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 #1641 : Calculus Ii

Find the derivative of the function:

$y=sin(e^{3x})$

Possible Answers:

$y'=3e^{3x}cos(e^{3x}})$

$y'=e^{3x}sin(e^{3x})$

$y'=e^{3x}cos(e^{3x})$

y'=3e^xcos(e^x)

$y'=3e^{3x}sin(e^{3x})$

Correct answer:

$y'=3e^{3x}cos(e^{3x}})$

Explanation:

to find the derivative of this function we need to use the chain rule:

$\frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}$

let

$u=e^{3x}$ and y=sin(u) $\therefore \frac{dy}{du}=cos(u)=cos(e^{3x})$

and

$\frac{du}{dx}=3e^{3x}$

and

$\frac{dy}{du}=3e^{3x}cos(e^3x)$ $\frac{dy}{dx}=3e^{3x}cos(e^{3x})$

Report an Error

Example Question #1 : Euler's Method

Suppose we have the following differential equation with the initial condition:

$\frac{\partial p }{\partial x} = 0.5x(1-x), \ p(0) = 2$

Use Euler's method to approximate p(2) , using a step size of .

Possible Answers:

Correct answer:

Explanation:

We start at x = 0 and move to x=2, with a step size of 1. Essentially, we approximate the next step by using the formula:

$p(x + \Delta x) = p(x) + p'(x)\cdot (\Delta x)$ .

So applying Euler's method, we evaluate using derivative:

$p'(x) = 0.5x(1-x), \ p(0) = 2$

And two step sizes, at x = 1 and x = 2.

${}p(x=1) = p(0) + (1 - 0)p'(0) = 2 + 1\cdot p'(0) = 2 + 1 \cdot 0.5(0)(1-0) = 2 + 0 = 2$

${}p(x=2) = p(1) + (2-1)\cdot p'(1) = 2 + 1\cdot p'(1) = 2 + 0.5(1)(1-1) = 2 + 0 = 2$

And thus the evaluation of p at x = 2, using Euler's method, gives us p(2) = 2.

Report an Error

Example Question #2 : Euler's Method

Approximate $\small e^{\frac{1}{2}}$ by using Euler's method on the differential equation

$\small y'=y$

with initial condition $\small y(0)=1$ (which has the solution $\small y(t)=e^t$ ) and time step $\small \small h=\frac{1}{4}$ .

Possible Answers:

$\small 1.5625$

$\small 1.56$

$\small 1.649$

$\small 1.5$

Correct answer:

$\small 1.5625$

Explanation:

Using Euler's method with $\small h=1/4$ means that we use two iterations to get the approximation. The general iterative formula is

$\small y_{i+1}=y_i+h\cdot f(t_i,y_i)$

where each $\small t_i$ is

$\small t_0=0$

$\small t_1=\frac{1}{4}$

$\small t_2=\frac{1}{2}$

$\small y_i$ is an approximation of $\small y(t_i)$ , and y'=f(t,y)=y , for this differential equation. So we have

$\small \small \small \small y_1=y_0+\frac{1}{4}f(t_0,y_0)=1+\frac{1}{4}\cdot 1=1.25$

$\small \small \small \small e^{1/2}= y(\frac{1}{2})\approx y_2=y_1+\frac{1}{4}f(t_1,y_1)=1.25+\frac{1}{4}\cdot 1.25=1.5625$

So our approximation of $\small e^{1/2}$ is

$\small e^{1/2}\approx 1.5625$

Report an Error

Example Question #3 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=3x+4y$ at x=1 with the initial condition y(0)=0 and step size h=0.25 .

Possible Answers:

(1,0)

(1,2.0625)

(1,0.1875)

(1,0.75)

Correct answer:

(1,2.0625)

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=3x+4y$

h=0.25

Starting at the initial point (x_0,y_0)=(0,0)

$x_{1}=x_0+h=0+0.25=0.25$

$y_{1}=y_0+h*f(x_0,y_0)=0+0.25*(3*0+4*0)=0$

(x_1,y_1)=(0.25,0)

$x_{2}=x_1+h=0.25+0.25=0.5$

$y_{2}=y_1+h*f(x_1,y_1)=0+0.25*(3*0.25+4*0)=0.1875$

(x_2,y_2)=(0.5,0.1875)

We continue using Euler's method until x=1 . The results of Euler's method are in the table below.

Screen shot 2015 11 19 at 3.48.44 pm

Note: Solving this differential equation analytically gives a different answer, 9.2997 . Future problems will explain this discrepancy.

Report an Error

Example Question #4 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=3x+4y$ at x=1 with the initial condition y(0)=0 and step size h=0.1 .

Possible Answers:

(1,3.0114)

(1,4.4860)

(1,0.03)

(1,0)

Correct answer:

(1,4.4860)

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=3x+4y$

h=0.1

Starting at the initial point (x_0,y_0)=(0,0)

$x_{1}=x_0+h=0+0.1=0.1$

$y_{1}=y_0+h*f(x_0,y_0)=0+0.1*(3*0+4*0)=0$

(x_1,y_1)=(0.1,0)

$x_{2}=x_1+h=0.1+0.1=0.2$

$y_{2}=y_1+h*f(x_1,y_1)=0+0.1*(3*0.1+4*0)=0.03$

(x_2,y_2)=(0.2,0.03)

We continue using Euler's method until x=1 . The results of Euler's method are in the table below.

Problem 2

Note: Solving this differential equation analytically gives a different answer, 9.2997 . As the step size gets larger, Euler's method gives a more accurate answer.

Report an Error

Example Question #5 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=\frac{y}{x}$ at x=2 with the initial condition y(1)=1 and step size h=0.25 .

Possible Answers:

(2,1)

(2,3)

(2,2)

(2,1.75)

Correct answer:

(2,2)

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=\frac{y}{x}$

h=0.25

Starting at the initial point (x_0,y_0)=(1,1)

$x_{1}=x_0+h=1+0.25=1.25$

$y_{1}=y_0+h*f(x_0,y_0)=1+0.25*(1/1)=1.25$

(x_1,y_1)=(1.25,1.25)

$x_{2}=x_1+h=1.25+0.25=1.5$

$y_{2}=y_1+h*f(x_1,y_1)=1.25+0.25*(1.25/1.25)=1.5$

(x_2,y_2)=(1.5,1.5)

We continue using Euler's method until x=2 . The results of Euler's method are in the table below.

Problem 3

Report an Error

Example Question #6 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=\frac{y}{x}$ at x=2 with the initial condition y(1)=1 and step size h=0.1 .

Possible Answers:

(2,1)

(2,3)

(2,2)

(2,1.75)

Correct answer:

(2,2)

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=\frac{y}{x}$

h=0.1

Starting at the initial point (x_0,y_0)=(1,1)

$x_{1}=x_0+h=1+0.1=1.1$

$y_{1}=y_0+h*f(x_0,y_0)=1+0.1*(1/1)=1.1$

(x_1,y_1)=(1.1,1.1)

$x_{2}=x_1+h=1.1+0.1=1.2$

$y_{2}=y_1+h*f(x_1,y_1)=1.1+0.1*(1.1/1.1)=1.2$

(x_2,y_2)=(1.2,1.2)

We continue using Euler's method until x=2 . The results of Euler's method are in the table below.

Problem 4

Note: Due to the simplicity of the differential equation, Euler's method finds the exact solution, even with a large step size, Using a smaller step size is unnecessary and more time consuming.

Report an Error

Example Question #7 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=cos(x)$ at $x=\pi/2$ with the initial condition y(0)=0 and step size $h= \pi/10$ .

Possible Answers:

$(\pi/2,0.314)$

$(\pi/2,1.57)$

$(\pi/2,1.25)$

$(\pi/2,1.14)$

Correct answer:

$(\pi/2,1.14)$

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=cos(x)$

$h= \pi/10$

Starting at the initial point (x_0,y_0)=(0,0)

$x_{1}=x_0+h=0+\pi/10=\pi/10$

$y_{1}=y_0+h*f(x_0,y_0)=0+(\pi/10)*cos(0)=\pi/10$

$(x_1,y_1)=(\pi/10,\pi/10)$

$x_{2}=x_1+h=\pi/10+\pi/10=\pi/5$

$y_{2}=y_1+h*f(x_1,y_1)=\pi/10+(\pi/10)*cos(\pi/10)=\pi/5$

$(x_2,y_2)=(\pi/5,\pi/5)$

We continue using Euler's method until $x=\pi/2$ . The results of Euler's method are in the table below.

Problem 7

Report an Error

Example Question #2 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=cos(x)$ at $x=\pi/2$ with the initial condition y(0)=0 and step size $h= \pi/20$ .

Possible Answers:

$(\pi/2,2)$

$(\pi/2,1.25)$

$(\pi/2,0.314)$

$(\pi/2,1.07)$

Correct answer:

$(\pi/2,1.07)$

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=cos(x)$

$h= \pi/20$

Starting at the initial point (x_0,y_0)=(0,0)

$x_{1}=x_0+h=0+\pi/20=\pi/20$

$y_{1}=y_0+h*f(x_0,y_0)=0+(\pi/20)*cos(0)=\pi/20$

$(x_1,y_1)=(\pi/20,\pi/20)$

$x_{2}=x_1+h=\pi/20+\pi/20=\pi/10$

$y_{2}=y_1+h*f(x_1,y_1)=\pi/20+(\pi/20)*cos(\pi/20)=\pi/10$

$(x_2,y_2)=(\pi/10,\pi/10)$

We continue using Euler's method until $x=\pi/2$ . The results of Euler's method are in the table below.

Problem 8

Report an Error

Example Question #9 : Euler's Method

Use Euler's method to find the solution to the differential equation $\frac{dy}{dx}=y^2e^{x}$ at x=6 with the initial condition y(0)=0.01 and step size h= 1 .

Possible Answers:

(6,1.0157)

(6,0.0239)

(6,0.1089)

(6,0.1013)

Correct answer:

(6,0.1089)

Explanation:

Euler's method uses iterative equations to find a numerical solution to a differential equation. The following equations

$x_{n+1}=x_n+h$

$y_{n+1}=y_n+h*f(x_n,y_n)$

are solved starting at the initial condition and ending at the desired value. f(x_n,y_n) is the solution to the differential equation.

In this problem,

$f(x,y)=\frac{dy}{dx}=y^2e^{x}$

h= 1

Starting at the initial point (x_0,y_0)=(0,0.01)

$x_{1}=x_0+h=0+1=1$

$y_{1}=y_0+h*f(x_0,y_0)=0.01+1*(0.01)^2*e^{0}=0.0101$

(x_1,y_1)=(1,0.0101)

$x_{2}=x_1+h=1+1=2$

$y_{2}=y_1+h*f(x_1,y_1)=0.0101+1*(0.0101)*e^{1}=0.01038$

(x_2,y_2)=(2,0.01038)

We continue using Euler's method until x=6 . The results of Euler's method are in the table below.

Problem 11

Report an Error

View Calculus 2 Tutors

Satweka
Certified Tutor

The University of Texas at Dallas, Master of Science, Biomedical Engineering.

View Calculus 2 Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

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.

All Calculus 2 Resources

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

Popular Subjects

Biology Tutors in Philadelphia, Calculus Tutors in Philadelphia, SSAT Tutors in Los Angeles, GRE Tutors in Phoenix, Chemistry Tutors in Miami, Physics Tutors in Los Angeles, Reading Tutors in Houston, GMAT Tutors in Dallas Fort Worth, LSAT Tutors in Houston, GMAT Tutors in Phoenix

Popular Courses & Classes

MCAT Courses & Classes in Philadelphia, LSAT Courses & Classes in Washington DC, GMAT Courses & Classes in Philadelphia, ACT Courses & Classes in New York City, MCAT Courses & Classes in Boston, Spanish Courses & Classes in Dallas Fort Worth, GRE Courses & Classes in Houston, SSAT Courses & Classes in Washington DC, GMAT Courses & Classes in Atlanta, SAT Courses & Classes in Dallas Fort Worth

Popular Test Prep

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

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 #1641 : Calculus Ii

Example Question #1 : Euler's Method

Example Question #2 : Euler's Method

Example Question #3 : Euler's Method

Example Question #4 : Euler's Method

Example Question #5 : Euler's Method

Example Question #6 : Euler's Method

Example Question #7 : Euler's Method

Example Question #2 : Euler's Method

Example Question #9 : Euler's Method

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: