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math 237 assignment 4

# math 237 assignment 4 - t given by x,y,z = t 2,t sin t cos...

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MATH 237 Assignment 4 Due Oct. 14, 10am Hand into your assigned drop box outside MC4066. 1. Use linear approximation to approximate arctan(0 . 99(1 . 01) 2 ). Indicate why your approximation is reasonable. 2. For each of the functions given below, which have domains R 2 , discuss where each is differentiable. You must give reasoning for all of your conclusions. (a) f ( x, y ) = ( x - 1) | y - 2 | [Hint. Use the definition of partial derivatives to determine points for which f y ( x, 2) exists.] (b) f ( x, y ) = ( x 3 + y 3 ) 2 / 3 [Hint. Verify the handy inequality | x 3 + y 3 | ≤ 2( x 2 + y 2 ) 3 / 2 .] Remark. Observe, in (b), that neither of ∂f ∂x or ∂f ∂y are continuous at (0 , 0). As an exercise for yourself you may wish to prove this. Why does this not contradict Theorem 3 on p. 50 of the Lecture Notes? [You do NOT need to hand this in; this remark is for information purposes, only.] 3. A fly passes through a room with position, at time t
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Unformatted text preview: t , given by ( x,y,z ) = ( t 2 ,t sin t, cos t ). The temperature T = T ( x,y,z ) satisﬁes ∇ T ( π 2 , ,-1) = (1 , 2 ,-1). [Recall that ∇ T = ( T x ,T y ,T z ).] Compute the rate at which the ﬂy’s temperature is changing at t = π . State any assumptions required in your computation. 4. Let f ( x,y ) = xe xy , and suppose x = x ( s,t ) = ( s + 2 t ) 2 and y = y ( s,t ) = sin( st ). (a) Find f s and f t . (b) Find f ss and f tt . 5. Let f,g : R → R be twice diﬀeretiable, and α ∈ R . Show that u ( x,t ) = f ( x-αt ) + g ( x + αt ) satisﬁes the wave equation u tt = α 2 u xx 6. Let g : R → R , and f ( u,v ) = g ( uv 2 ), where u = u ( x,y ) = ( x + y ) 3 and v = v ( x ) = 1 x . Calculate ∂ 2 f ∂y∂x . State any assumptions required in your computation....
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