100年 - 100 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#122190

科目：研究所、轉學考(插大)-微積分 | 年份：100年 | 選擇題數：0 | 申論題數：12

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所屬科目：研究所、轉學考(插大)-微積分

選擇題 (0)

申論題 (12)

(1) Let F(x) =dt. Find F'(1).

(2) Evaluate , where a>0.

(1) Let , x> 0. Find the minimum value of a for which f(x) is strictly decreasing on [a, ∞).

(2)Which is greater
, where n>8. Why?

(1) Find the value of the integral dx

(2) Evaluate the improper integral

(1)Find the volume of the solid formed by resolving the region bounded by the graphs of y = x³ + x + 1, y = 1, and x = 1 about the line x = 2.

(2)Find the volume of the torus which was formed by region the circular region bounded by (x-2)² + y² = 1about the y-axis.

5. The tangent line to the curve y = x³ at the point A(1,1) intersects the curve at another point B. Let R be the area of the region bounded by the curve and the tangent line. The tangent line at B intersects the curve at another point C. Let S be the area of the region bounded by the curve and the second tangent line. Find the values of R and S.

6.Find the relative extrema of f(x,y) = x³ - 4xy + 2y²- 1.

7.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.

8.Find the solid region Q where the triple integral (1 - x² - y² - z²) dV is a maximum. What is the exact maximum value?