MATH 281 SC Level of Curves Partial Derivatives & Differentials Calculus Exercises

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fuehr

Mathematics

Math 281

Southwestern College

MATH

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Due March 20, 2020, 11:59 PM Name 1. Let f(x,y) = Vxy. a) Find f(-3,-1). Math 281 - Test 2 Show all work. b) Find the domain off. c) Carefully sketch the level curves off for k= 0, 1, 2, and 3. 2. Find all first partial derivatives. a) f(x,y,z) = xye' + 3xz2 = arctan х b) f(x,y) 3. Find fer» fry > fx , and fyy, for f(x,y) = 4x’ – xy2. و لال Due March 20, 2020, 11:59 PM Math 281 - Test 2 Show all work. page 2 of 4 -1 4. Find the differentials, dz or dw for the following functions. a) 2 = x² + y b) w = 2z'ysin x 5. Use the differential dz to approximate the change in z X + x2 to (3.01, 3.97) so that (Ax, Ay) = (0.01, - 0.03). Do not use a calculator. the 6. Use partial derivatives to find dy/dx if y = f(x) is defined implicitly by xe' - cos(x - y) = 0. Math 281 - Test 2 Show all work. page 3 of 4 Due March 20, 2020, 11:59 PM 7. The radius, r, of a right circular cone is increasing at a rate, dr dr, of 3 cm per second, and the height, h, is increasing at a rate, dh/dt, of 4 cm per second. Use a chain rule to determine the rate of change of the volume, dv/dt, when r = 6 cm and h=9 cm. Volume of a cone: V = h. 8. Let f(x,y) = 2x2 - y2 a. Find the gradient Vf(x,y). b. Evaluate the gradient Vf(x,y) at the point (1,2). c. Find the directional derivative Duf at (1, 2) for the in the direction of u = ži- j. Due March 20, 2020, 11:59 PM page 4 of 4 Math 281 - Test 2 Show all work. 9. Let F(x,y,z) = x + 2y2 - 3z. a. Find the gradient VF(x, y,z). b. Evaluate the gradient Vf(x,y) at the point (x, y, z) = (2,-1, 1). c. Find an equation for the tangent plane to the hyperboloid x² + 2y2 – 3z= 3 at the point (x, y, z) = (2,-1, 1). 10. Find all critical points of f(x,y) = x° - 12xy + 8y and find locations and values of all local maxima, minima, or saddle points.
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I was having a hard time with this subject, and this was a great help.

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