108年 - 108 國立嘉義大學_碩士班(甲組)招生考試試題_應用數學系：微積分#145617

1.

2. Evaluate $\lim_{x\to 1}\frac{1}{x-1}\int_{x+1}^{x^2+x}\ln(t^2+3)\,dt$.

(a) $\sum_{n=1}^{\infty}\left(\frac{\ln 2}{2^n}+\frac{\ln 3}{3^n}\right)$.

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

4. Evaluate the double integral $\iint_D e^{y^2}\,dA$, where
                                        $D=\{(x,y)\mid 0\leq x\leq 1, x\leq y\leq 1\}$.

5. Let $x^3+y^3+z^3+6xyz=1$. Find $\frac{\partial z}{\partial x}$ and $\frac{\partial z}{\partial y}$.

6. Find the vertical asymptotes of the graph of the function $f(x)=\frac{x-3}{x^4-81}$. Justify your answer.

7. Let $f(x)=\int_0^x e^{-t^2}\,dt$. Find the derivatives f'(2) and f''(2).

8. Find an equation of the tangent line to the graph of
                                                      $xy^3+\tan(x+y)=3x$
at the point $(0,0)$.

9. Evaluate the integral $\int_0^{2\pi}e^x\cos(3x)\,dx$.

10. Evaluate the integral $\int_0^2\int_{3y}^6\cos(x^2)\,dxdy$.