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Calculus III

Learn about partial derivatives of f(x, y), the tangent plane, and the differential in Calculus III Chapter 14. Understand the properties of the gradient and the Taylor polynomial approximation for f(x, y) near a point. Explore the concept of differentiability and its relation to local linear approximation.

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Calculus III

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  1. Calculus III Chapter 14

  2. Partial Derivatives of f(x,y) • . • .

  3. Tangent Plane and the Differential • The tangent plane to the surface z = f(x,y): • The tangent plane approximation: • The differential: • For a function z = f(x,y), the differential, df, at a point (a,b) is the linear function of dx and dy given by the formula:

  4. The Gradient of z = f(x,y) • If f is a differentiable function at the point (a,b) and f (a,b)  0, then: • The direction of f (a,b) is • Perpendicular to the contour of f through (a,b) • In the direction of increasing f • The magnitude of the gradient vector, || f ||, is • The maximum rate of change of at that point • Large when the contours are close together and small when they are far apart.

  5. The Gradient of w = f(x,y,z) Properties: • The direction of the gradient vector is the direction in which f is increasing at the greatest rate, if it exists. • The magnitude, ||grad f||, is the rate of change of f in that direction. • If the directional derivative of f at (a,b,c) is zero in all directions then the grad f is defined to be 0.

  6. Taylor Polynomial of Deg 2 Approximation for f(x,y) near (a,b). • If f has continuous second-order partial derivatives, then:

  7. Differentiability • For a function f at a point (a,b), let E(x,y) be the error in the local linear approximation, that is the absolute value of the difference between the left and right hand sides, and let d(x,y) be the distance between (x,y) and (a,b). Then f is said to be locally linear, or differentiable at (a,b) if we can make the ratio E(x,y)/d(x,y) as small as we like by restricting (x,y) to a small enough non-zero distance from (a,b).

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