3.8 Directional Derivatives and the Gradient

Directional Derivatives and the Gradient

Introduction

Directional Derivatives

Efficiently Computing the Directional Derivative

The Gradient

Below is a contour plot showing the elevations for a region of a nearby park. We will be referring to $h(x,y)$ as the function of two variables that gives the elevation as a function of $x$-coordinate (location in the east-west or horizontal direction) and $y$-coordinates (location in the north-south or vertical direction).

Using the contour plot and treating the elevation as a multivariable function $h(x,y)$, state whether each of the following is positive, negative, or zero. Write a sentence to justify your reasoning. -$h_x(A)$-$h_y(A)$-$h_x(B)$-$h_y(B)$-$h_x(C)$-$h_y(C)$

Work through this exercise and explain your reasoning step by step.

Calculate $f_x(x,y)$ and $f_y(x,y)$.

Use equation to determine $D_{\vi} f(x,y)$ and $D_{\vj} f(x,y)$. Write a couple of sentences to describe what familiar functions $D_{\vi} f$ and $D_{\vj} f$ are.

Use equation to find the derivative of $f$ in the direction of the vector $\vv = \langle 2, 3 \rangle$ at the point $(1,-1)$.

Find the derivative of $f$ in the direction of the vector $\vv = \langle 4, 6 \rangle$ at the point $(1,-1)$.

Use equation to find the derivative of $f$ in the direction of the vector $\vv = \langle -2, -3 \rangle$ at the point $(1,-1)$. Write a couple of sentences to explain why this result is different from your answer to the previous two tasks, even though the direction vectors are parallel.

Find the gradient $\nabla f (x,y)$.

For each of the following points $(x_0,y_0)$, evaluate the gradient $\nabla f(x_0,y_0)$ and sketch the gradient vector with its tail at $(x_0,y_0)$. Some of the vectors are too long to fit onto the plot. To draw them to scale, you should scale each vector by a factor of $1/2$. -$(x_0,y_0) = (2,0)$-$(x_0,y_0) = (0,2)$-$(x_0,y_0) = (2,2)$-$(x_0,y_0) = (2,1)$-$(x_0,y_0) = (-3,2)$-$(x_0,y_0) = (-2,-4)$-$(x_0,y_0) = (0,0)$

Write a few sentences about how the direction of the gradient at each of these points is related to the the contour passing through that point.

Does the output of $f$ increase or decrease in the direction of $\nabla f(x_0,y_0)$? Use examples from the points above to write a couple of sentences that justify your answer.