3.9 Higher Dimensions

Higher Dimensions

Introduction

Inputs and Outputs with Functions of Three or More Variables

Visualizing Functions of Three or More Variables

Measuring Change with Functions of Three or More Variables

Directional Derivatives and Gradients

State the equation and shape of the level curves for the function $f(x,y)=x^2+y^2$ for the values $-2,-1,0,1,2$.

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

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

In this activity, we will look at level surfaces created by two three-variable functions and consider the direction in which the output is increasing or decreasing as fast as possible.

In this activity, we will look at level surfaces created by two three-variable functions and consider the direction in which the output is increasing or decreasing as fast as possible.

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

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

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

Calculate $\nabla f$ and $\nabla g$.

Sketch the level surfaces of $f$ for the values $k=\{-2,-1,0,1,2\}$. Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces change in terms of the value of $k$.

Write a few sentences about how the direction and magnitude of $\nabla f$ is related to the level surfaces from the previous part.

Sketch the level surfaces of $g$ for the values $k=\{-2,-1,0,1,2\}$. Write a few sentences about the shape of each of these level surfaces and describe how the level surfaces change in terms of the value of $k$. You may find it helpful to notice that each of the level surfaces can be expressed with $z$ as a function of $x$ and $y$.

Write a few sentences about how the direction and magnitude of $\nabla g$ is related to the level surfaces from the previous part.

Find the equation of the tangent plane to the surface given by $xz+2x^2y+y^2z^3=11$ at the point $(2,1,1)$.

Suppose that $\nabla f_P =\langle 2,-4,4\rangle$. Is $f$ increasing or decreasing at $P$ in the direction $\langle2,1,3\rangle$?