115年 - 115 國立臺北大學_日間學士班轉學生招生考試試題_資訊工程學系日間學士班2年級：微積分#147413

(a) \( (5\%) \) Show that \( E(x) = f(x) + f(-x) \) is an even function.

(b)(5%)Show that \( O(x) = f(x) - f(-x) \) is an odd function.

(c) \( (5\%) \) Prove that every \( f(x) \) can be written as a sum of an even function and an odd function.

(d)(5%)Express the function \( f(x) = 2^x + (x - 3)^2 \) as a sum of an even function and an odd function.

(a)(20%)Find the slope using different definitions of limits: $$m = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}, \quad m = \lim_{h \to 0} \frac{f(a+h)-f(a)}{h}$$

(b)(10%)Solve equation of the tangent line.

3. (20%)Use Fundamental Theorem to evaluate the integral. $$\int_0^1 (u+2)(u-3)du$$

(a) \( (20\%) \) Find parametric equations for the set of all points \( \mathbf{P} \) as shown in the figure such that \( |\mathrm{OP}| = |\mathrm{AB}| \).

(b)(10%)Using the parametric equations to graph the curve of P.