Question
upstudy study bank question image url

A function and a point are given. Let correspond to the direction of the directional derivative. Complete parts a. through .
(Type any angles in radians between 0 and . Type an exact answer, using as needed. Use a comma to separate answers as needed.)
What angles are associated with the direction of zero change?
(Type any angles in radians between 0 and . Type an exact answer, using as needed. Use a comma to separate answers as needed.)
c. Write the directional derivative at P as a function of ; call this function .
(Type an exact answer, using radicals as needed.)
d. Find the value of that maximizes and find the maximum value. What value of maximizes ?
(Type any angles in radians between 0 and . Type an exact answer, using as needed.)

Ask by Bowman Sanders. in the United States
Mar 21,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The angle that maximizes is .

Solution

We start with the given function and point:
At the point , the gradient is computed by differentiating . Write
so by the chain rule,
At we have:
and thus
The directional derivative in the direction given by the unit vector
is given by
To find the value of that maximizes , recall that for any expression of the form
the maximum value is attained when the unit vector points in the direction of . In our case, we compare with:
Thus the maximum is reached when
We compute the magnitude:
The direction of the vector is given by
Since and the vector is in the fourth quadrant. The basic angle satisfying
is
Being in the fourth quadrant we have
Thus, the value of that maximizes is:

Answered by UpStudy AI and reviewed by a Professional Tutor

error msg
Explain
Simplify this solution

Beyond the Answer

To find the angle that maximizes , we look for the critical points by taking the derivative and setting it equal to zero. The expression will be maximized when is as large as possible relative to , which occurs when . Therefore, .
To find the maximum value of at , we substitute this back into the function :
Thus, the value of that maximizes is:

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy