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112年 - 112 國立臺灣大學_碩士班招生考試題：微積分(D)#130254
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(8) Evaluate
cos(y
5
) dy dx =
(8)
.
其他申論題
(4) The area of the region inside the polar curve r = 5 + 3 cos θ is (4) .
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(5) The 4th nonzero term of the Maclaurin series of the function f(x) = √(9 + x²) is (5) .
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(6) The coefficient of in the Maclaurin series of g(x) = is (6) .
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(7) The tangent plane of the surface xy²z³ = 8 at the point (2, 2, 1) is given by the equation (7) = 0.
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(9) Evaluate dx dy = (9) .
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(10) Let E be a solid in the first octant. The largest possible value of (9 - x² - y² - z²) dV is (10) .
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(11) Sketch the curve y = and its asymptotes. Find the intervals of increase/decrease and concavity. Label local extrema and inflection points if any.
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(12) Evaluate the definite integral xln ln(x² - 4x + 5) dx.
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(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.)
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(14) Find the extreme values of f(x, y, z) = z subject to the constraints x² + y² = z² and x + 2y + z = (16)
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cos(y5) dy dx = (8) .
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cos(y5) dy dx = (8) .