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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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(12) Evaluate the definite integral
xln ln(x² - 4x + 5) dx.
其他申論題
(8) Evaluate cos(y5) dy dx = (8) .
#552095
(9) Evaluate dx dy = (9) .
#552096
(10) Let E be a solid in the first octant. The largest possible value of (9 - x² - y² - z²) dV is (10) .
#552097
(11) Sketch the curve y = and its asymptotes. Find the intervals of increase/decrease and concavity. Label local extrema and inflection points if any.
#552098
(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.)
#552100
(14) Find the extreme values of f(x, y, z) = z subject to the constraints x² + y² = z² and x + 2y + z = (16)
#552101
1. State and prove the mean value theorem.
#552102
2. Use ε-δ argument to show that = 0.
#552103
(a)
#552104
(b)
#552105