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國立中山大學 101 學年度第二學期 海工系/海洋學程 102.4.26 微積分(二) 期中考 Attempt ALL the problems. Part I has 13 blanks each of which carries 5 points. Show the details of your work for Part II, which carries 40 points. The total is 105. I. Fill in the blank. 1. R∞ 2 e−x dx = . 2. Determine the convergence/divergence of the following series. Give simple reasons. (a) ∞ X 1 ; n3 n! (b) n n=1 n n=1 ∞ X . 3. For the function f (x) = 4x2 + x − 1, the Taylor polynomial of order 3 at x0 = 0 is given by P2 (x) = . 4. The Maclaurin series for the function ex is . 5. The first three terms of the Taylor series for f (x) = (1 + x)5/2 at x = 1 2 is . 6. Let f (x, y) = xey + sin(xy). Then fx = ; fxy = 7. For the above problem, the gradient vector ∇f (π, 1) = . . And when u = ( √12 , √12 ), the directional derivative Du f (π, 1) = . One direction v ∈ R2 such that Dv f (π, 1) is maximum is given by v= . 8. Let w = ln(x2 + y 2 ) with x = u sin v and y = uev . When (u, v) = (3, 0), ∂w the partial derivative = . ∂u 9. For the above function w = g(u, v), the tangent plane at (u, v) = (3, 0) is . 1 II. Show details of your work. 10. (10%) Find the interval of convergence for the power series ∞ X xn . n=0 (n + 1)! 11. (15%) (a) Find the Maclaurin series for the function 1 . 1+x (b) Integrate the above series to give the Maclaurin series for ln(1 + x). (c) Hence or otherwise, express ln 2 in terms of a series. 12. (10%) Let f (x, y) = 2x2 + 2xy + y 2 − 2x. (a) Find all the critical points of the function. (b) Determine whether each critical point is a relative maximum, relative minimum or saddle point. 13. (5%) Consider the function f (x, y) = xy x2 +y 2 0 (x, y) 6= (0, 0) . (x, y) = (0, 0) Is f continuous at every point (x0 , y0 ) ∈ R2 ? Give argument for your answer. End of Paper 2 國立中山大學 101 學年度第二學期 海工系/海洋學程 102.6.21 微積分(二) 期末考 Attempt ALL the problems. The total is 102. I. True or False (T/F) (3%) 1(a) If a function f : Rn → R is differentiable at x0 ∈ Rn , then it is continuous at x0 . If f is continuous at x0 , then it has limit at x0 . (b) Every function can have a Taylor series at some point x0 in its domain. (c) The gradient vector ∇f (x) is a vector which is tangent to the level curve f (x) = c for some constant c. II. Fill in the blank. (52%) 2. Determine the convergence/divergence of the following series. Give simple reasons. ∞ X n +1 n=1 ∞ X n! (b) n n=1 10 (a) ; n2 . 3. The Maclaurin series for the function e3x is . 4. The Taylor polynomial of order 3 for f (x) = (1 + x)5/2 at x0 = 0, P3 (x) = . 5. Let f (x, y) = x2 y + xy . Then ∇f (x, y) = , and fxxx = . 6. For the above problem, when u = ( 35 , 45 ), the directional derivative Du f (2, 1) = . 7. Let w = ln(x2 + y 2 ) with x = 2uv and y = u2 − v 2 . For the above function w = g(u, v), the tangent plane at (u, v) = (3, 0) is 8. Let R = [0, 2]×[0, 1], then the double integral ZZ R . 1 xy 2 dA = . 9. Let D is the solid sphere x2 + y 2 + z 2 ≤ 1, then ZZZ √ D x2 1 dV = + y2 + z2 . 10. For the transformation x = u2 − v 2 , y = 2uv for a double integral, then the area elements have the ratio dxdy = g(u, v)dudv, where . g(u, v) = 11. Let C be the curve {(t, t2 ) : t ∈ [1, 3]}.. The line integral Z xy −2 dx + C yx−2 dy = . 12. Let S be the surface of the unit sphere x2 +y 2 +z 2 = 1 such that x, y, z ≥ 0. ZZ The surface integral z 2 dS = . C III. Show details of your work. (47%) 13. (a) Find the Maclaurin series for the function 1 . 1−x (b) Integrate the above series to give the Maclaurin series for ln(1 − x). (c) Hence or otherwise, express ln 2 in terms of a series. 14. Find the maximum and minimum values of the function f (x, y, z) = x − y + z on the sphere x2 + y 2 + z 2 = 100. 15. Evaluate the following double integrals. (a) ZZ xy dA where R is the triangle with endpoints (0, 0), (2, 0) and (1, 3). R (b) Find the volume of the region that is bounded above by the elliptic x2 x2 + y, on the sides by the cylinder + y 2 = 1 and paraboloid z = 9 9 below by the xy-plane. 16. (a) Evaluate R C x dx+y dy , x2 +y 2 where C is the unit circle traversed counterclockwise. (b) Show that if C is any closed curve, then Z (x − 3y) dx + (2x − y 2 ) dy = 5A , C where A is the area of the region D enclosed by C. Hint: Use Green’s Theorem. 2 17. (7%) (a) Prove the following Taylor expansion theorem at x0 = 0. 1 00 1 000 1 Z x 0000 2 3 f (x) = f (0) + f (0)x + f (0)x + f (0)x + f (t)(x − t)3 dt . 2 3! 3! 0 0 1 Z x 0000 Hint: Integrate by parts for the term R = f (t)(x − t)3 dt. 3! 0 (b) Is this R exactly the same as the remainder term in Taylor expansion? Why? End of Paper 3