107年 - 107 國立臺北大學_碩士班一般入學考試試題_財政學系：微積分#146654

1. (10%) (a) What is the definition of the *Definite Integral* of a function $f(x)$ over $[a, b]$?

(b) What is the definition of *The Fundamental Theorem of Calculus*?

(a) $\lim_{x \to 1} \frac{x^{50} - 1}{x - 1}$.

(b) $\lim_{x \to \infty} \frac{x + \sin x + 2\sqrt{x}}{x + \sin x}$.

(a) Show that f(x) is continuous at x = 0.

(b) Show that f(x) is not differentiable at x = 0.

4. (10%) Let $f(x) = \frac{x - 1}{x + 1}$. Use the definition of the derivative of a function to find the instantaneous rate of change of f(x) at x = -1/2.

5. (10%) Find dy/dx where $y = \int_{2}^{x^2} \sin(t^3) \, dt$.

6. (15%) Sketch the graph of the function $f(x) = \frac{x^2 - 9}{x^2 + 1}$. ($1/\sqrt{3} \approx 0.58$) (identify relative extrema, inflection points, concavity, and asymptotes if any)

7. (10%) Find the values of $p$ for which the integral $\int_{2}^{\infty} \frac{dx}{x(\ln x)^p}$ converges.

(a) $\sum_{n=1}^{\infty} \frac{4^n n! n!}{(2n)!}$.

(b) $\sum_{n=0}^{\infty} ar^n$, where a is a nonzero real number.

9. (10%) Find the volume of the solid that lies under the surface $z = xy^2$ and above the triangle with vertices $(0, 0), (1, 0)$ and $(1, 1)$.