MT2_review - Midterm 2 Review Edward Carter April 13, 2005...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon
Midterm 2 Review Edward Carter April 13, 2005 1 Partial Derivatives 1.1 Things to Know Definition of the limit of a multivariable function. Definitions of partial derivatives. Clairaut’s Theorem: Suppose f is defined on a disk D that contains the point ( a, b ). If the functions f xy and f yx are both continuous on D , then f xy ( a, b ) = f yx ( a, b ). Formula for tangent plane. Definition of differentiability of a multivariable function. If the partial derivatives f x and f y exist near ( a, b ) and are continuous at ( a, b ), then f is differentiable at ( a, b ). If z = f ( x, y ), x = g ( t ), and y = h ( t ), then dz dt = ∂f ∂x dx dt + ∂f ∂y dy dt . If z = f ( x, y ), x = g ( s, t ), and y = h ( s, t ), then ∂z ∂s = ∂z ∂x ∂x ∂s + ∂z ∂y ∂y ∂s and ∂z ∂t = ∂z ∂x ∂x ∂t + ∂z ∂y ∂y ∂t . If u = f ( x 1 , . . . , x n ) and each x i is a differentiable function of t 1 , . . . , t m , then ∂u ∂t i = n X j =1 ∂u ∂x j ∂x j ∂t i . Let f be a function of x and y . The gradient of f at ( a, b ), f ( a, b ), is defined to be the vector h f x ( a, b ) , f y ( a, b ) i . The definition is analogous for functions with more than two variables. The gradient vector points in the direction of steepest ascent for f . The vector f ( a, b ) is always perpendicular to the level curve passing through ( a, b ). 1
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
The directional derivative of a function of two variables is the derivative of the one variable function you get by taking a slice of the two variable function along a certain direction. It can be computed by taking the dot product of a unit vector for the direction you’re looking at and the gradient of the function at that point. A function f ( x, y, z ) of three variables has level surfaces rather than level curves. At any point ( a, b, c ), the normal vector for the plane tangent to the level surface of f passing through ( a, b, c ) is given by f ( a, b, c ). At any local minimum or maximum of any function, the gradient either does not exist or is equal to the zero vector. Points where the gradient vector does not exist or is equal to the zero vector are called critical points, like in 1A. Let f ( x, y ) have continuous partial derivatives on a disk centered at ( a, b ), and suppose f x ( a, b ) = f y ( a, b ) = 0. Let D = f xx ( a, b ) f yy ( a, b ) - [ f xy ( a, b )] 2 . Then if D > 0 and f xx ( a, b ) > 0, then ( a, b ) is a local maximum for f . If D > 0 and f xx ( a, b ) < 0, then ( a, b ) is a local minimum. If D < 0, then ( a, b ) is neither a local maximum nor a local minimum. Recall from 1A that if you have a continuous function on a closed and bounded set, it must achieve
Background image of page 2
Image of page 3
This is the end of the preview. Sign up to access the rest of the document.

Page1 / 6

MT2_review - Midterm 2 Review Edward Carter April 13, 2005...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online