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 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 x2 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 x2 y 2 Showed more examples of computer plots z x2 y 2 z y 2 x2 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 f x0 x y0 f x0 y0 fx 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 xing y y0 xing 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 18 02 Lecture 9 Thu Sept 27 2007 Handouts PS3 solutions PS4 Linear approximation Interpretation of fx fy as slopes of slices of the graph by planes parallel to xz and yz planes Linear approximation formula f fx x fy y Justi cation fx and fy give slopes of two lines tangent to the graph y y0 z z0 fx x0 y0 x x0 and x x0 z z0 fy x0 y0 y y0 We can use this to get the equation of the tangent plane to the graph z z0 fx x0 y0 x x0 fy x0 y0 y y0 Approximation formula the graph is close to its tangent plane Min max problems At a local max or min fx 0 and fy 0 since x0 y0 is a local max or min of the slice Because 2 lines determine tangent plane this is enough to ensure that tangent plane is horizontal approximation formula f 0 or rather f x y Def of critical point x0 y0 where fx 0 and fy 0 A critical point may be a local min local max or saddle Example f x y x2 2xy 3y 2 2x 2y Critical point fx 2x 2y 2 0 fy 2x 6y 2 0 gives x0 y0 1 0 only one critical point 1 2 Is it a max min or saddle pictures shown of each type Systematic answer next lecture For today observe f x y 2 2y 2 2x 2y x y 1 2 2y 2 1 1 so minimum Least squares Set up problem experimental data xi yi i 1 n want to nd a best t line y ax b the unknowns here are a b not x y Deviations yi axi b want to minimize the total square deviation D i yi axi b 2 D D 0 and 0 leads to a 2 2 linear system for a and b done in detail as in Notes LS a b x2i a xi b xi yi xi a nb yi Least squares setup also works in other cases e g exponential laws y ceax taking logarithms ln y ln c ax so setting b ln c we reduce to linear case or quadratic laws y ax2 bx c minimizing total square deviation leads to a 3 3 linear system for a b c Example Moore s Law number of transistors on a computer chip increases exponentially with time showed interpolation line on a log plot 18 02 Lecture 10 Fri Sept 28 2007 Second derivative test Recall critical points can be local min w x2 y 2 local max w x2 y 2 saddle w 2 y x2 slides shown of each type Goal determine type of a critical point and nd the global min max Note global min max may be either at a critical point or on the boundary of the domain at in nity We start with the case of w ax2 bxy cy 2 at 0 0 Example from Tuesday w x2 2xy 3y 2 completing the square w x y 2 2y 2 minimum b b b2 1 b If a 0 then w a x2 xy cy 2 a x y 2 c y 2 4a2 x y 2 4ac b2 y 2 a 2a 4a 2a 4a 3 cases if 4ac b2 0 same signs if a 0 then minimum if a 0 then maximum if 4ac b2 0 opposite signs saddle if 4ac b2 0 degenerate case x x This is related to the quadratic formula w y 2 a 2 b c y y 2 2 If b 4ac 0 then no roots so at bt c has a constant sign and w is either always nonnegative or always nonpositive min or max If b2 4ac 0 then at2 bt c crosses zero and changes sign so w can have both signs saddle General case second derivative test 2f We look at second derivatives fxx fxy fyx fyy Fact fxy fyx x2 Given f and a critical point x0 y0 set A fxx x0 y0 B fxy x0 y0 C fyy x0 y0 then if AC B 2 0 then if A 0 or C local min if A 0 local max if AC B 2 0 then saddle 3 if AC B 2 0 then can t conclude Checked quadratic case fxx 2a A fxy b B fyy 2c C then AC B 2 4ac b2 General justi cation quadratic approximation formula Taylor series at order 2 f fx x x0 fy y y0 12 fxx x x0 2 fxy x x0 y y0 12 fyy y y0 2 At a critical point f A2 x x0 2 B x x0 y y0 C2 y y0 2 In degenerate case would need higher order derivatives to conclude NOTE the global min max of a function is not necessarily at a critical point Need to check boundary in nity 1 xy for x 0 y 0 fx 1 x12 y 0 fy 1 xy1 2 0 So x2 y 1 xy 2 1 only critical fxx 2 x3 y fxy 1 x2 y 2 fyy 2 xy 3 So A 2 B 1 C 2 Question type of critical point Answer AC B 2 2 2 1 0 A Example f x y x y point is 1 1 2 0 local min What about the maximum Answer f near boundary x 0 or y 0 and at in nity
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