# Ex let f x y xe xy i find the rate of change of f at

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- ex. Let f(x, y) = xexy. (i) Find the rate of change of fat the point (3, 0) in the direction of 2ij. (ii) Find the equation of the tangent plane to the surface z= f(x, y) at the point (3, 0). Know the geometric properties of the gradient vector - In particular, the gradient f(a, b) is perpendicular to the contour of fthrough (a, b) - The gradient is the direction of greatest change for a function f- The magnitude of the gradient vector, ||grad f||, is the maximum rate of change of fat the point (a, b) Know that a vector perpendicularto the gradient vector for a function f(x, y) at a point Pis a vector for which the change of f(x, y) is 0.
Math 10C Final Exam Review Outline8 Section 14.6: The Chain Rule Know the chain rule for z= f(x, y) if x= g(t) and y= h(t). Know the chain rule for z= f(x, y) if x= g(u, v) and y= h(u, v). Know how the above diagrams help you determine derivatives such as dz/dtor z/u. - ex. Suppose z= xsin(y), x= 5t, y= t2– 2. Compute dz/dt. - ex. Suppose z= ln(x+ y), xuv=+, y= ev+ u. Compute z/uand z/v. Given a table of values, know how to find the value of the partial derivative using the chain rule - ex. Suppose fis a differentiable function of xand y, where x= rcosqand y= rsinq. Use the formulas for xand y, and the table of values to the right to calculate f/rand f/∑qwhen r= 2 and q= p/2. Section 14.7: Second-Order Partial Derivatives Know how to take second order partial derivatives Note: If fxyand fyxare continuous, then fxy= fyx- ex. If 22( ,)f x yxy=+, compute fxx(0, 1), fxy(0, 1), fyy(0, 1). Know how to use partial derivatives to construct Taylor polynomials (of degree 1 and 2) - Taylor Polynomial of Degree 1 near (a, b): f(x, y) ºf(a, b) + fx(a, b)(xa) + fy(a, b)(yb) - Note: This is just the tangent plane equation discussed in Section 14.3 - Taylor Polynomial of Degree 2 near (a, b): 22( ,)( , )( , )()( , )( , )( , )()( , )()()()22xyyyxxxyf x yf a bfa bxafa bfa bfa bxafa bxaybxb+++++- ex. Given 22( ,)f x yxy=+, find the Taylor Polynomial of Degree 2 near (0, 0). Section 15.1: Local Extrema Know how to find critical points of a function (This is analogous to what we did in 10A) - ex. Find the critical points of f(x) = x2+ 3xy+ y2. zz/xz/yxydx/dtdy/dttzz/xz/yxxyyvux/uy/vu v2( ,)(0,2)514(2,0)312(2,)224xyx yfffπ
Math 10C Final Exam Review Outline9 Know how to classify critical points using the second derivative test - D= [fxx(x0, y0)]ÿ[fyy(x0, y0)] – [fxy(x0, y0)]2+ if D> 0 and fxx(x0, y0) > 0, then fhas a local minimum at (x0, y0) + if D> 0 and fxx(x0, y0) < 0, then fhas a local maximum at (x0, y0) + if D< 0, then fhas a saddle point at (x0, y0) + if D= 0, the second derivative test provides no information - ex. Classify the critical points of f(x) = x2+ 3xy+ y2.
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