熱門科目
最新試題
- 2. (10 分) Let E be the solid bounded by the paraboloid z = x² + y² and the surface z = √(x² + y²) and let S be the boundary surface of E, given with positive (outward) orientation. Sketch the solid E and use the Divergence Theorem to find ∬<sub>S</sub> F ⋅ dS for the vector field F(x, y, z) = (x³, y³, 3xy).
- 1. (10 分) Use Lagrange multipliers to find the maximum and minimum values of f(x, y, z) = x² + y² + z² subject to the constraints y - z = 1 and z² - 2x² = 1.
- 5. Let F(x, y) be the vector field given by F(x, y) = (x² + y², x² + y²). Calculate ∮<sub>C</sub> F ⋅ dr, where C is the unit circle centered at the origin and oriented counterclockwise. ⇒ (e)
- 4. Let f(x, y) = 4 + x³ + y³ - 3xy. Which of the following must be true?
(A) f has a local maximum at (0,0).
(B) f has a local minimum at (1,1).
(C) At the point P(1,0), f have the maximum rate of change in the direction (1,-1).
(D) The maximum rate of change at the point P(1,0) is √2.
- 3. Suppose that series Σ a<sub>n</sub>x<sup>n</sup> converges when x = -3 and diverges when x = 5. Which of the following must be true?
n=0
(A) Σ a<sub>n</sub> converges.
n=0
(B) Σ a<sub>n</sub>(-2)<sup>n</sup> converges.
n=0
(C) Σ a<sub>n</sub>(-5)<sup>n</sup> diverges.
n=0
(D) Σ a<sub>n</sub>6<sup>n</sup> diverges.
n=0
- 2. Suppose that f'(x) > 0 for all x ∈ R and that f(2) = 0. Let g(x) = ∫<sub>0</sub><sup>x</sup> f(t) dt. Which of the following must be true?
(A) The graph of g is increasing on (0,2).
(B) g is a nonnegative continuous function.
(C) The graph of g has a horizontal tangent at x = 2.
(D) The graph of g does not have any inflection point.
最新情報
最新試卷
最新行事曆
其他資訊