Question

In Problems $23-26,$ find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=e^{2 x} \sin y ; \quad(0, \pi / 4) $$

Name: Problem 23, Chapter 13: Calculus, Early Transcendentals
Uploaded: 2021-11-23T22:20:37-08:00
Duration: 1 min 57 s
Channel: Lucas Finney
Description: In Problems 23-26, find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. f(x, y)=e^2 x siny ; (0, π/ 4)

   In Problems $23-26,$ find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate.
$$ f(x, y)=e^{2 x} \sin y ; \quad(0, \pi / 4) $$

Calculus, Early Transcendentals

Dennis G. Zill,… 4th Edition

Chapter 13, Problem 23

Instant Answer

Solved by Expert Lucas Finney

11/23/2021

Step 1

The gradient is given by the vector of partial derivatives with respect to $x$ and $y$. The partial derivative of $f$ with respect to $x$ is given by: \[ \frac{\partial f}{\partial x} = 2e^{2x} \sin y \] The partial derivative of $f$ with respect to $y$ is Show more…

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In Problems $23-26,$ find a vector that gives the direction in which the given function increases most rapidly at the indicated point. Find the maximum rate. $$ f(x, y)=e^{2 x} \sin y ; \quad(0, \pi / 4) $$

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Key Concepts

Gradient Vector

The gradient vector of a function is a vector composed of its partial derivatives with respect to each variable. It points in the direction of the greatest rate of increase of the function at a given point, making it a central concept in multivariable calculus for analyzing how functions change.

Directional Derivative

The directional derivative measures the rate of change of a function in any given direction. It is calculated as the dot product of the gradient vector and a unit vector in the specified direction, and it attains its maximum when the direction aligns with the gradient.

Maximum Rate of Increase

The maximum rate of increase of a function at a specific point is given by the magnitude of the gradient vector at that point. This maximum is achieved when moving in the direction of the gradient, where the function increases most rapidly.

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