2-Differential

Differential¶

Definition of differential¶

Solve differential¶

Implicit differential¶

• $(v/u)' = \frac{v'u-u'v}{u^2}$
• $(uv)'=u'v+v'u$
  

Composite function differentiation¶

Introduce new parameter¶

High order differential¶

Use differential to calculate $dy$¶

Lagrange mean value theorem (拉格朗日中值定理)¶

If f is a continuous fuction on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c in (a,b) such that the tangent at c (differential) is parallel to the secant line through the endpoins (a,f(a)) and (b,f(b)). $$ f'(c)=\frac{f(b)-f(a)}{b-a} $$

Rolle's theorem (罗尔中值定理)¶

if a function f is continuous on the closed interval [a, b] and differentiable on the open interval (a, b) such that f(a) = f(b), then f′(x) = 0 for some x with a ≤ x ≤ b.

Use differential to do approximate calculation¶

Theorem¶

$$ dy = f'(x_0) ∆x \to f'(x_0 + ∆x) = f(x_0)+f'(x_0)+∆x $$

Example¶

Determine the monotonic interval and find maximum and minimum¶

• Example
  

Determine the Convexity of functions¶

Definition¶

In the interval [a,b], if $f^{(2)}(x)>0$, then f(x) is a concave. If $f^{(2)} (x)<0$, then f(x) is a convex!

Example¶

• Construct the image of $y=x^4−2x^3+1$
  • $y^{''}=0→x_1=0,x_2=1 \to (−∞,0):concave, (0,1):convex, (1,∞):concave$

L'Hôpital's rule(洛必达法则)¶

Definition¶

Example¶

• Use Lhopital's Rule, we can prove. When x→∞ $$ e^x>x^2>\sqrt{x}>lnx>x $$

Taylor's-Formula¶

Definition¶

Prove¶

Example¶