98年 - 98 國立臺灣師範大學_學士班轉學生招生考試試題_物理系二年級：微積分#122285

(1) Find .

(2) Find the minimum value of a such that g(x) is decreasing in the interval [a, ∞).

(1) Find the values a, b such that (2-x)(2+x)(4+x²) dx is maximum. What is this maximum value?

(2) Find the value of the integral (x⁵ cosx + x¹⁰sin³x) dx.

(1) Find ∫cos⁴x dx.

(2) Find cos⁴xdx.

(1) Evaluate.

(2) Evaluate x² e^-xdx.

7.Find the length of the arc from θ= 0 to θ= 2π for the cardioid r = f( θ) = 2 - 2cosθ.

8.Find the gradient  ∇f(x,y,z) for the function given by f(x,y,z) = y² + z² - 4x, and also find the maximum value of the directional derivatives of f(x,y,z) at the point (1,2,-1).

9.Let R be the annular region lying between the circles x² + y² = 1and x² + y² = 4. Evaluate the double integral dA.