Lecture Note
University
Massachusetts Institute of TechnologyCourse
Multivariable CalculusPages
4
Academic year
2022
Sporkz
Views
18
Review 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, f of (x,y). F(x,y) The function is complicated, but if we take a little box in the xy- plane, then inside of thisbox 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 numbersand x + 2y + 4, near this. Inside this box, a function is well approximated by a simpler function. That is what calledlinear approximation. And here you can see the illustration of the example that we were thinking about.
On the left are level curves of the function x2 + y2. Not really complicated. However, thenwe were looking near the point (minus 1, 1). This graph shows a point (-1, 1). Zooming inon the point reveals a second 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 theoriginal function. On the right are the level curves of the linear approximation we have already found.
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 smallbox 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 willgain 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 anexample. 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 redvector, 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 vectorthat is perpendicular to the level lines.
ax+bx+c 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 thatred vector that's perpendicular to the line.
Linear Approximation Review: Integrating Functions and Vectors
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