MATH 281 Southwestern College Relative Maxima Math Questions

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fuehr

Mathematics

Math 281

Southwestern College

MATH

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In the last problem, you are allowed to use without proof that the volume inside a sphere of radius đť‘…R is 43đťś‹đť‘…343Ď€R3.


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Due May 26, 2020 Math 281 - Final Name__________________ Show your work for full credit. 1. Find and classify all critical points and any relative maxima, minima or saddle points of f (x,y) = x 3 + 3y 2 !12x . 2. Find r(t ) if r" (t ) = 2t i + 6e3t j + 4e 4 t k and r (0) = i + 5 j + 3 k . 3. Find an equation for the tangent plane to the hyperboloid x 2 + y 2 ! z 2 = 1 at (–2, 1, 2). Due May 26, 2020 4. Consider the integral Math 281 - Final 25 ! x 2 5 ## 0 0 page 2 of 4 xy x 2 + y 2 dy dx . a. Sketch the region of integration of the integral. b. Evaluate the integral, using polar coordinates. 5. a. Given that F( x, y) = x 2 + y 2 , 2 xy is conservative, find a function f such that $f = F . & x = !3 cos t . Use the Fundamental Theorem for b. Let C be the parametric curve ' ( y = 2 + 3 sin t, 0 % t % ) Line Integrals to evaluate # F * dr C (–3, 2) (3, 2) Due May 26, 2020 Math 281 - Final page 3 of 4 6. Let r(t ) = cos t + t sin t, sin t ! t cos t , 0 % t % ) 2 describe the position of an object moving in a plane. a. Find the velocity, r "(t ) , of the object. (Apply the Product Rule correctly!) b. Find the speed, r "(t ) , of the object.. b c. Find the arc length, L, of the object’s path from t = 0 to t = )2 . (Use L = # r"(t ) dt .) a 7. Use the Divergence Theorem to calculate # # F * n dS , where S is the boundary of the solid S cylindrical region E = {( x, y, z ) : x 2 + y 2 % 25, 0 % z % 4} with outward unit normal n, and F ( x, y, z ) = ! y i + x j + z 2 k , by evaluating a triple integral. Due May 26, 2020 Math 281 - Final # page 4 of 4 $ 8. Let E be the half ball of radius 8, E = ( x, y, z ) : 0 % z % 64 ! x 2 ! y 2 . Find the z-coordinate of the centroid, z = 1 +++ 3 +++ 3 1024 ) 2 --- z dV , given that V = volume of E = 3 ) 8 = 3 . --- z dV = 1024) ,,, V ,,, E Use spherical coordinates. E
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