Explanation Given information: f ( x , y , z ) = x 2 y z , P = ( 1 , 5 , 9 ) Calculation: Since the gradient of function is denoted by ∇ f = 〈 ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ y 〉 The given equation f ( x , y , z ) = x 2 y z , Differentiating f with respect to x we get, ∂ f ∂ x = ∂ ∂ x x 2 y z ⇒ ∂ f ∂ x = y z ∂ ∂ x x 2 ⇒ ∂ f ∂ x = 2 x y z Differentiating f with respect to y we get, ∂ f ∂ y = ∂ ∂ y x 2 y z ⇒ ∂ f ∂ y = x 2 z ∂ ∂ y y = x 2 z Differentiating f with respect to z we get, ∂ f ∂ z = ∂ ∂ z x 2 y z ⇒ ∂ f ∂ z = x 2 y ∂ ∂ z z = x 2 y 1 2 z = x 2 y 2

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

4th Edition

ISBN: 9781319312886

Author: Rogawski, Jon, Adams, Colin, FRANZOSA, Robert

Publisher: W. H. Freeman

See similar textbooks

Videos

Question

Chapter 14.5, Problem 36E

To determine

Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 14.5, Problem 35E

Chapter 14.5, Problem 37E

Students have asked these similar questions

Given lim x-4 f (x) = 1,limx-49 (x) = 10, and lim→-4 h (x) = -7 use the limit properties to find lim→-4 1 [2h (x) — h(x) + 7 f(x)] : - h(x)+7f(x) 3 O DNE

17. Suppose we know that the graph below is the graph of a solution to dy/dt = f(t). (a) How much of the slope field can you sketch from this information? [Hint: Note that the differential equation depends only on t.] (b) What can you say about the solu- tion with y(0) = 2? (For example, can you sketch the graph of this so- lution?) y(0) = 1 y AN

(b) Find the (instantaneous) rate of change of y at x = 5. In the previous part, we found the average rate of change for several intervals of decreasing size starting at x = 5. The instantaneous rate of change of fat x = 5 is the limit of the average rate of change over the interval [x, x + h] as h approaches 0. This is given by the derivative in the following limit. lim h→0 - f(x + h) − f(x) h The first step to find this limit is to compute f(x + h). Recall that this means replacing the input variable x with the expression x + h in the rule defining f. f(x + h) = (x + h)² - 5(x+ h) = 2xh+h2_ x² + 2xh + h² 5✔ - 5 )x - 5h Step 4 - The second step for finding the derivative of fat x is to find the difference f(x + h) − f(x). - f(x + h) f(x) = = (x² x² + 2xh + h² - ])- = 2x + h² - 5h ])x-5h) - (x² - 5x) = ]) (2x + h - 5) Macbook Pro

Chapter 14 Solutions

Loose-Leaf Version for Calculus: Early Transcendentals Single Variable

Ch. 14.1 - Prob. 1PQ Ch. 14.1 - Prob. 2PQ Ch. 14.1 - Prob. 3PQ Ch. 14.1 - Prob. 4PQ Ch. 14.1 - Prob. 5PQ Ch. 14.1 - Prob. 1E Ch. 14.1 - Prob. 2E Ch. 14.1 - Prob. 3E Ch. 14.1 - Prob. 4E Ch. 14.1 - Prob. 5E

Knowledge Booster

$Calculus$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.

Tofind: the direction in which f has maximum rate of increase from P , and give the rate of change in that direction.

Solution Summary: The author explains how to determine the direction in which f has maximum rate of increase from P.

Videos

Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

Want to see the full answer?

Chapter 14 Solutions

Tofind: the direction in which f has maximum rate of increase from P , and give the rate of change in that direction.

Solution Summary: The author explains how to determine the direction in which f has maximum rate of increase from P.

Videos

Tofind: the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.

Want to see the full answer?

Chapter 14 Solutions

Tofind:

the direction in which f has maximum rate of increase from P, and give the rate of change in that direction.