Math 2433 Section 26013 MW 1-2:30pm GAR 205 Bekki George firstname.lastname@example.org 639 PGH Office Hours: 11:00 - 11:45am MWF or by appointment Class webpage: http://www.math.uh.edu/~bekki/Math2433.html Quiz due dates and sections covered are now posted on class webpage! Popper07 1. Find the gradient of f (x, y, z) = 3x + ln z y 15.2 Gradients and Directional Derivatives Properties of gradients: Directional Derivatives: f 'u gives the directional derivative of f in the direction u. In other words, f u′ gives the rate of change of f in the direction of u. f 'u = ∇f (x) i u Example: 1. Find the directional derivative at the point P in the direction indicated. f (x, y) = x 2 + 3y 2 at P(1,1) in the direction of i − j 2 2 2.Find the directional derivative for f (x, y) = x + 3y at ⎛ 3⎞ R ⎜ 2, ⎟ ⎝ 2⎠ . ⎛ 1⎞ Q ⎜ −1, ⎟ 2⎠ ⎝ towards Note that the directional derivative in a direction u is the component of the gradient vector in that direction. Important: Example: Find a unit vector in the direction in which f increases most rapidly at P and give the rate of change of f in that direction; find a unit vector in the direction in which f decreases most rapidly at P and give the rate of change of f in that direction. f ( x, y) = y 2e2 x at P(0,1) Popper07 2. What is the direction in which the function f (x, y) = yx 2 − x y2 increases most f (x, y) = yx 2 − x y2 decreases rapidly at the point (-2, 1)? 3. What is the direction in which the function most rapidly at the point (-2, 1)? 15.3 - The Mean Value Theorem; Chain Rules What was the MVT for functions of one variable? 3 Example: Let f (x, y) = x − xy and let a = (0,1) and b = (1,3). Find a point c on the line segment ab for which the mean value theorem (for several variables) is satisfied. A nonempty open set U (in the plane or in three-space) is said to be connected if any two points of U can be joined by a polygonal path that lies entirely in U. Thm – Let U be an open connected set and let f be a differentiable function on U. If ∇ f (x) = 0 for all x in U, then f is constant on U. Thm - Let U be an open connected set and let f and g be differentiable functions on U. If ∇ f (x) = ∇ g(x) for all x in U, then f and g differ by a constant on U. Examples: 1. Find d ⎡ f (r(t)) ⎤⎦ dt ⎣ given f (x, y) = 6x + y, r(t) = t i + 7t j 2. Find the rate of change of f with respect to t along the given curve. f (x, y) = x2y, r(t) = e t i + e−t j 3. Find the rate of change of f with respect to t along the given curve. Other chain rules: If Then Example: 4. u = x − 3xy + 2 y 2 2 x(t) = cost du y(t) = sint . Find dt And if Then Example: 5. u = x − xy + z 2 2 du x = scost y = sin(t − s) z = t sin s . Find dt Popper07 4. Suppose that g(t) = f (r(t)) . How would you find g ′(t) ?