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Unformatted text preview: MATH 20C - Monday, October 18, 2010: ﬁrst midterm MATH 20C Lecture 11 - Wednesday, October 20, 2010
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 − x2 − y 2 , using slices by the coordinate planes. (derived carefully). Contour map: 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 − x2 − y 2 . Showed more examples of computer plots (z = y 2 − x2 , and another one). Contour map gives some qualitative info about how f varies when we change x, y. (shown an example where increasing x leads f to increase). MATH 20C Lecture 12 - Friday, October 22, 2010
Reviewed countour maps and level curves. Limits
By substitution: 3x2 + 2y − 5 cos(4(x + y )) 0+0−5 = =5 x−y xy − e 0−1 (x,y )→(0,0) lim Disclaimer: limits do not always exist! 2 For instance, take lim(x,y)→(0,0) x2x y2 . Direct substitution does not work. Drawn contour map. We + see that a bunch of level curves intersect at (0, 0), so the limit does not exist. xy On the other hand lim(x,y)→(0,0) sin(++) ) = 1. (x y Partial derivatives
fx = ∂f f (x0 + ∆x, y0 ) − f (x0 , y0 ) = lim ; same for fy . ∂x ∆x→0 ∆x Geometric interpretation: fx , fy are slopes of tangent lines of vertical slices of the graph of f (ﬁxing y = y0 ; ﬁxing x = x0 ). How to compute: treat x as variable, y as constant. Example: f (x, y ) = x3 y + y 2 , then fx = 3x2 y, fy = x3 + 2y. Another example: g (x, y ) = cos(x3 y + y 2 ). Use chain rule (version I) ∂F dF ∂u = ∂x du ∂x 1 Here F (u) = cos u and u = f , so get Product rule: ∂g = −(3x2 y ) sin(x3 y + y 2 ). ∂x ∂ (f g ) ∂f ∂g (x0 , y0 ) = g (x0 , y0 ) + f (x0 , y0 ) ∂x ∂x ∂x 2 ...
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