高数极限例题及详解 (求导)

2024-11-06 00:24:13
推荐回答(1个)
回答1:

a = [f''(0)]/2;b = f'(0);c = f(0)
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解析:

令 g(x) = ax^2 + bx + c;

则 g'(x) = 2ax + b
g''(x) = 2a

二阶可导,即二阶导数存在,因此:
f''(0) = lim(x→0) [g''(x)] = 2a
a = [f''(0)]/2

因为二阶导数存在,所以一阶导数 [存在] 且 [连续],因此:
f'(0) = lim(x→0) [g'(x)] = b

因为一阶导数存在,所以原函数 [连续],因此:
f(0) = lim(x→0) [g(x)] = c

所以 g(x) = [f''(0)]*(x^2)/2 + f'(0) + f(0)