lec_week4

lec_week4 - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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MIT OpenCourseWare http://ocw.mit.edu 18.02 Multivariable Calculus Fall 2007 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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18.02 Lecture 8. Tue, Sept 25, 2007 Functions of several variables. Recall: for a function of 1 variable, we can plot its graph, and the derivative is the slope of the tangent line to the graph. Plotting graphs of functions of 2 variables: examples z = y , z = 1 x 2 y 2 , using slices by the coordinate planes. (derived carefully). Contour plot: level curves f ( x, y ) = c . Amounts to slicing the graph by horizontal planes z = c . Showed 2 examples from “real life”: a topographical map, and a temperature map, then did the examples z = y and z = 1 x 2 y 2 . Showed more examples of computer plots ( z = x 2 + y 2 , z = y 2 x 2 , and another one). Contour plot gives some qualitative info about how f varies when we change x, y . (shown an example where increasing x leads f to increase). Partial derivatives. f x = ∂f = lim f ( x 0 + Δ x, y 0 ) f ( x 0 , y 0 ) ; same for f y . ∂x Δ x 0 Δ x Geometric interpretation: f x , f y are slopes of tangent lines of vertical slices of the graph of f (Fxing y = y 0 ; Fxing x = x 0 ). How to compute: treat x as variable, y as constant. Example:
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lec_week4 - MIT OpenCourseWare http:/ocw.mit.edu 18.02...

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