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題組內容

4. Let f(x) =on [1, 2].

(b) (12%) Find the volume of the solid generated by revolving R around the
x-axis.

其他申論題

(c) (8%) Determine the path of steepest descent along the surface z = f(x,y)from the point (1, 1, 3).

(a) (12%) Prove that ln(ab) = ln(a) + ln(b) for a, b ＞ 0.

(b) (8%) By (a), to prove ln(1/b) = -ln(b) and ln(a/b) = ln(a) - ln(b).

(a) (8%) Find the area of the region R bounded by the graph of f and thex-axis.

5. (20%) Verify that the serieshas interval of convergence [-1, 1].

1. Prove: If f is the function defined by thendoes not exist. (10 points)

(a) Findfor y = (x3+ 1)(3x² + 2x - 1). (10 points)

(b) Find (). (10 points)

(c) Find，hint: sec²x - tan²x = 1. (10 points)

3. The number is a root of the equation x² - 3 = 0. Please estimateby applying the Newton-Raphson method to the functionf(x) = x² - 3 starting at x₁ = 2. Hint: the Newton-Raphson formula is. (10 points)

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