Topics for the Second Hour Exam The short answer to what this hour exam will cover is 13 6 14 5 and 15 1 6 The longer more nuanced answer is I Decomposition of the acceleration vector into tangential and normal components with respect to the velocity vector i e a d2 r dv T v 2 N 2 dt dt with definitions and the derivation II Continuity The main points here are a All continuous functions of one variable are continuous as functions of more variables For instance f x y z 1 x is continuous at all points x y z where x 0 because g x 1 x is continuous at points where x 0 b Sums products and constant multiples of continuous functions are continuous Quotients of continuous functions are continuous at all points where the denominator is not zero c Compositions of continuous functions are continuous For instance log x2 y 2 is continuous at all points x y where x y because x2 y 2 a polynomial is continuous and log z is continuous when z 0 III Traces and level curves Remember that traces are simply the points on the graph of a function where one variable is fixed The parametric curves a t f a t and t b f t b are x and y traces To get level curves of a function f x y you use the z traces the intersections of the the surface z f x y with the planes z c for evenly spaced values of c but you plot all those curves in the x y plane Understanding functions of two variables from traces and level curves is important For functions of three variables you have level surfaces Most of the quadratic surfaces in 13 6 are described as level surfaces You need to know the basic 6 types cones ellipsoids the two types of hyperboloids and the elliptic and hyperbolic paraboloids IV Partial Derivatives and Their Uses This is the big topic Subtopics are a Tangent planes b Differentiability Know the definition and remember the main theorem A function which has partial derivatives is differentiable at all points where all of its partial derivatives are continuous 2 c Directional Derivatives Know the definition and the proof that when f is differentiable at a b D u f a b f f a b u1 a b u2 x y d The Gradient Know its relation to directional derivatives and level curves e The Chain Rule Make sure you understand the chain rule well enough to do the assigned exercises in Rogawski 15 5 6 and Webwork Five
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