人工智能教程 - 數學基礎課程1.1 - 數學分析（一）8-9 導數的應用,線性近似(逼近),二階近似,凹凸性

導數的應用 Applications of diff

線性近似(逼近)

Linear Approximations

${\color{Red} f(x) \approx f(x_0)+f'(x_0).(x-x_0)}$

General:

$ln(1+x) \approx x$

$(1+x)^r \approx 1+rx$

f’	f(0)	f’(0)
1/(1+x)	0	1
r(1+x)^(r-1)	1	r

EX2:

$ln1.1 \approx \frac{1}{10}$

$ln(1+x) \approx x$

EX3:Find Linear

approx near x=0 of

$\frac{e^{-3x}}{\sqrt{1+x}} = e^{-3x}.(1+x)^{\frac{1}{2}}$

$\approx (1-3x).(1-\frac{1}{2}x)$

$\approx 1-3x-\frac{1}{2}x+\frac{3}{2}x^2$

$\approx1-\frac{7}{2}x$

$drops \ \ x^2 \ term(s) \ +x^3 \ and \ higher$

EX4:

satellite
時間膨脹(timedilation)

$T' = \frac{T}{\sqrt{1-\frac{v^2}{t^2}}}$

$\because (1-u)^{-\frac{1}{2}} \approx 1+\frac{1}{2}u$

$\therefore T' \approx T.(1+\frac{1}{2}. \ \frac{v^2}{c^2})$

二階近似

(Quadratic approximation)

${\color{Red} f(x) \approx f(x_0) +f'(x_0).(x-x_0)+\frac{f''(x_0)}{2}(x-x_0)^2}$

$x \approx 0$,則：

$sinx \approx x$

$cosx \approx 1-\frac{1}{2}x^2$

$e^x \approx 1+x+\frac{1}{2}x^2$

f’’	f’’(0）
-sinx	0
-cosx	-1
$e^x$	1

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Quadratic approx

(use these when linear is not enough)

important rule!!!

${\color{Red} f(x) \approx f(0)+f'(0)x+\frac{f''(0)}{x}x^2 }$

逼近 rate of convergence

$(ln{a_k})-1\rightarrow 0$

曲線構圖 Curve sketching

Goal: draw graph of f using f’,f’’,positive/negative

$f'>0\Rightarrow f \ is \ increasing$
$f'<0\Rightarrow f \ is \ decreasing$
$f''>0\Rightarrow f' \ is \ increasing$

f concave up 凸

$f''<0\Rightarrow f' \ is \ decreasing$

f concave down 凹

• 駐點 critical point
• 駐點值 critical value
• 極值點 turning points
• 拐點 inflection point