= \sin \left(\pi x \right )$ .

Possible Answers:

$- \pi$

$\pi$

$\frac{\pi}{2}$

Correct answer:

$- \pi$

Explanation:

By the chain rule, we can show

$\frac{d}{dx} \sin \left(f(x) \right )= cos \left(f(x) \right ) \cdot f'(x)$ , or in this case,

$\frac{d}{dx} \sin \left( \pi x \right )= \cos \left( \pi x \right ) \cdot \pi\Rightarrow \cos \left(5 \pi \right ) \cdot \pi = -\pi$ .

Report an Error

Example Question #42 : Calculus Review

Find f'(x) given $f(x)=x^{x^2}$

Possible Answers:

$x^{x^2+1}$

$x^{x^2}\left(1+2 \ln x \right )$

$x^{x^2+1}\left(1+2 \ln x \right )$

$x^{x^2+1}\left(2 \ln x \right )$

Correct answer:

$x^{x^2+1}\left(1+2 \ln x \right )$

Explanation:

This derivative must be found using logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=x^{x^2} \\ \log_x f&= x^2\\ \frac{\ln f}{\ln x}&=x^2 \\ \ln f &= x^2 \ln x \\ \frac{f'}{f}&=x+2x \ln x \\ f'&= \left(x + 2x \ln x \right )x^{x^2}= f'(x) = x^{x^2+1}\left(1+2 \ln x \right ) \end{align*}$

Report an Error

Example Question #43 : Calculus Review

Find f'(x) given $f(x)= 7^{3x}$

Possible Answers:

$\ln 7^3$

$3 \cdot 7^{2x}$

$7^{3x} \cdot \ln 7^3$

$3 \cdot 7^{3x}$

Correct answer:

$7^{3x} \cdot \ln 7^3$

Explanation:

This derivative is most easily done by logarithmic differentiation. Consider the following

$\begin{align*} f(x)&=7^{3x} \\ \ln f&=\ln 7 \cdot 3x \\ \frac{f'}{f}&=\ln 7 \cdot 3 \\ f'(x)&=7^{3x} \cdot 3 \ln 7 \\ f'(x)&=7^{3x} \cdot \ln 7^3 \end{align*}$

Report an Error

Example Question #44 : Calculus Review

Find the derivative of: y=-cos(5x+2)

Possible Answers:

-5sin(5x+2)

-5sin(5x+2)cos(5x+2)

5sin(5x+2)

5sin(5x+2)cos(5x+2)

-sin(5x+2)

Correct answer:

5sin(5x+2)

Explanation:

Take the derivative of the cosine function. This will also require chain rule, which is the derivative of the inner function.

$\frac{dy}{dx} = -[-sin(5x+2)] \cdot 5$

The answer is: $\frac{dy}{dx}= 5sin(5x+2)$

Report an Error

Example Question #45 : Calculus Review

Find the derivative: ln(cos(2x))

Possible Answers:

tan(2x)

$-\frac{2}{sin(2x)}$

$\frac{1}{sin(2x)}$

-2tan(2x)

$\frac{1}{cos(2x)}$

Correct answer:

-2tan(2x)

Explanation:

This problem will require multiple chain rule. Take the derivative of natural log, and then apply the inner function of the natural log, cosine, and then also apply the chain rule for .

$\frac{d}{dx}ln(cos(2x)) = \frac{1}{cos(2x)} \cdot -sin(2x) \cdot 2$

Combine this into one term and simplify.

$-\frac{2sin(2x)}{cos(2x)} = -2tan(2x)$

The answer is: -2tan(2x)

Report an Error

Example Question #45 : Calculus Review

Evaluate $\frac{d}{dx}(yx+y)$

Possible Answers:

None of the other answers

y+x

y+1

x+1

Correct answer:

None of the other answers

Explanation:

The correct answer is (x+1)y'+y .

Since we are taking the (full) derivative of an expression involving and , we must use implicit differentiation. We proceed as follows

$\frac{d}{dx}(yx+y)$ start.

$=\frac{d}{dx}(yx)+\frac{d}{dx}(y)$

=(xy'+(1)y)+y' . Using $\frac{d}{dx}y=y'$ , and the Product Rule.

=(1+x)y' +y . Factor

Report an Error

Example Question #48 : Calculus Review

Evaluate the limit

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$

by interpreting the limit in terms of the definition of the derivative.

Possible Answers:

$\frac{9}{128}$

The limit does not exist.

Correct answer:

$\frac{9}{128}$

Explanation:

The definition of the derivative is

$\lim_{h \to 0}\frac{f(a+h)-f(a)}{h} = f'(a)$ .

Equating

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h}$ , to the definition of the derivative, we have

$\lim_{h \to 0}\frac{12(\frac{1}{2}+h)^{12}+12(\frac{1}{2})^{12}}{h} = \frac{d}{dx}(12x^{12})|_{x=1/2}=144x^{11}|_{x=1/2}$

$=144(\frac{1}{2})^{11}=\frac{144}{2048} =\frac{9}{128}$ .

Report an Error

Example Question #45 : Derivatives

What is the derivative of $5x^4+2x^2+\frac {3}{x}$ ?

Possible Answers:

$20x^3-4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x^2}$

$-20x^3+4x-\frac {3}{x^2}$

$20x^3+4x-\frac {3}{x}$

Correct answer:

$20x^3+4x-\frac {3}{x^2}$

Explanation:

Step 1: Take the derivative of the first term: 5x^4

$(5x^4)'=20x^{4-1}=20x^3$ via the power rule.
Step 2: Take the derivative of the next term: 2x^2
$(2x^2)'=4x^{2-1}=4x$ , again using the product rule.
Step 3: Take the derivative of the last term $\frac {3}{x}$ .

We will have to use Quotient Rule here.

Find the derivative of f(x) and g(x):

f'(x)=0 , g'(x)=1 .

Use the formula: $\frac {f'(x)g(x)-g'(x)f(x)}{[g(x)]^2}$

$\frac {0(x)-1(3)}{(x)^2}=\frac {-3}{x^2}$ .

Step 4: Put all the final terms from highest to lowest degree...

We get: $20x^3+4x-\frac {3}{x^2}$ .

Report an Error

Example Question #45 : Calculus Review

Find the derivative of

$\sin^n (x^m)$

Possible Answers:

$n(x^{m-1})\sin^{n-1}(x^m)$

$mn(x^{m-1})\sin^{n-1}(x^m)$

$(x^{m-1})\sin^{n-1}(x^m)$

$m^2n^2(x^{m})\sin^{n}(x^m)$

Correct answer:

$mn(x^{m-1})\sin^{n-1}(x^m)$

Explanation:

Untitled

Report an Error

View Calculus 3 Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Calculus 3 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Dallas Fort Worth, English Tutors in Boston, Computer Science Tutors in Miami, Math Tutors in San Diego, MCAT Tutors in Atlanta, Biology Tutors in Seattle, GMAT Tutors in Dallas Fort Worth, GRE Tutors in Atlanta, Spanish Tutors in San Diego, MCAT Tutors in Miami

Popular Courses & Classes

GRE Courses & Classes in San Diego, ACT Courses & Classes in Chicago, ISEE Courses & Classes in Chicago, SAT Courses & Classes in Phoenix, GRE Courses & Classes in Boston, ACT Courses & Classes in Phoenix, ACT Courses & Classes in Philadelphia, GRE Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Atlanta, MCAT Courses & Classes in Washington DC

Popular Test Prep

ISEE Test Prep in San Diego, LSAT Test Prep in San Francisco-Bay Area, ACT Test Prep in San Diego, GMAT Test Prep in Denver, GRE Test Prep in Seattle, GMAT Test Prep in Los Angeles, ISEE Test Prep in Seattle, GMAT Test Prep in Washington DC, SAT Test Prep in Miami, ISEE 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

Calculus 3 : Calculus Review

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #228 : Calculus 3

Example Question #41 : Calculus Review

Example Question #42 : Calculus Review

Example Question #43 : Calculus Review

Example Question #44 : Calculus Review

Example Question #45 : Calculus Review

Example Question #45 : Calculus Review

Example Question #48 : Calculus Review

Example Question #45 : Derivatives

Example Question #45 : Calculus Review

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: