PART 16：例題-可微分與連續性 【96淡江財金所】

Let \(f(x) = \left\{ {\begin{array}{*{20}{c}}{\;\;\;{x^2}\quad \;if\;\;x \le 1}\\{ax + b\quad if\;\;x > 1}\end{array}\;\;} \right.\)
Find the value of  \(a\) and  \(b\) such that \(f\) is continuous and differentiable at \(x = 1\). sketch the graph of  \(f\)

題目內容提到函數 \(f\) 在 \(x = 1\) 連續且可微分，要我們求出 \(a\) 與 \(b\) 的值。

SOL:

(1)連續性

左極限 = \(\lim\limits_{x \to {1^ - }} f(x) = \lim\limits_{x \to {1^ - }} {x^2} = 1\) ，
右極限 = \(\lim\limits_{x \to {1^ + }} f(x) = \lim\limits_{x \to {1^ + }} (ax + b) = a + b\)
符合連續性要求須 \(a + b = 1\)

(2)可微分

左導數 = \({f_ - }^\prime (1) = {\left. {2x} \right|_{x = 1}} = 2\)，
右導數 = \({f_ + }^\prime (1) = {\left. a \right|_{x = 1}} = a\)
符合可微分要求須 \(a = 2\)

(3)綜合以上結果　

\(a = 2\)，\(b = - 1\)

(4)繪出 \(f\) 圖形如圖