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MATH 251 – LECTURE 11 JENS FORSGÅRD http://www.math.tamu.edu/~jensf/ This week: 12.4–6 webAssign: 12.4–6, due 2/15 11:55 p.m. Friday: Kevin. Next week: 12.7 webAssign: 12.7, opens 2/15 12 a.m. Friday 2/19: Midterm 1: Covering chapters 11 and 12. Help Sessions: M W 5.30–8 p.m. in BLOC 161 Office Hours: BLOC 641C M 12:30–2:30 p.m. W 2–3 p.m. or by appointment. Directional derivatives The surface z = f (x, y) can be viewed as a level curve of the function F (x, y, z) = z − f (x, y). Exercise 1. If you are at the point (x0, y0, f (x0, y0)), in which direction should you move so that the function F increases the fastest? Exercise 2. Recall from previous lectures: what is the normal vector of the surface z = f (x, y) at the point (x0, y0, f (x0, y0))? The Gradient The gradient of F (x, y, z) is always a normal vector to the level curves of the function F . Exercise 3. Let f (x, y) = x2 + y 2. Sketch the level curve of f passing through the point (1, 1). What is ∇f (1, 1)? Local maximum and minimum Definition 4. A function f (x, y) is said to have a local maxima at a point (a, b) if f (a, b) ≥ f (x, y) for all (x, y) close to (a, b). Definition 5. A function f (x, y) is said to have a local minima at a point (a, b) if f (a, b) ≤ f (x, y) for all (x, y) close to (a, b). Theorem 6. If f (x, y) is differentiable at a local maximum (or minimum) (a, b), then ∇f (a, b) = h0, 0i. Local maximum and minimum Definition 7. A point (a, b) is said to be a critical point of f (x, y) if either ∇f (a, b) = h0, 0i, or one of the first partial derivatives of f does not exists at (a, b). Exercise 8. Find all critical points of the function f (x, y) = x2 + xy + 4y. Local maximum and minimum Exercise 9. Find all critical points of the function f (x, y) = ex 3 +4xy . Local maximum and minimum Assuming that f is smooth at the critical point (a, b), then there are three types of critical points. Local maxima Local minima Saddle points Local maximum and minimum Theorem 10 (Second derivative test). Assume that f is twice differentiable at a critical point (a, b). Let 00 00 fxx(a, b) fxy (a, b) 00 00 00 D(a, b) = 00 = fxx (a, b)fyy (a, b) − (fxy (a, b))2 00 fyx(a, b) fyy (a, b) 00 1) if D(a, b) > 0 and fxx (a, b) < 0 then (a, b) is a local maximum. 00 2) if D(a, b) > 0 and fyy (a, b) > 0 then (a, b) is a local minimum. 3) if D(a, b) < 0 then (a, b) is a saddle point. Exercise 11. Find and classify the critical points of f (x, y) = x2 + y 2. Local maximum and minimum Exercise 12. Find and classify the critical points of f (x, y) = −x2 − y 2. Exercise 13. Find and classify the critical points of f (x, y) = x2 − y 2. Local maximum and minimum Exercise 14. Find the point on the plane x − y + z = 7 that is closest to (1, 4, 6).