Linear Approximation and Perpendicular Vectors

Linear approximation

You already know about about functions of several variables and linear approximation
Also, you are familiar with vectors. Now we'll unite those concepts.
Let's review linear approximation.
Here is some complicated function, fof (x,y).
F(x,y)
The function is complicated, but if we take a little box in the xy- plane, then inside of this
box the function is well approximated by something way less difficult.
It is almost equal to an expression of the form ax + by + c, where a, b, and c are numbers
and x + 2y + 4, near this.

[Graph]
Inside this box, a function is well approximated by a simpler function. That is what called
linear approximation.
And here you can see the illustration of the example that we were thinking about.
[Graph]
On the left are level curves of the function ×2 + y2. Not really complicated. However, then
we were looking near the point (minus 1, 1). This graph shows a point (-1, 1).

Zooming in
on the point reveals a second graph.
[Graph]
In the second picture, the level curves look simpler.
The lines almost look like they are straight, and they almost run parallel to each other.
In that picture, the function ax + by + c has level curves almost equal to those of the
original function.
On the right are the level curves of the linear approximation we have already found.
[Graph]
And if we zoomed in further, the two images would look nearly identical.
Therefore, whatever interesting question we may have about our function f(x) on a small
box can usually be answered if we can find an answer for simple functions such as ax + by
+ c.
With practicing exploring the behaviors of specific functions, such as ax + by = c, you will
gain an understanding of what any function is doing in its little box. That's the goal.
How can we better understand the value of the ax plus by plus c equation? Here's an
example.
[Graph]
The graphs in the image show the level curves of the function x plus one-half y.
We know that they're parallel lines.
One thing we might like to know is the direction perpendicular to those lines. Here is a red
vector, which is perpendicular to the lines.
Let's find that vector. As we are studying ax plus by plus c, for example let's find a vector
that is perpendicular to the level lines.
ax+bxto
So that's our goal.
And that starts to tie in with previous stuff.
You remember vectors, dot product, perpendicular vectors. All of that is useful to find that
red vector that's perpendicular to the line.