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2313-in-class-Sept28

# 2313-in-class-Sept28 - 2 i t − 3 2 j t − 4 3 k a...

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MAC 2313, Fall 2010 — In-class problems 9/28 These problems are solely for your ediFcation, and will not be graded. You are encouraged to work on them in groups. Problems 1–3 are from exams I have given in the past. Problems 4–6 are due to Professor Stephen GreenFeld, and are probably too involved to put on an exam. 1 Let r ( t ) = a t 2 + 5 , 4 t 3 / 2 3 , t 6 A be the position of a particle in 3-space. Compute the distance the particle travels from t = 0 to t = 1. 2 List all third order partial derivatives of the function f ( x, y ) = x 3 2 xy 2 . 3 What is the maximum value of a directional derivative of the function f ( x, y ) = 1 1 + x 2 + y 2 at the point (1 , 1 , 1 / 3 )? 4 A spaceship maneuvering in space, far from any gravitational inFuences, is executing a predetermined acceleration program which yields a position vector r ( t ) for the ship, relative to a small space beacon, given by r ( t ) = ( t
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Unformatted text preview: 2) i + ( t − 3) 2 j + ( t − 4) 3 k . a. Suppose that the captain shuts down the engines at time t . ±ind the subsequent motion of the ship. b. Show that if t is chosen appropriately then the ship will hit the beacon. 5 A certain function f ( x, y ) is known to have partial derivatives of the form ∂f ∂x = 2 xy + g ( y ) ∂f ∂y = x 2 + 3 x. Note that g is a function of y only. Use the Clairaut’s Theorem to ²nd the function g up to a constant of integration. Then ²nd all functions f which satisfy these conditions. 6 Rewrite the function f ( x, y ) = x y as a composition of standard functions (hint: think about exponentials and logarithms). Then: a. Determine the domain of f ( x, y ). b. Compute both partial derivatives of f ( x, y ) in its domain....
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