Chain rule (multivariable calculus)

[Poetria]

Homework Statement

Let $f(x,y)=x^3+y^2+x*y$
Suppose that a point is moving through the plane. At time t , the point is at $(x(t), y(t))=(t^2, e^{t-1})$. Use linear approximation to estimate the change in f as t goes from 1 to 1.1 . In other words, approximate

Relevant Equations

Multivariable chain rule

$f_x=3*x^2+y$
$f_y=2*y+x$

$(3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1}$
Well, I am not sure how to evaluate it.
I got a wrong result by multiplying by 0.1, i.e.
$((3*(t^2)^2+e^{t-1})*2*t+(2*e^{t-1}+t^2)*e^{t-1})*0.1$

I guess it is trivial but I am lost. :(

 

[fresh_42]

We are interested in the change along $t$. So why don't you write $f(x,y)=f(x(t),y(t))=f(t)$ in the first place? Then we have two points at $t=1$ and $t=1.1$. The linear approximation is the secant through these points:
$$
y=\underbrace{\dfrac{f(1)-f(1.1)}{1-1.1}}_{=:m}\cdot x + b\;\wedge\;f(1)=m\cdot 1+b
$$
Finally, you can check the quality of this approximation by calculating the tangent at $t=1$:
$$
y=\left. \dfrac{d}{dt}\right|_{t=1}f(t) \cdot x + c =f'(1)x+c\;\wedge\;f(1)=f'(1)\cdot 1 +c
$$
which is the linear approximation if only one point is given.

 

[Poetria]

Many thanks. But I can't read this:
"The linear approximation is the secant through these points:
You can't use 'macro parameter character #' in math mode ?

A nice quote from Leibniz. :) Émilie du Châtelet (my avatar) was Leibnizian. :)

 

[fresh_42]

Many thanks. But I can't read this

You need to give me a second to correct my format. Please reload.

"He who hasn't tasted bitter things hasn't earned sweet things." (G. Leibniz)

"Let us choose for ourselves our path in life, and let us try to strew that path with flowers." (E. du Châtelet)

$t\longmapsto f(t)$ is the path, and you should plant the flowers at $t=1$ and $t=1.1.$

 

[Poetria]

You need to give me a second to correct my format. Please reload.

"He who hasn't tasted bitter things hasn't earned sweet things." (G. Leibniz)

"Let us choose for ourselves our path in life, and let us try to strew that path with flowers." (E. du Châtelet)

$t\longmapsto f(t)$ is the path, and you should plant the flowers at $t=1$ and $t=1.1.$

I got:
-10.3022 + 13.3022*x = y
x=1.1
Approximation of the change:
4.33022-3=1.33022

A beautiful metaphor: $t\longmapsto f(t)$ is the path, and you should plant the flowers at $t=1$ and $t=1.1.$

 

[Poetria]

Perhaps a gradient would be also a good idea:
Gradient $\vec (0.21, 0.105171)$
A tangent plane: 0.21*x + 0.105171*y-0.32

Slope of the gradient: 0.500814

 

[fresh_42]

A beautiful metaphor: $t\longmapsto f(t)$ is the path, and you should plant the flowers at $t=1$ and $t=1.1.$

Yes, but a little bit too fast posted. The path is actually $t\longmapsto (x(t),y(t))$ and $f(t)$ the function values along the path.

 

[Poetria]

Yes, but a little bit too fast posted. The path is actually $t\longmapsto (x(t),y(t))$ and $f(t)$ the function values along the path.

You may want to benefit from licentia poetica. :)
A long time ago I have written a post about Émilie du Châtelet: https://blogs.otago.ac.nz/emxphi/emilie-du-chatelet-and-experimental-philosophy-i/
Elinor is my avatar's name in Second Life.