Call Now to Set Up Tutoring:

(888) 888-0446

High School Math : High School Math

Study concepts, example questions & explanations for High School Math

Create An Account

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Finding First And Second Derivatives

Find the second derivative of f(x).

$f(x)=\frac{1}{4} e^{2x}-cosx$

Possible Answers:

$e^{2x}+cosx$

$e^{2x}-cosx$

$\frac{1}{2}e^{2x}+cosx$

$\frac{1}{2}e^{2x}+sinx$

$e^{2x}+sinx$

Correct answer:

$e^{2x}+cosx$

Explanation:

First we should find the first derivative of f(x) . Remember the derivative of $e^{2x}$ is $2e^{2x}$ and the derivative of cosx is -sinx :

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

The second derivative is just the derivative of the first derivative:

$f''(x)=e^{2x}+cosx$

Report an Error

Example Question #2 : Finding First And Second Derivatives

Find the derivative of the function

f(x)=tan^4x-tan^3x+tan^2x .

Possible Answers:

$\frac{4tan^3x+3tan^2x+2tanx}{sec^2x}$

$(4tan^3x-3tan^2x+2tanx)\cdot csc^2x$

$(4tan^3x-3tan^2x+2tanx)\cdot secx$

$(4tan^3x-3tan^2x+2tanx)\cdot sec^2x$

$\frac{4tan^3x-3tan^2x+2tanx}{csc^2x}$

Correct answer:

$(4tan^3x-3tan^2x+2tanx)\cdot sec^2x$

Explanation:

We can use the Chain Rule:

Let u=tanx , so that f(x)=u^4-u^3+u^2 .

$\frac{df}{dx}=\frac{df}{du}.\frac{du}{dx}=(4u^3-3u^2+2u)\cdot sec^2x=(4tan^3x-3tan^2x+2tanx)\cdot sec^2x$

Report an Error

Example Question #1 : Derivative Interpreted As An Instantaneous Rate Of Change

Evaluate the following limit:

$\lim_{x\rightarrow 0} (\frac{1-cos2x}{sinx})$

Possible Answers:

$\infty$

Correct answer:

Explanation:

When approaches 0 both sinx and (1-cos2x) will approach . Therefore, L’Hopital’s Rule can be applied here. Take the derivatives of the numerator and denominator and try the limit again:

$\lim_{x\rightarrow 0} (\frac{1-cos2x}{sinx})= \lim_{x\rightarrow 0} (\frac{2sin2x}{cosx})=\frac{0}{1}=0$

Report an Error

Example Question #1 : Integrals

Evaluate:

$\int_{1}^{e^{100}} \frac{1}{x}dx$

Possible Answers:

$\frac{100}{e}$

$\ln 100$

100e

$\frac{1}{100}$

100

Correct answer:

100

Explanation:

$\int_{1}^{e^{100}} \frac{dx}{x} = \ln e^{100} - \ln 1 = 100-0 = 100$

Report an Error

Example Question #1 : Integrals

Find $\int_{0}^{\pi} (\cos^{2}x+\sin^{2}x)dx$

Possible Answers:

$\pi$

$2\pi$

Correct answer:

$\pi$

Explanation:

This is most easily solved by recognizing that $\sin^{2}x+\cos^{2}x=1$ .

Report an Error

Example Question #1 : Finding Definite Integrals

$\int_{\pi}^{2\pi}\sin(2x){\mathrm{d} x}=?$

Possible Answers:

$\frac{1}{2}$

$\pi$

Correct answer:

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our $f(x)=\sin(2x)$ , we can't use the power rule. Instead we end up with:

$\int f(x){\mathrm{d} x}=\int \sin(2x){\mathrm{d} x}=\frac{-\cos(2x)}{2}+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\frac{-\cos(2b)}{2}+c)-(\frac{-\cos(2a)}{2}+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(\frac{-\cos(4\pi)}{2})-(\frac{-\cos(2\pi)}{2})$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=(\frac{-1}{2})-(\frac{-1}{2})$

$\int_{\pi}^{2\pi}f(x){\mathrm{d} x}=0$

Report an Error

Example Question #3 : Integrals

$\int_{2}^{4}x^2{\mathrm{d} x}=?$

Possible Answers:

$18\frac{2}{3}$

$28\tfrac{2}{3}$

$5\tfrac{1}{2}$

$11\tfrac{3}{4}$

$22\tfrac{1}{3}$

Correct answer:

$18\frac{2}{3}$

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our f(x)=x^2 , we can use the reverse power rule to find the indefinite integral or anti-derivative of our function:

$\int (x^2){\mathrm{d} x}=\frac{1}{3}x^3+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\frac{1}{3}b^3+c)-(\frac{1}{3}a^3+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{2}^{4}f(x){\mathrm{d} x}=(\frac{1}{3}\cdot 4^3)-(\frac{1}{3}\cdot 8^3)$

$\int_{2}^{4}f(x){\mathrm{d} x}=(\frac{64}{3})-(\frac{8}{3})$

$\int_{2}^{4}f(x){\mathrm{d} x}=\frac{56}{3}=18\frac{2}{3}$

Report an Error

Example Question #2187 : High School Math

$\int_{1}^{2}\frac{1}{x}{\mathrm{d} x}=?$

Possible Answers:

1.2

0.81

0.75

0.30

0.69

Correct answer:

0.69

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

As it turns out, since our $f(x)=\frac{1}{x}$ , the power rule really doesn't help us. $\frac{1}{x}$ has a special anti derivative: $\ln{x}$ .

$\int \frac{1}{x}{\mathrm{d} x}=\ln{x}+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\ln{b}+c)-(\ln{a}+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{1}^{2}f(x){\mathrm{d} x}=(\ln(2))-(\ln(1))$

$\int_{1}^{2}f(x){\mathrm{d} x}=(\ln(2))-(0)$

$\int_{1}^{2}f(x){\mathrm{d} x}\approx 0.69$

Report an Error

Example Question #5 : Finding Definite Integrals

$\int_{3}^{40}e^x{\mathrm{d} x}=?$

Possible Answers:

$2.35\cdot 10^{19}$

$2.35\cdot 10^{9}$

$1.05\cdot 10^{17}$

$2.35\cdot 10^{15}$

$2.35\cdot 10^{17}$

Correct answer:

$2.35\cdot 10^{17}$

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

As it turns out, since our f(x)=e^x , the power rule really doesn't help us. e^x is the only function that is it's OWN anti-derivative. That means we're still going to be working with e^x .

$\int e^x{\mathrm{d} x}=e^x+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(e^b+c)-(e^a+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{3}^{40}f(x){\mathrm{d} x}=(e^{40})-(e^3)$

$\int_{3}^{40}f(x){\mathrm{d} x}=(2.35\cdot 10^{17})-(20.09)$

Because e^3 is so small in comparison to the value we got for $e^{40}$ , our answer will end up being $2.35\cdot 10^{17}$

Report an Error

Example Question #6 : Finding Definite Integrals

$\int_2^6(5x^2+3x+2){\mathrm{d} x}=?$

Possible Answers:

$402\tfrac{2}{3}$

$-12\tfrac{2}{3}$

$26\tfrac{2}{3}$

$401\tfrac{1}{3}$

$204\tfrac{2}{3}$

Correct answer:

$402\tfrac{2}{3}$

Explanation:

Remember the fundamental theorem of calculus!

$\int_{a}^{b}f(x){\mathrm{d} x}=\int f(b){\mathrm{d} x}-\int f(a){\mathrm{d} x}$

Since our f(x)=5x^2+3x+2 , we can use the power rule for all of the terms involved to find our anti-derivative:

$\int (5x^2+3x+2){\mathrm{d} x}=\frac{5}{3}x^3+\frac{3}{2}x^2+2x+c$

Remember to include the + c for any anti-derivative or integral taken!

Now we can plug that equation into our FToC equation:

$\int_{a}^{b}f(x){\mathrm{d} x}=(\frac{5}{3}b^3+\frac{3}{2}b^2+2b+c)-(\frac{5}{3}a^3+\frac{3}{2}a^2+2a+c)$

Notice that the c's cancel out. Plug in the given values for a and b and solve:

$\int_{2}^{6}f(x){\mathrm{d} x}=(\frac{5}{3}(6)^3+\frac{3}{2}(6)^2+2(6))-(\frac{5}{3}(2)^3+\frac{3}{2}(2)^2+2(2))$

$\int_{2}^{6}f(x){\mathrm{d} x}=(360+54+12)-(\frac{40}{3}+6+4)$

$\int_{2}^{6}f(x){\mathrm{d} x}=(426)-(23\tfrac{1}{3})$

$\int_{2}^{6}f(x){\mathrm{d} x}=402\tfrac{2}{3}$

Report an Error

View Tutors

Samantha
Certified Tutor

Eastern Kentucky University, Bachelor of Science, Psychology.

View Tutors

Philipp
Certified Tutor

University of Nevada-Las Vegas, Bachelor of Science, Mechanical Engineering.

View Tutors

Kayla
Certified Tutor

Northern Virginia Community College, Associate in Arts, Social Sciences.

All High School Math Resources

8 Diagnostic Tests 613 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Reading Tutors in Atlanta, ISEE Tutors in Boston, GRE Tutors in Dallas Fort Worth, English Tutors in Atlanta, Spanish Tutors in New York City, French Tutors in Los Angeles, Biology Tutors in Boston, MCAT Tutors in New York City, Biology Tutors in San Diego, English Tutors in Los Angeles

Popular Courses & Classes

GMAT Courses & Classes in San Diego, GRE Courses & Classes in Phoenix, MCAT Courses & Classes in Washington DC, GRE Courses & Classes in Denver, LSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Washington DC, MCAT Courses & Classes in San Diego, MCAT Courses & Classes in Miami, SAT Courses & Classes in Boston, MCAT Courses & Classes in Los Angeles

Popular Test Prep

ISEE Test Prep in Miami, ACT Test Prep in Atlanta, ISEE Test Prep in Chicago, GMAT Test Prep in Atlanta, LSAT Test Prep in Denver, LSAT Test Prep in Houston, MCAT Test Prep in New York City, MCAT Test Prep in Denver, GRE Test Prep in Denver, GRE 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

High School Math : High School Math

Study concepts, example questions & explanations for High School Math

All High School Math Resources

Example Questions

Example Question #1 : Finding First And Second Derivatives

Example Question #2 : Finding First And Second Derivatives

Example Question #1 : Derivative Interpreted As An Instantaneous Rate Of Change

Example Question #1 : Integrals

Example Question #1 : Integrals

Example Question #1 : Finding Definite Integrals

Example Question #3 : Integrals

Example Question #2187 : High School Math

Example Question #5 : Finding Definite Integrals

Example Question #6 : Finding Definite Integrals

All High School Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: