102年 - 102 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#146285

(a) \( f(x) = \ln(\tan^{-1} x) \). (5%)

(b) \( g(x) = \csc^{-1}(e^x) \). (5%)

(a) Show that \( -1/2 \leq x/(1+x^2) \leq 1/2 \) for every value of \( x \). (5%)

(b)Suppose that \( f \) is a function whose derivative is \( f'(x)=x/(1+x^2) \). Use the result in part (a) to show that \( |f(b)-f(a)| \le \frac{1}{2}|b-a| \) for any \( a \) and \( b \). (5%)

3. For what value or values of \( a \) does \[ \int_{1}^{\infty} \left( \frac{ax}{x^2+1} - \frac{1}{2x} \right) dx \] converge? Evaluate the corresponding integral(s). (10 %)

4. Find the local maxima, local minima, and saddle points of the function \( f(x, y) = y \sin x \). (10%)

5. Show that the curve \( r(t) = (\ln t)i + (t \ln t)j + tk \) is tangent to the surface \( xz^2 - yz + \cos xy = 1 \) at \( (0, 0, 1) \). (10%)

6. Let the function \( f(x) = \begin{cases} a^2x^2 + bx - 3, & x < 1 \\ ax + b, & x \geq 1 \end{cases} \). Find \( a \) and \( b \) so that \( \lim_{x \to 1} f(x) = 3 \). (10%)

(a)\[ \lim_{x \to 0} \lim_{y \to 0} \frac{x^2 - 4y^2}{x^2 + 4y^2} \] (5%)

(b) \[ \lim_{(x, y) \to (0, 0)} \frac{\sqrt{x^2 + y^2 + 4} - 2}{x^2 + y^2} \](5%)

8. Find the interval of convergence for the power series \[ \sum_{n=1}^{\infty} \frac{5^n}{n} (2x - 1)^n . (10\%) \]

9.Let the function \( f(x) = \frac{x - 1}{4 - x} \). Find the Maclaurin series of \( f(x) \) and the value of the derivative \( f^{(10)}(0) \). (10%)

10. Let \( w = f(u, v) \), \( u = 3x + 3y \), \( v = 5x - 5y \). Show that \( \displaystyle \frac{\partial w}{\partial x} \cdot \frac{\partial w}{\partial y} = 9\left(\frac{\partial f}{\partial u}\right)^2 - 25\left(\frac{\partial f}{\partial v}\right)^2 \). (10 %)