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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.

其他申論題

(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) .

(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)

1. State and prove the mean value theorem.

2. Use ε-δ argument to show that = 0.

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