We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : How to write equations

Study concepts, example questions & explanations for Calculus 1

Create An Account

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #195 : Writing Equations

My friend has 100 dollars at time t = 0 days. She earns dollars every day, and spends dollars every day (so at day , she has 103-2=101 dollars). Write an equation for how much money she has at time .

Possible Answers:

y = 100 + 3t

y = 100 + t

y = t

y = 100 - t

y = 100 + 1

Correct answer:

y = 100 + t

Explanation:

My friend starts off with 100 dollars, and every day, she makes a net dollar. This is because she earns three dollars and spends one dollar.

Net=Earn-Spent

1=3-2

So, we need an equation that reflects that she has 100 +1 *(number of days), which is given by

y = 100 + t .

Report an Error

Example Question #191 : Writing Equations

Identify the inner and the outer functions of the following equation (let f(x) be the outer equation and g(x) be the inner equation:

$y=\sqrt[3]{1-4x}$

Possible Answers:

f(x)=x

$g(x)=\sqrt[3]{1-4x}$

$g(x)=\sqrt[3]{x}$

f(x)=1-4x

$f(x)=\sqrt[3]{1-4x}$

g(x)=x

$f(x)=\sqrt[3]{x}$

g(x)=1-4x

Correct answer:

$f(x)=\sqrt[3]{x}$

g(x)=1-4x

Explanation:

The first section of the equation to be resolved (in this case 1-4x) is the innermost function and the second section to solve is the outer function. So:

$f(x)=\sqrt[3]{x}$

g(x)=1-4x

becuase g(x) must be performed first before plugging it into f(x).

Report an Error

Example Question #201 : Writing Equations

Identify the inner and the outer functions of the following equation (let f(x) be the outer equation and g(x) be the inner equation:

y=(2x^3+5)^4

Possible Answers:

f(x)=2x^3+5

g(x)=x^4

f(x)=2x^3

g(x)=x+5

Cannot be determined.

f(x)=x^4

g(x)=2x^3+5

Correct answer:

f(x)=x^4

g(x)=2x^3+5

Explanation:

The first section of the equation to be resolved (in this case 2x^3+5) is the innermost function and the second section to solve is the outer function.

So:

f(x)=x^4

g(x)=2x^3+5

becuase g(x) must be performed first before plugging it into f(x).

Report an Error

Example Question #202 : Writing Equations

Identify the inner and the outer functions of the following equation (let f(x) be the outer equation and g(x) be the inner equation:

$y=\tan\pi x$

Possible Answers:

$f(x)=\tan(x)$

$g(x)=\pi x$

$f(x)=\tan (\pi )$

g(x)=x

Cannot be determined.

$g(x)=\tan(x)$

$f(x)=\pi x$

Correct answer:

$f(x)=\tan(x)$

$g(x)=\pi x$

Explanation:

The first section of the equation to be resolved (in this case pi * x) is the innermost function and the second section to solve is the outer function.

So:

$f(x)=\tan(x)$

$g(x)=\pi x$

becuase g(x) must be performed first before plugging it into f(x).

Report an Error

Example Question #203 : Writing Equations

Derive the equation for the function of the cot(x) .

Possible Answers:

None of these

$\tan (x)$

$-\csc(x)^{2}$

$\sec (x)^{2}$

$-\csc (x)$

Correct answer:

$-\csc(x)^{2}$

Explanation:

$\cot(x)=\frac{\cos (x)}{\sin (x)}$

By the quotient rule, the derivative of

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

The derivative of sin is cos and the derivative of cos is -sin. Thus the derivative of cot(x) is

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

$=-(1+\cot (x)^2)$

By the trigonometric identities this is equal to $-\csc(x)^{2}$ .

Report an Error

Example Question #11 : How To Write Equations

Suppose Charlie deposits $\$15$ per month into an account that contains a starting balance of $\$60$ .

Which of the following is a correct initial condition that when coupled with the differential equation from the previous question, will yield a specific solution to the scenario described above?

Possible Answers:

M(0)=0`

M(0)=60

M(0)=15

M(12)=180

M(12)=60

Correct answer:

M(0)=60

Explanation:

Here we are looking for an initial condition that describes the balance of the account at a specific time. We are given the information that the account starts with $\$60$ initially. So at time t=0 years, we know that the amount of money in the account is $\$60$ . Therefore, we can write M(0)=60

Report an Error

Example Question #204 : Writing Equations

Suppose Charlie deposits $\$15$ per month into an account that contains a starting balance of $\$60$ .

Which of the following is a differential equation that best models the amount of money, , in the account after years?

Possible Answers:

$\frac{dM}{dt}=15$

$\frac{dM}{dt}=180$

$\frac{dM}{dt}=180t$

$\frac{dM}{dt}=180t+60$

$\frac{dM}{dt}=15t+60$

Correct answer:

$\frac{dM}{dt}=180$

Explanation:

Here we are writing a differential equation in the units of dollars per year. This means that we need to reconcile our units to get the amount deposited each year. $\$15/month = \$180/year$ ,

so the yearly rate of change of money in the account,

$\frac{dM}{dt}=180$ .

The starting balance is our initial condition and does not tell us about the change in money in the account, so is not included in our differential equation.

Report an Error

Example Question #2285 : Calculus

Find the implicit derivative $\frac{\mathrm{d} y}{\mathrm{d} x}$ of 2x^2y+yx-y^3=2x-y at the point (0,1){} .

Possible Answers:

$-\frac{3}{2}$

$-\frac{1}{2}$

$-\frac{3}{4}$

$-\frac{2}{3}$

Correct answer:

$-\frac{1}{2}$

Explanation:

Use implicit differentiation to find $\frac{\mathrm{d} y}{\mathrm{d} x}$ of 2x^2y+yx-y^3=2x-y which means to take the derivative of each term in the function with its respective part.

It is simpfied to

$4xy+2x^2*\frac{\mathrm{d} y}{\mathrm{d} x} + y+x*\frac{\mathrm{d} y}{\mathrm{d} x} - 3y^2*\frac{\mathrm{d} y}{\mathrm{d} x}=2 -\frac{\mathrm{d} y}{\mathrm{d} x}$ .

It is then further simplified to

$2x^2*\frac{\mathrm{d} y}{\mathrm{d} x}+x*\frac{\mathrm{d} y}{\mathrm{d} x}-3y^2*\frac{\mathrm{d} y}{\mathrm{d} x}+\frac{\mathrm{d} y}{\mathrm{d} x}=2-4xy-y$ ,

then to

$\frac{\mathrm{d} y}{\mathrm{d} x}(2x^2+x-3y^2+1)=2-4xy-y$ ,

then to

$\frac{\mathrm{d} y}{\mathrm{d} x}=\frac{2-4xy-y}{2x^2+x-3y^2+1}$ .

Plugging in (0,1){} into the equation gives the value of $-\frac{1}{2}$ .

Report an Error

Example Question #2286 : Calculus

If f(x)=3g(x) and g(x)=f(x) , then what is f''(x^2) ?

Possible Answers:

3g(x^2)+12xf(x^2)

3g(x^2)+6xf(x^2)

6g(x^2)+12x^2f(x^2)

6g(x^2)+12xf(x^2)

6g(x)+12f(x)

Correct answer:

6g(x^2)+12x^2f(x^2)

Explanation:

First we need to find f'(x^2) , which is f'(x^2)=3g(x^2)2x=6xg(x^2) .

Then we find f''(x^2) , which equals to

$f''(x^2)=\frac{\mathrm{d} (6xg(x^2))}{\mathrm{d} x}=6g(x^2)+6xf(x^2)2x=6g(x^2)+12x^2f(x^2)$ .

We used the power rule and the product rule to find the derivatives.

Power Rule: $x^n \rightarrow nx^{n-1}$

Product Rule: $f(x)g(x)\rightarrow f(x)g'(x)+g(x)f'(x)$

Report an Error

Example Question #2287 : Calculus

$y=x^{x^3+2x^2}$ , find $\frac{\mathrm{d} y}{\mathrm{d} x}$ .

Possible Answers:

$\frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^3+4x)]*(x^{x^3+2x^2})$

$\frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^2+2x)]*(x^{x^3+2x^2})$

$\frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^3+2x^2)]*(x^{x^3+2x^2})$

$\frac{\mathrm{d} y}{\mathrm{d} x}=2[(3x^2+4x)\ln (x)+(x^2+2x)]*(x^{x^3+2x^2})$

$\frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^2+2x)]$

Correct answer:

$\frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^2+2x)]*(x^{x^3+2x^2})$

Explanation:

Our starting function is $y=x^{x^3+2x^2}$ in order to create an easier function we will want to take the natural log of both sides. Taking the natural log will bring the exponet infront creating a product for which we can more easily differentiate.

Next take the natural log of both sides which results in the following,

$\ln(y)=\ln (x)^{x^3+2x^2}=(x^3+2x^2)\ln (x)$ .

Now differentiate both sides with respect to x.

$\frac{1}{y}*\frac{\mathrm{d} y}{\mathrm{d} x}=(3x^2+4x)\ln (x)+\frac{1}{x}*(x^3+2x^2)=(3x^2+4x)\ln (x)+(x^2+2x)$ , and when one isolates $\frac{\mathrm{d} y}{\mathrm{d} x}$ .

After isolating $\frac{\mathrm{d} y}{\mathrm{d} x}$ make sure to replace the y term in terms of x.

$\\ \frac{\mathrm{d} y}{\mathrm{d} x}=[(3x^2+4x)\ln (x)+(x^2+2x)]*y\\ \\=[(3x^2+4x)\ln (x)+(x^2+2x)]*(x^{x^3+2x^2})$

Report an Error

View Calculus Tutors

Dossevi
Certified Tutor

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

View Calculus Tutors

Keith
Certified Tutor

Xavier University, Bachelor in Business Administration, Accounting. Ohio University-Main Campus, Masters in Education, Mathem...

View Calculus Tutors

Blossom
Certified Tutor

University of Maryland-Baltimore County, Bachelor of Science, Computer Science.

All Calculus 1 Resources

10 Diagnostic Tests 438 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Seattle, Biology Tutors in San Diego, French Tutors in Washington DC, ACT Tutors in Phoenix, SAT Tutors in Chicago, French Tutors in Houston, Biology Tutors in Chicago, French Tutors in Miami, SSAT Tutors in Dallas Fort Worth, Computer Science Tutors in San Diego

Popular Courses & Classes

GRE Courses & Classes in Phoenix, Spanish Courses & Classes in Los Angeles, ACT Courses & Classes in Atlanta, SAT Courses & Classes in Denver, SAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in San Diego, MCAT Courses & Classes in Washington DC, ISEE Courses & Classes in San Diego, GRE Courses & Classes in Denver, LSAT Courses & Classes in Philadelphia

Popular Test Prep

ISEE Test Prep in Denver, MCAT Test Prep in San Diego, GRE Test Prep in Atlanta, SAT Test Prep in Philadelphia, SAT Test Prep in Washington DC, ACT Test Prep in Philadelphia, GMAT Test Prep in Philadelphia, GMAT Test Prep in New York City, GRE Test Prep in Miami, LSAT Test Prep in Philadelphia

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 1 : How to write equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #195 : Writing Equations

Example Question #191 : Writing Equations

Example Question #201 : Writing Equations

Example Question #202 : Writing Equations

Example Question #203 : Writing Equations

Example Question #11 : How To Write Equations

Example Question #204 : Writing Equations

Example Question #2285 : Calculus

Example Question #2286 : Calculus

Example Question #2287 : Calculus

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: