101年 - 101 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#122188

科目：研究所、轉學考(插大)-微積分 | 年份：101年 | 選擇題數：0 | 申論題數：16

試卷資訊

所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (16)

(1) Find the value of the integral .

(2) Find the value of the integralcos⁴x dx.

(3) Find dx.

(4) Find ∫ x²e^x dx.

(5) If f(x) =, find f'(1).

(1) Let y = ax + b be the equation of the tangent line to the graph of the function y = sin x at the point (0, 0). Find the values of a, b.

(2) Let z = ax + by + c be the equation of the tangent plane to the graph of the function z = x² + y² at the point (1, 2, 5). Find the values of a, b, c.

(1) Find the values of a, b, c.

(2) Find the maximum and the minimum of the cubic function f(x) = x³- 2x + 1 for a ≤ x ≤ c.

(3) Sketch the graph of the function y = f(x) for a ≤ x ≤ c.

(4) Find the value of the integralf(x) dx.

(1) Find the arc length of the graph of y = on the interval [, 2].

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

5. (10 分)
Find the relative extrema of f(x, y) = x³ - 4xy + 2y².

6. (12 分) Let f(x, y, z) = x² + y² - 4z. Find the direction of maximum increase of f at the point (2, -1, 1). What is the maximum value of the directional derivative of f at this point?

7. (10 分) Let R be the annular region lying between the two circles x² + y² = 1 and

x²+ y² = 5. Evaluate the double integral (x² + y²) dA.