32A Stovall Midterm 2 Name November 9 Section Tu Th Duncan Melissa I certify that the work appearing on this exam is completely my own Signature There are 5 problems and a total of 8 pages Please make sure that you have all pages Please show your work You will receive little or no credit for a correct answer to a problem which is not accompanied by sufficient explanations No materials other than the exam and a writing and or erasing implement on your desk Out of consideration for your class mates try to be still and quiet Turn off all noise and or music making devices cellphones music players 1 2 3 4 5 10 10 10 10 10 50 1 Find the equation for the tangent plane to z x2 xy y 2 at 1 1 3 2 Four positive numbers each less than 10 are rounded to the first decimal place and then multiplied together Use differentials to estimate the maximum possible error in the computed product that might result from the rounding 3 a 7 pts Show that the limit as x y approaches 0 0 of x2 yey x4 4y 2 does not exist b 3 pts Use the definition of limit to show that p lim 5 x2 y 2 0 x y 0 0 4 Let r t ht cos t sin ti a Find the curvature at h0 1 0i b Find an equation for the osculating plane at this point 5 Mark each item as True or False You do not have to justify your answers 2 points for each correct answer 3 points for each incorrect answer i e there is a guessing penalty The minimum possible score on this problem is 0 a The following is a reasonable sketch of the surface given by x2 4x y 2 2y z 4 0 b Let f be a function whose domain is all of R2 Let a R2 Then lim f x L x a if for every positive number and every positive number f x L whenever 0 x a c Let f be a function whose domain is all of R2 Assume that the partial derivatives of f exist and are continuous on all of R2 Then f is differentiable at 0 d Let f be a function whose domain is all of R2 and assume that fxy and fyx exist on all of R2 Then fxy 0 0 fyx 0 0 e Let f be a function whose domain is all of R2 and assume that f is differentiable at x0 y0 R2 Then f x0 x y0 y f x0 y0 fx x0 y0 x fy x0 y0 y Extra scratch paper Extra scratch paper
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