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265Lesson11Problems(3)

# 265Lesson11Problems(3) - d T d t the rate of change of the...

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MATH 265: MULTIVARIABLE CALCULUS: LESSON 11 PROBLEMS Problems are worth 20 points each. (1) Let f ( u, v ) = u + v + uv 2 and u = u ( x, y ) = e x + y , v = v ( x, y ) = xy . Use the Chain Rule to compute ∂f ∂x and ∂f ∂y . (2) Repeat problem 1, but this time first finding the formula for the composite function f ( g ( x, y )), then computing ∂f ∂x and ∂f ∂y directly. Check that these partial derivatives are equal to the ones found in problem 1. (3) Let z = ln( x 2 + y ), x = re t . and y = te r . Use the Chain Rule to compute ∂z ∂r and ∂z ∂t at the point where ( r, t ) = (1 , 2). (4) Suppose a moth is flying in a circle about a candle flame so that its position at time t is given by x = cos t , y = sin t . Suppose that the air temperature is given by T ( x, y ) = x 2 e y - xy 3 . (Do not ask how we ever managed to discover such a formula for the air temperature!) Use the Chain Rule to find a formula for
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Unformatted text preview: d T d t , the rate of change of the temperature the moth feels. (5) Compute ∂z ∂x using the Chain Rule trick described in this section if sin xyz + x + 2 y + 3 z = 0. (6) (Bonus question: worth 10 points. Total points for assignment not to exceed 100.) A rectangular box has its edges changing length as time passes. At a particular instant, the sides have lengths a = 150 feet, b = 80 feet, and c = 50 feet. At that instant, a is increasing at 100 feet/sec, b is decreasing 20 feet/sec, and c is increasing at 5 feet/sec. Determine if the volume of the box is increasing, decreasing, or not changing at all, at that instant. 1...
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