Resourcesdown chevron
Testimonials
Plans
Sign in
Sign up
Resourcesright arrow
Testimonials
Plans
Bytelearn - cat image with glassesAI tutor

Welcome to Bytelearn!

Let’s check out your problem:

[0//1 Points ] DETAILS MY NOTES SCALCET9 16.3.007. Determine whether or not F is a coliservative vector fleld. If it is, find a function f such that F=grad f. (If the vector field is not conservative, enter DNE.) F(x,y)=(ye^(x)+sin(y))i+(e^(x)+x cos(y))j f(x,y)=

44. [0/1 [0 / 1 Points ] ] \newlineDETAILS\newlineMY NOTES\newlineSCALCET99 1616.33.007007.\newlineDetermine whether or not F \mathrm{F} is a coliservative vector fleld. If it is, find a function f f such that \mathrm{F}=\(\newlineabla f \). (If the vector field is not conservative, enter DNE.)\newlineF(x,y)=(yex+sin⁡(y))i+(ex+xcos⁡(y))j \mathbf{F}(x, y)=\left(y e^{x}+\sin (y)\right) \mathbf{i}+\left(e^{x}+x \cos (y)\right) \mathbf{j} \newlinef(x,y)= f(x, y)=

View step-by-step helpView step-by-step help
Homeright chevron icon
Math Problemsright chevron icon
Algebra 2right chevron icon
Write equations of cosine functions using propertiesright chevron icon

Full solution

Q. 44. [0/1 [0 / 1 Points ] ] \newlineDETAILS\newlineMY NOTES\newlineSCALCET99 1616.33.007007.\newlineDetermine whether or not F \mathrm{F} is a coliservative vector fleld. If it is, find a function f f such that \mathrm{F}=\(\newlineabla f \). (If the vector field is not conservative, enter DNE.)\newlineF(x,y)=(yex+sin⁡(y))i+(ex+xcos⁡(y))j \mathbf{F}(x, y)=\left(y e^{x}+\sin (y)\right) \mathbf{i}+\left(e^{x}+x \cos (y)\right) \mathbf{j} \newlinef(x,y)= f(x, y)=
  1. Check Curl of F: To check if FF is conservative, we need to see if the curl of FF is zero. The vector field F(x,y)=(yex+sin⁡(y))i+(ex+xcos⁡(y))jF(x,y) = (y e^x + \sin(y))i + (e^x + x \cos(y))j. Calculate the partial derivatives: ∂∂y\frac{\partial}{\partial y} of the first component: ∂∂y(yex+sin⁡(y))=ex+cos⁡(y)\frac{\partial}{\partial y} (y e^x + \sin(y)) = e^x + \cos(y). ∂∂x\frac{\partial}{\partial x} of the second component: ∂∂x(ex+xcos⁡(y))=ex+cos⁡(y)\frac{\partial}{\partial x} (e^x + x \cos(y)) = e^x + \cos(y).
  2. Verify Conservative Field: Since the partial derivatives are equal, the curl of FF is zero, indicating that FF is a conservative vector field.
  3. Find Potential Function: Now, find a potential function ff such that F = \(\newlineabla f\). Start by integrating the first component of FF with respect to xx:f(x,y)=∫(yex+sin⁡(y)) dx=yex+xsin⁡(y)+g(y)f(x,y) = \int (y e^x + \sin(y)) \, dx = y e^x + x \sin(y) + g(y), where g(y)g(y) is a function of yy only.
  4. Differentiate with Respect to y: Next, differentiate this potential function with respect to yy to match the second component of FF:∂f∂y=ex+xcos⁡(y)+g′(y).\frac{\partial f}{\partial y} = e^x + x \cos(y) + g'(y).Since ∂f∂y\frac{\partial f}{\partial y} should equal the second component of FF, which is ex+xcos⁡(y)e^x + x \cos(y), we set g′(y)=0g'(y) = 0.
  5. Integrate g′(y)=0g'(y) = 0: Integrate g′(y)=0g'(y) = 0 to find g(y)g(y):g(y)=Cg(y) = C, where CC is a constant.
  6. Substitute g(y)=Cg(y) = C: Substitute g(y)=Cg(y) = C into the potential function:\newlinef(x,y)=yex+xsin⁡(y)+Cf(x,y) = y e^x + x \sin(y) + C.\newlineSince the constant CC does not affect the gradient, we can set C=0C = 0 for simplicity.

More problems from Write equations of cosine functions using properties

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (π,−1)\left(\pi, -1\right). \newlineWrite your answer in the form f(x)=Acos⁡(Bx+C)+Df(x)=A\cos(Bx+C)+D, where AA, BB, CC, and DD are real numbers.\newline f(x)=f(x)=
Get tutor helpright-arrow

Posted 1 month ago

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (−π2,−1)\left(-\frac{\pi}{2},-1\right). Write your answer in the form f(x)=Asin⁡(Bx+C)+Df(x) = A\sin(Bx + C) + D, where AA, BB, CC, and DD are real numbers. `f(x)` =______
Get tutor helpright-arrow

Posted 2 months ago

Question
Find an equation for a sinusoidal function that has period 2π2\pi, amplitude 11, and contains the point (0,−1)(0,-1). \newlineWrite your answer in the form f(x)=Acos⁡(Bx+C)+Df(x) = A\cos(Bx + C) + D, where AA, BB, CC, and DD are real numbers. \newlinef(x)=f(x) = _____
Get tutor helpright-arrow

Posted 2 months ago

Question
Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought 1010 cans of food, and each student brought 22 cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought 33 cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class yy in terms of the number of students xx.
Get tutor helpright-arrow

Posted 2 months ago

Question
Franklin Middle School is having a canned food drive. In Mr. Lao's class, Mr. Lao brought 1010 cans of food, and each student brought 22 cans. In Ms. Roger's class, Ms. Rogers did not bring any cans, but each student brought 33 cans of food. Write two equations in slope-intercept form to represent the number of cans of food for each class yy in terms of the number of students xx.
Get tutor helpright-arrow

Posted 2 months ago

Question
Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is 1212 feet long, and each strand of his Christmes lights will wrap around a 7070-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?\newlineA) 66\newlineB) 77\newlineC) 88\newlineD) 99
Get tutor helpright-arrow

Posted 2 months ago

Question
Kris is wrapping Christmas lights around the railing that runs around three sides of his square porch If one side of his porch is 1212 feet long, and each strand of his Christmas lights will wrap around a 7070-inch length of the railing, what is the minimum number of strands Kris needs to completely wrap the porch railing?\newlineA) 66\newlineB) 77\newlineC) 88\newlineD) 99
Get tutor helpright-arrow

Posted 2 months ago

Question
For a given input value b b , the function f f outputs a value a a to satisfy the following equation.\newline4a+7b=−52 4a+7b=-52 \newlineWrite a formula for f(b) f(b) in terms of b b .\newlinef(b)=□ f(b)=\square
Get tutor helpright-arrow

Posted 2 months ago

Question
For a given input value xx, the function gg outputs a value yy to satisfy the following equation.\newline−4x−6=−5y+2-4x-6=-5y+2\newlineWrite a formula for g(x)g(x) in terms of xx.\newlineg(x)=□g(x)=\square
Get tutor helpright-arrow

Posted 4 months ago

Question
For a given input value aa, the function ff outputs a value bb to satisfy the following equation.\newline−3a+6b=a+4b-3a+6b=a+4b\newlineWrite a formula for f(a)f(a) in terms of aa.\newlinef(a)=□f(a)=\square
Get tutor helpright-arrow

Posted 4 months ago

View all (107)