111年 - 111 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#145431

1. \( \frac{d}{dx} \int_{0}^{x^2} \sqrt{t^2 + 1} dt \)= ?

2. Let \(f(x, y) = e^{x\sin(y)}\). Then \(\nabla f(x, y) = ?\)

3. Find the integral \(\int \frac{1+x}{1+x^2} dx = ?\)

4. Evaluate \(\iint_{R} (x+2y)dA\), where \(R = \{(x,y): 0 \le x \le 1, 0 \le y \le x\}\).

5. Determine whether the series \(\sum_{n=1}^{\infty} \frac{n}{n^2+1}\) converges or diverges.

6. Prove that the following the equation of \(x^5 + ax^3 + bx + c = 0\) for \(a > 0\) and \(b > 0\) have exactly one solution.

7. Define the function $\tanh(x) = \frac{e^x - e^{-x}}{e^x + e^{-x}}$. Prove that $\tanh x$ is increasing.

8. Show that \( f(x) = x^3 + ax^2 + bx + c \) is an increasing function if \( a^2 \le 3b \).

9. Compute the following integral \(\int_{-2}^{1} x\sqrt{2-x}dx\).

10. .Let \( F(x) = \int_{2}^{\sin(x)} \cos(t) dt \). Compute \( \frac{d}{dx} F(x) \).