Name: Final Exam Review. (b) Reparameterize r(t) with respect to arc length measured for the point (1, 0, 1) in the direction of increasing t.

Transcription

1 MATH 127 ALULU III Name: 1. Let r(t) = e t i + e t sin t j + e t cos t k (a) Find r (t) Final Exam Review (b) Reparameterize r(t) with respect to arc length measured for the point (1,, 1) in the direction of increasing t. 2. Find a parametric representation of the conical surface x 2 + y 2 = z 2, 1 z 1 3. Let f(x, y) be a function with values and derivatives in the table. Use linear approximation to estimate f(2.1,.9). 4. Find an equation of the tangent plane to the parametric surface x = u + v, y = 3u 2, v = u v at (2, 3, ). 5. Find a vector normal to the tangents of both the curves v(t) =< te t, t 1, t 2 > and v(s) =< s 2, s 2 5, ln(s 1) > at the point (,-1,). 6. onsider the curve which is the intersection of the surfaces x 2 +y 2 = 9 and z = 1 y 2. (a) Find the parametric equations that represent the curve. (b) Find the equation of the tangent line to the curve at point (, 3, 8). 7. For r(t) = 2 sin t, 5t, 2 cos t (a) Find the unit tangent vector. (b) Find the unit normal vector. (c) Find the curvature. 8. The helix R(t) =< cos t, sin t, 2t > has constant curvature. Find this constant. 9. Find the it if it exists. 1

2 (a) (b) (c) 2x 2 + 3xy + 4y 2 (x,y) (,) 3x 2 + 5y 2 (x,y) (,) 4x 2 y x 2 + y 2 2x 2 + y 2 (x,y) (,) x 2 + 2y 2 1. Find the equation of the plane tangent to the surface f(x, y) = x x+y at (2,7). 11. Find the equation of the the plane tangent to x 2 +y 2 +z 2 = 169 at the point (-3,12,-4). 12. uppose a particle has acceleration vector a(t) = tî 9.8ˆk with initial velocity 5ĵ +1ˆk and position at t =. What is the position of the particle at t = 2? 13. Find the velocity and position of an object at any time t, given that its acceleration is a(t) = 6t, 12t, e t and its initial velocity is v() =< 2,, 1 > and initial position s() =<, 3, 5 > 14. Find the maximum and minimum values of f(x, y) = 6x + 8y subject to the constraint x 2 + y 2 = For f(x, y) = x 2 + y 2 and A =< 4, 3 >: (a) Find f(x, y) (b) Find f(a) (c) In what direction û is Dûf(A) most negative. 16. Find the directional derivative of the function g(x, y, z) = xyz at the point (3, 2, 6) in the direction f the vector v = i 2 j 2 k 17. Identify the critical points of f(x, y) = 5x 2 + 3y 2 + 6y + 1. Find the absolute maximum and minimum values of f on D = {(x, y) x 2 + y 2 4} 18. Find the local maximum, local minimum and saddle point(s) for the function. f(x, y) = x 3 + 6xy 3y 2 + 4x 9x 19. Isaac Newton is cbing a hill whose crest is 5m high. With the top of the hill at the origin, a function giving altitude around the hill is f(x, y) = 5.1x 4.1y 2 where x, y and f(x, y) are given in meters. At what rate is Newton cbing if he is standing at (2, 1) and moving towards (14, 18)? 2. Find positive x and y such that g(x, y) = 2x + y = 4 [constraint equation] which maximizes f(x, y) = xy [objective function]. 21. Use a double integral to find the volume of the given solid. (et up but do not evaluate) 2

3 (a) The solid enclosed by the paraboloid z = 3 + x 2 + (y 2) 2 and the planes z = 1, x = 1, x = 1, y =, and y = 3 (b) The solid that lies above the cone z = x 2 + y 2 and below the sphere x 2 + y 2 + z 2 = Evaluate the following double integrals. (a) (b) R xy x dv ; R = {(x, y) : x 1, 3 y 3} 2 2x x 2 x2 + y 2 dydx (c) 9 3 x 6 y dydx 23. For I = 1 1 x ey2 dydx: (a) ketch the region; (b) Rewrite as...dxdy. Answer: 1 y ey2 dxdy (c) Evaluate the the iterated integral from (b). = 1 yey2 dy = eu du = e Find the volume of the region over the triangle (, ),(2, ),(, 3) in the xy-plane and under the surface z = x 2 y Find the surface area of the part of the hyperbolic paraboloid z = y 2 x 2 that lies between the cylinders x 2 + y 2 = 1 and x 2 + y 2 = For : x 2 + y 2 + z 2 9 and x and y and z : Express I = f(z) dv as spherical coordinates integral. 27. Evaluate D x2 + 9y2 4 da for D = {(x, y) 4x 2 + 9y 2 36}. 28. Use the transformation u = x y, v = x + y to evaluate x y da R where R is the square with vertices (,2), (1,1), (2,2), and (1,3). (The graph of R and the equations of the four edges are given below) 3

4 29. F (x, y) =< 4x + 5y, x 2y >, and path starts at (2,2) and ended at (3,-2). Evaluate I = F d r. 3. For F (x, y) =< 2y, 5x >, and is the path from A =< 1, > to B =< 5, 5 > to < 3, > then back to A enclosing the triangular region R: (a) ketch (with directional arrows) and R (b) Express I = F dr with a double integral (using Green's Theorem). (c) Evaluate the integral from (b).2qbv 16Th 31. Evaluate the integral 3xy dx + (x yz) dy + (y 2z) dz where is the line segment from (1,-1,1) to (2,3,). 32. Find the surface area of the section of the hemisphere z = 4 x 2 y 2 which lies over the cone z = x 2 + y Use Green's Theorem to evaluate the line integral sin y dx + x cos y dy where is the ellipse x 2 + xy + y 2 = Use Green's Theorem to evaluate F r where F (x, y) =< y 2 cos x, x 2 + 2y sin x > where is the triangle from (,) to (2,6) to (2,) to (,). 35. Evaluate the following surface integrals. (a) y d, is the helicon with vector equation r(u, v) =< u cos v, u sin v, v >, u 1, v π (b) y2 d, is the part of the sphere x 2 + y 2 + z 2 = 4 that lies inside the cylinder x 2 + y 2 = 1 and above the xy-plane. 36. Use Green's Theorem to evaluate the line integral sin y dx + x cos y dy where is the ellipse x 2 + xy + y 2 = Use Green's Theorem to evaluate F r where F (x, y) =< y 2 cos x, x 2 + 2y sin x > where is the triangle from (,) to (2,6) to (2,) to (,). 38. Evaluate the following surface integrals. (a) y d, is the helicon with vector equation r(u, v) =< u cos v, u sin v, v >, u 1, v π 4

5 (b) y2 d, is the part of the sphere x 2 + y 2 + z 2 = 4 that lies inside the cylinder x 2 + y 2 = 1 and above the xy-plane. 39. Use toke's Theorem to evaluate F d r (a) F (x, y, z) = (x+y 2 )î+(y +z 2 )ĵ +(z +x 2 )ˆk; is the triangle with vertices (1,,), (,1,) and (,,1). (b) F (x, y, z) = xyî + 2zĵ + 3yˆk; is the curve of intersection of the plane x + z = 5 and the cylinder x 2 + y 2 = Use the Divergence Theorem to calculate the surface integral F d, that is the flux of F across. (a) F (x, y, z) = (x 3 + y 3 )î + (y 3 + z 3 )ĵ + (z 3 + x 3 )ˆk; where is the sphere with center the origin and radius 2 (b) F (x, y, z) = x 4 î x 3 z 2 ĵ + 4xy 2 zˆk; where is the surface of the solid bounded by the paraboloid z = x 2 + y 2 and the plane z = 4 5