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Homework3

Homework3 - a maximum 6(10 Points Section 3.4 Exercise 2...

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1 Mathematics 1c: Solutions, Homework Set 3 Due: Monday, April 19th by 10am. 1. (10 Points) Section 3.1, Exercise 16 Let w = f ( x, y ) be a function of two variables, and let x = u + v, y = u - v. Show that 2 w ∂u∂v = 2 w ∂x 2 - 2 w ∂y 2 . 2. (10 Points) Section 3.1, Exercise 22 (a) Show that the function g ( x, t ) = 2 + e - t sin x satisfies the heat equation: g t = g xx . [ Here g ( x, t ) represents the temper- ature in a metal rod at position x and time t . ] (b) Sketch the graph of g for t 0 . ( Hint: Look at sections by the planes t = 0 , t = 1 , and t = 2 . ) (c) What happens to g ( x, t ) as t → ∞ ? Interpret this limit in terms of the behavior of heat in the rod. 3. (10 Points) Section 3.2, Exercise 6 Determine the second-order Taylor for- mula for the function f ( x, y ) = e ( x - 1) 2 cos y expanded about the point x 0 = 1 , y 0 = 0. 4. (10 Points) Section 3.3, Exercise 7 Find the critical points for the function f ( x, y ) = 3 x 2 + 2 xy + 2 x + y 2 + y + 4 and determine if they are maxima, minima or saddle points. 5. (10 Points) Section 3.3, Exercise 25 Write the number 120 as a sum of

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Unformatted text preview: a maximum. 6. (10 Points) Section 3.4, Exercise 2 Find the extrema of f ( x,y ) = x-y subject to the constraint x 2-y 2 = 2 . 7. (10 Points) Section 3.4, Exercise 20 A light ray travels from point A to point B crossing a boundary between two media ( see Figure 3.4.7 of the text ) . In the ﬁrst medium its speed is v 1 and in the second v 2 . Show that the trip is made in minimum time when Snell’s law holds: sin θ 1 sin θ 2 = v 1 v 2 . 2 8. (10 Points) Section 3.4, Exercise 22 Let P be a point on a surface S in R 3 deﬁned by the equation f ( x,y,z ) = 1 , where f is of class C 1 . Suppose that P is a point where the distance from the origin to S is maximized. Show that the vector emanating from the origin and ending at P is perpendicular to S ....
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Homework3 - a maximum 6(10 Points Section 3.4 Exercise 2...

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