/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 14 Computing gradients Compute the ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 13: Problem 14

Computing gradients Compute the gradient of the following functions and evaluate it at the given point \(P\). $$f(x, y)=\sin (3 x+2 y) ; P(\pi, 3 \pi / 2)$$

Short Answer

Expert verified
The gradient of the function at the point P is \(\nabla f(\pi, \frac{3\pi}{2}) = \left[3, 2\right]\).

Step by step solution

01

Find the partial derivatives

First, we need to compute the partial derivatives of the function \(f(x, y)\) with respect to x and y: $$ \begin{aligned} \frac{\partial f(x, y)}{\partial x} &= \frac{\partial}{\partial x}\left( \sin(3x + 2y) \right) \\ \frac{\partial f(x, y)}{\partial y} &= \frac{\partial}{\partial y}\left( \sin(3x + 2y) \right) \end{aligned} $$
02

Calculate the partial derivatives

Now, let's calculate the partial derivatives using the chain rule: $$ \begin{aligned} \frac{\partial f(x, y)}{\partial x} &= \cos(3x + 2y) \cdot \frac{\partial}{\partial x}(3x + 2y) = 3 \cos(3x + 2y) \\ \frac{\partial f(x, y)}{\partial y} &= \cos(3x + 2y) \cdot \frac{\partial}{\partial y}(3x + 2y) = 2 \cos(3x + 2y) \end{aligned} $$
03

Evaluate the gradient at point P

Finally, let's evaluate the gradient at the given point \(P(\pi, \frac{3\pi}{2})\): $$ \begin{aligned} \nabla f(\pi, \frac{3\pi}{2}) &= \left[3 \cos\left(3(\pi)+2\left(\frac{3\pi}{2}\right)\right), 2 \cos\left(3(\pi)+2\left(\frac{3\pi}{2}\right)\right)\right] \\ &= \left[3 \cos\left(\frac{15\pi}{2}\right), 2 \cos\left(\frac{15\pi}{2}\right)\right] \\ &= \left[3 , 2 \right] \end{aligned} $$ The gradient of the function \(f(x, y) = \sin(3x + 2y)\) at the point \(P(\pi, \frac{3\pi}{2})\) is \(\nabla f(\pi, \frac{3\pi}{2}) = \left[3, 2\right]\).

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Show that $$\lim _{(x, y) \rightarrow(0,0)} \frac{a x^{2(p-n)} y^{n}}{b x^{2 p}+c y^{p}} \text { does }$$ not exist when \(a, b,\) and \(c\) are nonzero real numbers and \(n\) and \(p\) are positive integers with \(p \geq n\)

Identify and briefly describe the surfaces defined by the following equations. $$z^{2}+4 y^{2}-x^{2}=1$$

Use the gradient rules of Exercise 81 to find the gradient of the following functions. $$f(x, y, z)=(x+y+z) e^{x y z}$$

Identify and briefly describe the surfaces defined by the following equations. $$9 x^{2}+y^{2}-4 z^{2}+2 y=0$$

Evaluate the following limits. $$a.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin (x+y)}{x+y}$$ $$b.\lim _{(x, y) \rightarrow(0,0)} \frac{\sin x+\sin y}{x+y}$$
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.