Q. 16 - Calculus

Question

16. Sketch level curves $z=1,4,9$, and $16$ for the function $z=\frac{x^2}{4}+\frac{y^2}{9}$. Include the graphs of three gradient vectors on each level curve. What do you observe?

Step-by-Step Solution

Answer

The graphs of three gradient vectors on each level curve is included and it is observed that the gradient and tangent vectors are othogonal

Introduction

The given is the function $z=\frac{x^2}{4}+\frac{y^2}{9}$with level curve $z=1,4,9$

The objective is to include the graphs of three gradient vectors and observe the function

Step 1

Let the function be $z=\frac{x^2}{4}+\frac{y^2}{9}$.

The goal is to draw the function's level curves $z=1,4,9$ and $z=16$.

 The function  $z=\frac{x^2}{4}+\frac{y^2}{9}$ has an ellipse as its graph.

The gradient is,

$z=\frac{x^2}{4}+\frac{y^2}{9}$

$\nabla z=\frac{x}{2}\mathbf{i}+\frac{2y}{9}\mathbf{j}$

As a result, the gradient vectors are orthogonal to the level curves and point to the $x$ and $y$ axes, respectively. The magnitude of the gradient vectors grows.

Step 2

The level curves $z=\frac{x^2}{4}+\frac{y^2}{9}$ for $z=1,4,9$ and 16 is,

$$\begin{gathered} 1=\frac{x^2}{4}+\frac{y^2}{9} \\ 4=\frac{x^2}{4}+\frac{y^2}{9} \\ 9=\frac{x^2}{4}+\frac{y^2}{9} \\ 16=\frac{x^2}{4}+\frac{y^2}{9} \end{gathered}$$

The graph of level curves $z=\frac{x^2}{4}+\frac{y^2}{9}$ for $z=1,4,9$ and 16 is as shown below.

As a result, the tangent and gradient vectors are orthogonal.