112年 - 112 國立臺灣大學_碩士班招生考試題：微積分(D)#130254

(1) Evaluate =      (1)     .

(2) Let y = f(x) be a function defined implicitly by the equation 2(x² + y²)² = 25(x² - y²) near x = 3, y = 1. By linear approximation f(2.87) ≈       (2)      .

(3) Let R be the region described by {(x, y) | cos x ≤ y ≤ sec²x, 0 ≤ x ≤ }. The volume obtained by rotating R about the y-axis is      (3)     .

(4) The area of the region inside the polar curve r = 5 + 3 cos θ is      (4)     .

(5) The 4th nonzero term of the Maclaurin series of the function f(x) = √(9 + x²) is       (5)      .

(6) The coefficient of in the Maclaurin series of g(x) = is      (6)     .

(7) The tangent plane of the surface xy²z³ = 8 at the point (2, 2, 1) is given by the equation      (7)      = 0.

(8) Evaluate cos(y⁵) dy dx =       (8)     .

(9) Evaluate dx dy =      (9)     .

(10) Let E be a solid in the first octant. The largest possible value of (9 - x² - y² - z²) dV is      (10)     .

(11) Sketch the curve y = and its asymptotes. Find the intervals of increase/decrease and concavity. Label local extrema and inflection points if any.

(12) Evaluate the definite integral xln ln(x² - 4x + 5) dx.

(13) A logistic population model with relative growth rate 0.1 per year and carrying capacity 50 thousand can be expressed by the differential equation =0.1P(1 - ), with P in thousands and t in years. Given that the initial population is 9 thousand. Find the population size after 20 years. (If you memorized the formula, then you need to derive it for this problem.)

(14) Find the extreme values of f(x, y, z) = z subject to the constraints x² + y² = z² and x + 2y + z =   (16)