Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 1 : Writing 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 #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 #1252 : Functions

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)=60

M(12)=60

M(0)=15

M(12)=180

M(0)=0`

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

Example Question #2288 : Calculus

Find the limit:

${} \lim_{x\rightarrow \infty} \frac{3x^2+5x+2}{2x^2+x+1}$

Possible Answers:

$\large \infty$

undefined

$\large \frac{3}{2}$

$\large \frac{5}{2}$

Correct answer:

$\large \frac{3}{2}$

Explanation:

This is an indeterminate limit of the form ${} \frac{\infty}{\infty}$ so we can use L'Hopital's rule:

$\lim_{x\to c}\frac{f(x)}{g(x)} = \lim_{x\to c}\frac{f'(x)}{g'(x)}$

(this only works when our original limit is an indeterminate form!)

We take the derivative of the top and bottom and this becomes

${} \lim_{x \to \infty}\frac{6x+5}{4x+1}$ .

But yet again, this is indeterminate, of the same type. So we apply L'Hopital's rule one more time and get

${} \lim_{x \to \infty}\frac{6}{4}$

which reduces to 3/2.

Report an Error

Example Question #2289 : Calculus

Given the acceleration of a baseball is given by $a(t)=8t^{2}+2t$ , what is the average value of the acceleration function over the interval t=1 to t=5 ?

Possible Answers:

$35 \frac{meters}{sec^{2}}$

$88.6667 \frac{meters}{sec^{2}}$

$-14.2 \frac{meters}{sec^{2}}$

$19.86 \frac{meters}{sec^{2}}$

$29.33333 \frac{meters}{sec^{2}}$

Correct answer:

$88.6667 \frac{meters}{sec^{2}}$

Explanation:

To find the average value of this function, we must use the formula:

$Average Value=\frac{\int_{a}^{b}a(t)}{b-a}=\frac{\int_{1}^{5}8t^{2}+2t}{5-1}$

To integrate this function, we must use these formulas:

$\int cf(x)dx=c\int f(x)dx, \int x^{n}dx=\frac{1}{n+1}x^{n+1}$

$\\ \int_{1}^{5} a(t)dt=\frac{8t^{3}}{3}+\frac{2t}{2}\\=\left [ \frac{8t^{3}}{3}+t\right ]^{5}_{1} =\left(\frac{8(5)^{3}}{3}+5\right)-\left(\frac{8(1)^{3}}{3}+1\right)$

=358.33-3.67=354.667

$=\frac{354.667}{4}=88.67 \frac{meters}{sec^{2}}$

Report an Error

View Calculus Tutors

Jordyn
Certified Tutor

University of Illinois at Chicago, Bachelor of Science, Biological and Physical Sciences.

View Calculus Tutors

Shallu
Certified Tutor

Hans Raj Mahila Maha Vidyalaya, Bachelor of Science, Computer Science. Jiwaji University, Master of Science, Computer Science.

View Calculus Tutors

Abhisek
Certified Tutor

University of Florida, Bachelor of Engineering, Computer Science.

All Calculus 1 Resources

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

Popular Subjects

Physics Tutors in Phoenix, Physics Tutors in Chicago, Biology Tutors in Houston, Algebra Tutors in Dallas Fort Worth, Statistics Tutors in Philadelphia, SAT Tutors in Miami, Physics Tutors in Miami, Computer Science Tutors in Atlanta, Spanish Tutors in Phoenix, Reading Tutors in Dallas Fort Worth

Popular Courses & Classes

GRE Courses & Classes in Miami, Spanish Courses & Classes in Atlanta, MCAT Courses & Classes in Atlanta, ISEE Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in San Diego, ACT Courses & Classes in Houston, GMAT Courses & Classes in Miami, ACT Courses & Classes in Phoenix, GMAT Courses & Classes in New York City, Spanish Courses & Classes in Denver

Popular Test Prep

ACT Test Prep in Phoenix, GRE Test Prep in Washington DC, SSAT Test Prep in Seattle, SAT Test Prep in Washington DC, MCAT Test Prep in Chicago, GMAT Test Prep in Los Angeles, MCAT Test Prep in Seattle, GRE Test Prep in Atlanta, GRE Test Prep in Miami, LSAT Test Prep in Los Angeles

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 : Writing Equations

Study concepts, example questions & explanations for Calculus 1

All Calculus 1 Resources

Example Questions

Example Question #201 : Writing Equations

Example Question #202 : Writing Equations

Example Question #203 : Writing Equations

Example Question #1252 : Functions

Example Question #204 : Writing Equations

Example Question #2285 : Calculus

Example Question #2286 : Calculus

Example Question #2287 : Calculus

Example Question #2288 : Calculus

Example Question #2289 : Calculus

All Calculus 1 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: