世新大學九十三學年度碩博士班考試試題卷

學 系 別	考 試 科 目
圖文傳播暨數位出版學系	微積分

※考生請於答案卷內作答

1. (10 points, 5 points each)

Prove or disprove the following statements, respectively.

(a) If a function f(x) is continuous at x = a, then f(x) is differentiable at x = a.

(b) If a function f(x) is differentiable at x = a, then f(x) is continuous at x = a.

2. (25 points, 5 points each)

Find the derivatives of the following functions.

(a) (b) ln(tan(3x²)) (c)

(d) e^{cos x} sec(2x+3) (e) (6x⁷-5x³+x)⁹⁹

3. (15 points, 5points each)

Evaluate the following limits.

(a) (b) (c)

4. (20 points)

Graph the function f(x)= x⁴+4x³+1 using the first and the second derivative tests. Your answer must show all significant features, such as intervals of increase and decrease, all local extrema, the intervals of concavity, all points of inflection, and vertical asymptotes and/or horizontal asymptotes (if any).

5. (20 points, 4 points each)

Evaluate the following integrals.

(a) (b) (c)

(d) (e)

6. (10 points)

Find the volume of the solid resulting from revolving the region under the curve on the interval [0,4] about the x-axis.