MHB How do I draw the contour line and gradient of a function at a specific point?

[mathmari]

Hey!

I want to draw the contour line of the function $\displaystyle{y=f(x_1, x_2)=-0,1x_1^2-0,4x_2^2}$ at $y=-2,5$ and at the point $(3,-2)$ I want to draw the gradient.

We have the following: \begin{equation*} y=-2,5 \Rightarrow -0,1x_1^2-0,4x_2^2=-2,5 \Rightarrow -10\cdot \left (-0,1x_1^2-0,4x_2^2\right )=-10\cdot \left (-2,5\right )\Rightarrow x_1^2+4x_2^2=25 \Rightarrow x_1^2=25-4x_2^2 \\ \Rightarrow x_1=\pm \sqrt{25-4x_2^2} \end{equation*}

So that the root is well-defined it must be \begin{align*}25-4x_2^2\geq 0 &\Rightarrow 5^2-(2x_2)^2\geq 0 \\ &\Rightarrow (5-2x_2)(5+2x_2)\geq 0 \\ &\Rightarrow \left (5-2x_2>0 \text{ and } 5+2x_2 >0 \right )\ \ \text{ or } \ \ \left (5-2x_2<0 \text{ and } 5+2x_2< 0 \right ) \\ &\Rightarrow \left (x_2<\frac{5}{2} \text{ and } x_2 >-\frac{5}{2}\right ) \ \ \text{ or } \ \ \left (x_2>\frac{5}{2} \text{ and } x_2<-\frac{5}{2}\right )\end{align*}
Since it cannot be that $\left (x_2>\frac{5}{2} \text{ and } x_2<-\frac{5}{2}\right )$ it must be $\left (x_2<\frac{5}{2} \text{ and } x_2 >-\frac{5}{2}\right )$.

We get the following table:

$\begin{matrix}
x_2 & -2 & -1 & 0 & 1 & 2\\
x_1 & \pm 3 & \pm \sqrt{21} & \pm 5 & \pm \sqrt{21} & \pm 3
\end{matrix}$
So, the contour line is the following:
[desmos="-10,10,-10,10"]\sqrt{25-4x^2};-\sqrt{25-4x^2};[/desmos]

Is this correct? (Wondering) We have that $f_{x_1}=-0,2x_1$ and $f_{x_2}=-0,8x_2$. So, the gradient is $\nabla f=\begin{pmatrix}-0,2x_1\\ -0,8x_2\end{pmatrix}$.

At the point $(3,-2)$ it is $\nabla f(3,-2)=\begin{pmatrix}-0,6\\ 1,6\end{pmatrix}$.

How can we draw it at the graph above? (Wondering)

 

[Dethrone]

Hi mathmari,

Your work looks correct to me, assuming $x_1$ is the vertical axis and $x_2$ is the horizontal axis. Alternatively, you could have recognized that $x_1^2+4x_2^2=25$ is an ellipses with major and minor axis of 5 and 2.5, respectively, to graph it directly.

To draw $(-0.6, 1.6)$, recall that the gradient of a function at a point is perpendicular to the contour lines at that point.

 

[mathmari]

Your work looks correct to me, assuming $x_1$ is the vertical axis and $x_2$ is the horizontal axis. Alternatively, you could have recognized that $x_1^2+4x_2^2=25$ is an ellipses with major and minor axis of 5 and 2.5, respectively, to graph it directly.

So, it must be as follows, or not? (Wondering) [desmos="-10,10,-10,10"]x=\sqrt{25-y^2};x=-\sqrt{25-y^2};\left(-0.6,\space1.6\right);[/desmos]

To draw $(-0.6, 1.6)$, recall that the gradient of a function at a point is perpendicular to the contour lines at that point.

This point isn't on the contour line. Is this correct? (Wondering)

 

[Dethrone]

So, it must be as follows, or not? (Wondering)

Yup, I agree with your answer. Your first plot was correct also -- it doesn't matter which axis you set as $x_1$ or $x_2$.

This point isn't on the contour line. Is this correct? (Wondering)

Well, the gradient is a vector. Since we are finding the gradient at a point, namely, $(3,-2)$, we would draw the vector starting from that point, rather than the origin.

 

[mathmari]

Well, the gradient is a vector. Since we are finding the gradient at a point, namely, $(3,-2)$, we would draw the vector starting from that point, rather than the origin.

Ah ok!

[desmos="-5,5,-5,5"]x=\sqrt{25-4y^2};x=-\sqrt{25-4y^2};\left(3,-2\right);\left(3-0.6t,\space-2+1.6t\right);[/desmos]Is this correct? (Wondering)

 

[Dethrone]

Ah ok!

Is this correct? (Wondering)

Good job! :)

 

[mathmari]

For the function $\displaystyle{y=f(x_1, x_2)=4-x_1-\frac{1}{2}x_2}$ we want to draw the contour line $y=2$.

We have that \begin{equation*} y=2 \Rightarrow 4-x_1-\frac{1}{2}x_2=2 \Rightarrow \frac{1}{2}x_2=4-x_1-2 \Rightarrow \frac{1}{2}x_2=2-x_1 \Rightarrow x_2=4-2x_1\end{equation*}

So, the contour line is
[desmos="-8,8,-5.5,5.5"]4-2x;[/desmos]

We have that \begin{equation*}f_{x_1}(x_1, x_2)=\frac{\partial{f(x_1, x_2)}}{\partial{x_1}}=-1\end{equation*}
and \begin{equation*}f_{x_2}(x_1, x_2)=\frac{\partial{f(x_1, x_2)}}{\partial{x_2}}=-\frac{1}{2}\end{equation*}

Therefore, \begin{equation*}\nabla f=\begin{pmatrix}f_{x_1}\\ f_{x_2}\end{pmatrix}=\begin{pmatrix}-1\\ -\frac{1}{2}\end{pmatrix}\end{equation*}
and at the point $(1,2)$ we have that \begin{equation*}\nabla f(1,2)=\begin{pmatrix}f_{x_1}(1,2)\\ f_{x_2}(1,2)\end{pmatrix}=\begin{pmatrix}-1\ -\frac{1}{2}\end{pmatrix}\end{equation*}

The gradient at the graph is:
[desmos="-8,8,-5.5,5.5"]4-2x;\left(1-t,\space2-\frac{t}{2}\right);[/desmos]

Is this correct? (Wondering)

 

[Dethrone]

Is this correct? (Wondering)

Perfect :)

 

[mathmari]

Great! Thank you very much! (Happy)