mid1bpractice

mid1bpractice - PRACTICE PROBLEMS FOR MIDTERM I Problem 1...

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PRACTICE PROBLEMS FOR MIDTERM I Problem 1. Find the limits (i) lim x,y 0 x 2 + y 6 x 6 + y 2 ; (ii) lim x,y,z 0 x 6 + y 6 + z 6 x 2 + y 2 + z 2 . (iii) lim x,y 0 sin( x 2 + y 2 ) x 2 +2 y 2 Problem 2. Consider the function f ( x,y ) = ( x - 1) 2 + ( y - 1) 2 . (i) Draw the level diagram for f . (ii) Draw the graph of f . (iii) Using the ± - δ definition, prove that f is continuous at the origin. Problem 3. Consider the function f ( x,y ) = x 2 y 4 . (i) Carefully draw the level curves of f . (ii) Single out the level curve passing through (1 , - 1) and draw the gradient of the function at (1 , - 1). (iii) Find the equation of the tangent line to the level curve through (1 , - 1) at the point (1 , - 1). (iv) Compute the directional derivative of f at (1 , - 1) in the direction u = ( 4 5 , 3 5 ) . Use this calculation to estimate f ((1 , - 1) + . 01 u ) . (v) Find the unit direction v of steepest descent for the function f at (1 , - 1). What is the rate of decrease in that direction? (vi) Find the two unit directions w for which the derivative f w (1 , - 1) = 0. Problem 4. Consider the function f ( x,y ) = p ln( e 2 x y 3 ) . (i) Write down the tangent plane to the graph of f at (2 , 1). (ii) Find the approximate value of the number p ln( e 4 . 1 (1 . 02) 3 ) . Problem 5. (i) Assume
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mid1bpractice - PRACTICE PROBLEMS FOR MIDTERM I Problem 1...

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