We are open Saturday and Sunday!

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 #2 : Parametric Calculations

Calculate $\frac{dy}{dx}$ at the point (1,2) on the curve defined by the parametric equations

x(t) = e^t , y(t)=ln(t+1)+t^2 +2

Possible Answers:

ln(2)

None of the other answers

ln(3)

Correct answer:

None of the other answers

Explanation:

The correct answer is .

We use the equation

$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}} = \frac{\frac{1}{t+1}+2t+0}{e^t}$

But we need a value for to substitute into our derivative. We can obtain such a by setting x(t) =1, y(t) =2 as our given point suggests.

1 = e^t $\Rightarrow t =0$
$2 = ln(t+1)+t^2+2 \Rightarrow t=0$
Since our values of match, t=0 is our correcct value. Substituting this into the derivative and simplifying gives us our answer of

Report an Error

Example Question #2 : Parametric Calculations

Which of the answers below is the equation obtained by eliminating the parametric from the following set of parametric equations?

x=t^2+3t+5

$y=\frac{3}{2}t-6$

Possible Answers:

$y=\frac{4}{9}x^2+\frac{22}{3}x+33$

$x=\frac{3}{2}y^2+16y+\frac{4}{3}$

$y=-\frac{14}{3}x^2+\frac{4}{7}x-4$

$y=\frac{3}{2}x^2+16x+\frac{4}{3}$

$x=\frac{4}{9}y^2+\frac{22}{3}y+33$

Correct answer:

$x=\frac{4}{9}y^2+\frac{22}{3}y+33$

Explanation:

When the problem asks us to eliminate the parametric, that means we want to somehow get rid of our variable t and be left with an equation that is only in terms of x and y. While the equation for x is a polynomial, making it more difficult to solve for t, we can see that the equation for y can easily be solved for t:

$y=\frac{3}{2}t-6$

$t=\frac{2}{3}y+4$

Now that we have an expression for t that is only in terms of y, we can plug this into our equation for x and simplify, and we will be left with an equation that is only in terms of x and y:

$x=t^2+3t+5=(\frac{2}{3}y+4)^2+3(\frac{2}{3}y+4)+5$

$x=\frac{4}{9}y^2+\frac{16}{3}y+16+2y+12+5$

$x=\frac{4}{9}y^2+\frac{22}{3}y+33$

Report an Error

Example Question #131 : Parametric, Polar, And Vector

Suppose x=cos(t) and y=sin(t) . Find the arc length from $0\leq t \leq \pi$ .

Possible Answers:

$\pi$

$\frac{\pi}{2}$

$\sqrt{\pi}$

$\sqrt{\frac{\pi}{2}}$

$2\pi$

Correct answer:

$\pi$

Explanation:

Write the arc length formula for parametric curves.

$L=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^2+\left(\frac{dy}{dt}\right)^2}\:dt$

Find the derivatives. The bounds are given in the problem statement.

$\frac{dx}{dt}=-sin(t)$

$\frac{dy}{dt}=cos(t)$

$L=\int_{0}^{\pi}\sqrt{(-sin(t))^2+(cos(t))^2}\:dt$

$L=\int_{0}^{\pi}\sqrt{sin(t)^2+cos(t)^2}\:dt$

$L=\int_{0}^{\pi}1\:dt =[t]_{0}^{\pi}= \pi$

Report an Error

Example Question #5 : Parametric Calculations

Solve for $\frac{dy}{dx}$ if $y=7t^{2}-12$ and $x=5t^{2}+2$ .

Possible Answers:

$-\frac{5}{7}$

$-\frac{7}{5}$

None of the above

$\frac{5}{7}$

$\frac{7}{5}$

Correct answer:

$\frac{7}{5}$

Explanation:

Given equations for and in terms of , we can find the derivative of parametric equations as follows:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ , as the terms will cancel out.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ and given $y=7t^{2}-12$ and $x=5t^{2}+2$ :

$\frac{dy}{dt}=(2)7t^{2-1}-(0)12=14t$

$\frac{dx}{dt}=(2)5t^{2-1}+(0)2=10t$

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{14t}{10t}=\frac{7}{5}$ .

Report an Error

Example Question #3 : Parametric Calculations

Solve for $\frac{dy}{dx}$ if $y=9t^{2}-13$ and $x=t^{2}+15$ .

Possible Answers:

$-\frac{1}{9}$

$\frac{1}{9}$

None of the above

Correct answer:

Explanation:

Given equations for and in terms of , we can find the derivative of parametric equations as follows:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ , as the terms will cancel out.

Using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ and given $y=9t^{2}-13$ and $x=t^{2}+15$ :

$\frac{dy}{dt}=(2)9t^{2-1}-(0)13=18t$

$\frac{dx}{dt}=(2)t^{2-1}+15=2t$

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{18t}{2t}=9$

Report an Error

Example Question #4 : Parametric Calculations

Solve for $\frac{dy}{dx}$ if y=2t-9 and x=5t+10 .

Possible Answers:

None of the above

$\frac{5}{2}$

$\frac{2}{5}$

$-\frac{2}{5}$

$-\frac{5}{2}$

Correct answer:

$\frac{2}{5}$

Explanation:

Since we have two equations y=2t-9 and x=5t+10 , we can find $\frac{dy}{dx}$ by dividing the derivatives of the two equations - thus:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ since the terms cancel out by standard rules of division of fractions.

In order to find the derivatives of and , let's use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ :

$y=2t-9\rightarrow \frac{dy}{dt}=(1)2t^{1-1}-0(9)=2$

$x=5t+10\rightarrow\frac{dx}{dt}=(1)5t^{1-1}+(0)10=5$

Therefore,

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{2}{5}$ .

Report an Error

Example Question #4 : Parametric Calculations

Solve for $\frac{dy}{dx}$ if y=7t+4 and x=7t-8 .

Possible Answers:

$\frac{1}{7}$

Correct answer:

Explanation:

Since we have two equations y=7t+4 and x=7t-8 , we can find $\frac{dy}{dx}$ by dividing the derivatives of the two equations - thus:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ since the terms cancel out by standard rules of division of fractions.

In order to find the derivatives of and , let's use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ :

$y=7t+4\rightarrow \frac{dy}{dt}=(1)7t^{1-1}+(0)4=7$

$x=7t-8\rightarrow\frac{dx}{dt}=(1)7t^{1-1}-(0)8=7$

Therefore,

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{7}{7}=1$ .

Report an Error

Example Question #1 : Parametric Calculations

Solve for $\frac{dy}{dx}$ if $y=5t^{2}+6t-9$ and $x=2t^{2}-6t+12$ .

Possible Answers:

$-\frac{4}{10}$

$\frac{4}{10}$

None of the above

$\frac{5t+3}{2t-3}$

$\frac{5t-3}{2t-3}$

Correct answer:

$\frac{5t+3}{2t-3}$

Explanation:

Since we have two equations $y=5t^{2}+6t-9$ and $x=2t^{2}-6t+12$ , we can find $\frac{dy}{dx}$ by dividing the derivatives of the two equations - thus:

$\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}$ (since the terms cancel out by standard rules of division of fractions).

In order to find the derivatives of and , let's use the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ :

$y=5t^{2}+6t-9\rightarrow \frac{dy}{dt}=(2)5t^{2-1}+(1)6t^{1-1}-(0)9=10t+6$

$x=2t^{2}-6t-12\rightarrow \frac{dx}{dt}=(2)2t^{2-1}-(1)6t^{1-1}-(0)12=4t-6$

Therefore, $\frac{dy}{dx}=\frac{\frac{dy}{dt}}{\frac{dx}{dt}}=\frac{10t+6}{4t-6}=\frac{2(5t+3)}{2(2t-3)}=\frac{5t+3}{2t-3}$ .

Report an Error

Example Question #141 : Parametric, Polar, And Vector

Given x=5t-1 and y=2t+9 , what is the length of the arc from $0\leq t\leq3$ ?

Possible Answers:

$4\sqrt{29}$

$3\sqrt{29}$

$2\sqrt{29}$

$\sqrt{29}$

Correct answer:

$3\sqrt{29}$

Explanation:

In order to find the arc length, we must use the arc length formula for parametric curves:

$L=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}dt$ .

Given x=5t-1 and y=2t+9 , we can use using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ , to derive

$\frac{dx}{dt}=(1)5t^{1-1}-(0)1=5$ and $\frac{dy}{dt}=(1)2t^{1-1}+(0)9=2$ .

Plugging these values and our boundary values for into the arc length equation, we get:

$L=\int_{0}^{3}\sqrt{(5)^{2}+(2)^{2}}dt$

$L=\int_{0}^{3}(\sqrt{25+4})dt$

$L=\int_{0}^{3}(\sqrt{29})dt$

Now, using the Power Rule for Integrals $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ , we can determine that:

$L=(t\sqrt{29})_{0}^{3}\textrm{}$

$L=3\sqrt{29}-0\sqrt{29}$

$L=3\sqrt{29}$

Report an Error

Example Question #11 : Parametric Calculations

Given x=7t+9 and y=t-1 , what is the length of the arc from $0\leq t\leq1$ ?

Possible Answers:

$6\sqrt{2}$

$5\sqrt{2}$

$7\sqrt{2}$

$2\sqrt{2}$

$4\sqrt{2}$

Correct answer:

$5\sqrt{2}$

Explanation:

In order to find the arc length, we must use the arc length formula for parametric curves:

$L=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}{dt}\right)^{2}}dt$ .

Given x=7t+9 and y=t-1 , we can use using the Power Rule

$\frac{d}{dx}x^{n}=nx^{n-1}$ for all $n\neq0$ , to derive

$\frac{dx}{dt}=(1)7t^{1-1}+(0)9=7$ and $\frac{dy}{dt}=(1)t^{1-1}-(0)1=1$ .

Plugging these values and our boundary values for into the arc length equation, we get:

$L=\int_{0}^{1}\sqrt{(7)^{2}+(1)^{2}}dt$

$L=\int_{0}^{1}(\sqrt{49+1})dt$

$L=\int_{0}^{1}(\sqrt{50})dt$

$L=\int_{0}^{1}(\sqrt{25\times2})dt$

$L=\int_{0}^{1}(5\sqrt{2})dt$

Now, using the Power Rule for Integrals $\int x^{n}dx=\frac{x^{n+1}}{n+1}$ for all $n\neq0$ , we can determine that:

$L=(5t\sqrt{2})_{0}^{1}\textrm{}$

$L=5(1)\sqrt{2}-5(0)\sqrt{2}$

$L=5\sqrt{2}$

Report an Error

View Calculus 2 Tutors

Muhammad
Certified Tutor

Quaid-E-Azam University, Master of Science, Applied Mathematics. Quaid-E-Azam University, Master of Philosophy, Applied Lingu...

View Calculus 2 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 2 Tutors

Varun
Certified Tutor

University of Arizona, Bachelor of Science, Aerospace Engineering.

All Calculus 2 Resources

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

Popular Subjects

MCAT Tutors in Chicago, LSAT Tutors in Houston, ACT Tutors in Dallas Fort Worth, Spanish Tutors in Phoenix, Statistics Tutors in Chicago, Biology Tutors in Seattle, Physics Tutors in New York City, SAT Tutors in San Francisco-Bay Area, ACT Tutors in Denver, Biology Tutors in Houston

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, ISEE Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Houston, MCAT Courses & Classes in Denver, GRE Courses & Classes in Miami, ACT Courses & Classes in Los Angeles, ACT Courses & Classes in Phoenix, LSAT Courses & Classes in Boston, Spanish Courses & Classes in Los Angeles, ISEE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

ISEE Test Prep in Dallas Fort Worth, ACT Test Prep in San Francisco-Bay Area, ACT Test Prep in Los Angeles, MCAT Test Prep in Atlanta, LSAT Test Prep in Washington DC, SAT Test Prep in Los Angeles, ISEE Test Prep in Denver, GRE Test Prep in Houston, GRE Test Prep in San Diego, SAT Test Prep in San Diego

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 #2 : Parametric Calculations

Example Question #2 : Parametric Calculations

Example Question #131 : Parametric, Polar, And Vector

Example Question #5 : Parametric Calculations

Example Question #3 : Parametric Calculations

Example Question #4 : Parametric Calculations

Example Question #4 : Parametric Calculations

Example Question #1 : Parametric Calculations

Example Question #141 : Parametric, Polar, And Vector

Example Question #11 : Parametric Calculations

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: