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Unformatted text preview: Mathematics 20E - Spring 2009 - Homework 1
April 1, 2009
Please complete the following problems and turn them in following the directions that will be announced in class and on the website by Friday, April 10 at 5:30pm. Only a subset of the problems will be graded; leave some blank at your own risk. 1. Compute the total derivative of the following functions. (a) f : 2 , (b) g : 2 , (c) h : 2 2 , 2. Give the linearization of h : 2 2 , near 1 0 . f(x) = x1 ex2 cos(x1 x2 ) f(x) = x1 ex2 cos(x1 x2 ) g(x) = x1 ex2 + cos(x1 x2 ) f(x) = xex cos(x2 ) 3. Let A be an arbitrary m n matrix and let f : f(x) = Ax. What is Df(x)? 1 n Rm be given by 4. Let c : [0, 2] 2 be given by c(t) = 2 sin t cos t . (a) Sketch the curve parametrized by c. (b) Find the velocity of c. (c) What is the equation of the tangent line to the image of c at (2, 0)? 5. Use the chain rule to compute D(f g)(x) when f(x, y, z) = and xy yz euv g(u, v) = cos(u + v) . u2 + v 2 f(x, y) = ex+y ex-y 6. Let and let c be a path such that c(0) = (0, 0) and c (0) = (1, 1). Define a new path d by d(t) = f(c(t)). What is the velocity of d when t = 0? 7. Let f (x, y, z) = xy + yz + xz. (a) Find the gradient of f . (b) Find the directional derivatie of f at (1, -1, 1) in the direction 1 1 ( 2 , 0, 2 ). 8. Find the maximum and minimum values of f along the path c where f (x, y) = x2 + y 2 and c(t) = 2 2 cos t sin t . 9. Find the arc length of the curve parametrized by (t, t sin t, t cos t) for 0 t . 10. Question 11 from Section 4.2 of the textbook. 3 ...
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