Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #181 : Derivatives

Find the derivative of the following function at the point x=0 :

$f(x)=\frac{2x^3+x^2+e^x}{3x+1}$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

$f'(x)=\frac{(3x+1)(6x^2+2x+e^x)-3(2x^3+x^2+e^x)}{(3x+1)^2}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x}( \frac{f(x)}{g(x)})=\frac{g(x)f'(x)-f(x)g'(x)}{(g(x))^2}$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ ,
$\frac{\mathrm{d} }{\mathrm{d} x}e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ .

Now, plug in the point x=0 into the above function:

$f'(x)=\frac{(3(0)+1)(6(0)^2+2(0)+e^{(0)})-3(2(0)^3+(0)^2+e^{(0)})}{(3(0)+1)^2}$

$f'(x)=\frac{(1)(1)-3(1)}{(1)^2}$

$f'(0)=\frac{-2}{1}=-2$

Report an Error

Example Question #185 : Derivative Review

Find the derivative of the following function at the point x=0 :

$f(x)=\tan(x)-\frac{3x}{4}$

Possible Answers:

$\frac{3}{4}$

$\frac{1}{4}$

$\frac{5}{4}$

Correct answer:

$\frac{1}{4}$

Explanation:

The derivative of the function is

$f'(x)=\sec^2(x)-\frac{3}{4}$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$ ,

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

Now, plug in the given point into the first derivative function:

$f'(0)=1-\frac{3}{4}=\frac{1}{4}$

Report an Error

Example Question #186 : Derivative Review

Find the derivative of the following function at x=0 :

$f(x)=x^3\cos(x)+\sin(3x)$

Possible Answers:

Correct answer:

Explanation:

The derivative of the function is

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

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)=f(x)g'(x)$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ ,

$\frac{\mathrm{d} }{\mathrm{d} x}\sin(x)=\cos(x)$ ,

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

Now, plug in the point we want into the derivative and solve:

f'(0)=0-0+3(1)=3

Report an Error

Example Question #187 : Derivative Review

Evaluate the derivative at the point $\theta=\pi$ :

$f(\theta)=\cos^2(\theta)+e^\theta$

Possible Answers:

$2\pi$

$e^\pi$

Correct answer:

$e^\pi$

Explanation:

The derivative of the function is

$f'(\theta)=-2\cos(\theta)\sin(\theta)+e^\theta$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(g(x))=f'(g(x))\cdot g'(x)$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} \cos(x)=-\sin(x)$ ,

$\frac{\mathrm{d} }{\mathrm{d} x} x^n=nx^{n-1}$ .

Now, plug in the point asked into the above function and solve:

$f'(\pi)=0+e^\pi=e^\pi$

Report an Error

Example Question #182 : Derivatives

Find the derivative of the following function at $\theta=\pi$ :

$f(\theta)=\tan(\theta)+\theta e^\theta$

Possible Answers:

$e^\pi (\pi+1)$

$e^\pi (\pi-1)$

$e^\pi (\pi+1)+1$

$e^\pi (\pi+1)-1$

Correct answer:

$e^\pi (\pi+1)-1$

Explanation:

The derivative of the function is

$f'(\theta)=\sec^2(\theta)+e^\theta+\theta e^\theta$

and was found using the following rules:

$\frac{\mathrm{d} }{\mathrm{d} x} f(x)g(x)=f'(x)g(x)+f(x)g'(x)$ , $\frac{\mathrm{d} }{\mathrm{d} x} e^u=e^u\frac{\mathrm{du} }{\mathrm{d} x}$ , $\frac{\mathrm{d} }{\mathrm{d} x} \tan(x)=\sec^2(x)$

Finally, plug in $\pi$ for $\theta$ and we get our final answer:

$f'(\theta)=e^\pi (\pi+1)-1$

Report an Error

Example Question #10 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

What is the rate of change of the function f(x)=1+2x+3x^4 at the point (1,6) ?

Possible Answers:

2594

Correct answer:

Explanation:

The rate of change of a function at a point is the value of the derivative at that point. First, take the derivative of f(x) using the power rule for each term.

Remember that the power rule is

$\frac{d}{dx}x^n=nx^{n-1}$ , and that the derivative of a constant is zero.

f'(x)=2+12x^3

Next, notice that the x-value of the point (1,6) is 1, so substitute 1 for x in the derivative.

f'(1)=2+12(1)^3=14

Therefore, the rate of change of f(x) at the point (1,6) is 14.

Report an Error

Example Question #301 : Ap Calculus Bc

Calculate the derivative of f(x)=4x^2-x+2 at the point x=1 .

Possible Answers:

Correct answer:

Explanation:

There are 2 steps to solving this problem.

First, take the derivative of f(x) .

Then, replace the value of x with the given point.

For example, if x=a , then we are looking for the value of f'(x=a) , or the derivative of f(x) at x=a .

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

Calculate f'(x)

Derivative rules that will be needed here:

Derivative of a constant is 0. For example, $\frac{\mathrm{d} }{\mathrm{d} t} 5 = 0$
Taking a derivative on a term, or using the power rule, can be done by doing the following: $\frac{\mathrm{d} }{\mathrm{d} t} t^n = n * t^{n-1}$

f'(x)=8x-1

Then, plug in the value of x and evaluate

f'(x=1)=8*1-1 = 7

Report an Error

Example Question #311 : Ap Calculus Bc

Evaluate the first derivative if

f(x)=2e^x and x=0 .

Possible Answers:

f'(0)=1

f'(0)=e

f'(0)=0

f'(0)=2

Correct answer:

f'(0)=2

Explanation:

First we must find the first derivative of the function.

Because the derivative of the exponential function is the exponential function itelf, or

$\frac{\mathrm{d} }{\mathrm{d} x}[e^x]=e^x$

and taking the derivative is a linear operation,

we have that

$f'(x)=\frac{\mathrm{d} }{\mathrm{d} x}[f(x)]$

$=\frac{\mathrm{d} }{\mathrm{d} x}[2e^x]$

$=2\frac{\mathrm{d} }{\mathrm{d} x}[e^x]$

=2e^x

Now setting x=0

f'(0)=2e^0=2*1=2

Thus

f'(0)=2

Report an Error

Example Question #1311 : Calculus Ii

What is the derivative of $(2+3cos(3x))^\pi$ ?

Possible Answers:

$-3\pi(2+cos(3x))^{\pi-1}$

$3\pi(2+cos(3x))^{\pi-1}cos(3x)$

$-3\pi(2+cos(3x))^{\pi-1}cos(3x)$

$3\pi(2+cos(3x))^{\pi-1}sin(3x)$

$-3\pi(2+cos(3x))^{\pi-1}sin(3x)$

Correct answer:

$-3\pi(2+cos(3x))^{\pi-1}sin(3x)$

Explanation:

Need to use the power rule which states: $\frac{d}{dx}u^n=nu^{n-1}\frac{du}{dx}$

In our problem $\frac{du}{dx}=-3sin(3x)$

Report an Error

Example Question #21 : Calculus Review

$f (x) = 7x^{2} + 6x - 9$

Give f'(x) .

Possible Answers:

f'(x) = 14x

$f'(x) = 14x^{2}+6x$

f'(x) = 14x+6

f'(x) = 7x+6

f'(x) = 14x+12

Correct answer:

f'(x) = 14x+6

Explanation:

$\frac{\mathrm{d} }{\mathrm{d} x} \left (ax^{n} \right )= n\cdot ax^{n-1}$ , and the derivative of a constant is 0, so

$f (x) = 7x^{2} + 6x - 9$

$f ' (x) = 7\cdot 2\cdot x^{2-1} + 6\cdot 1 + 0$

$f ' (x) = 14\cdot x^{1} + 6$

f ' (x) = 14x + 6

Report an Error

View Calculus 2 Tutors

Ping
Certified Tutor

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

View Calculus 2 Tutors

Fatemeh
Certified Tutor

University of Isfahan, Doctor of Engineering, Chemical Engineering. University of Windsor, Doctor of Engineering, Environment...

View Calculus 2 Tutors

MEER
Certified Tutor

IIT , Master's/Graduate, Microelectronics .

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Houston, ISEE Tutors in San Francisco-Bay Area, Reading Tutors in Boston, Algebra Tutors in New York City, Algebra Tutors in Denver, French Tutors in Washington DC, Chemistry Tutors in Miami, Chemistry Tutors in Boston, GMAT Tutors in Seattle, Physics Tutors in Phoenix

Popular Courses & Classes

ACT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Philadelphia, ACT Courses & Classes in Atlanta, LSAT Courses & Classes in Houston, LSAT Courses & Classes in Phoenix, LSAT Courses & Classes in Washington DC, SAT Courses & Classes in Los Angeles, ACT Courses & Classes in New York City, SAT Courses & Classes in Chicago, MCAT Courses & Classes in Washington DC

Popular Test Prep

GMAT Test Prep in New York City, SAT Test Prep in Denver, ACT Test Prep in Dallas Fort Worth, ACT Test Prep in Washington DC, ISEE Test Prep in Washington DC, MCAT Test Prep in Philadelphia, GMAT Test Prep in Philadelphia, GRE Test Prep in Seattle, GRE Test Prep in Houston, ACT Test Prep in Seattle

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 2 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #181 : Derivatives

Example Question #185 : Derivative Review

Example Question #186 : Derivative Review

Example Question #187 : Derivative Review

Example Question #182 : Derivatives

Example Question #10 : Rules Of Basic Functions: Power, Exponential Rule, Logarithmic, Trigonometric, And Inverse Trigonometric

Example Question #301 : Ap Calculus Bc

Example Question #311 : Ap Calculus Bc

Example Question #1311 : Calculus Ii

Example Question #21 : Calculus Review

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: