考上清华和中500万彩票哪个更难?李永乐老师讲解正态分布的应用(2018最新)

Key Points: Normal Distribution –

  1. Mean
  2. Standard Deviation

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Quora : How likely is it that a mathematics student can’t solve IMO problems?

How likely is it that a mathematics student can’t solve IMO problems?

Is there a fear of embarrassment in being a math Ph.D. who can’t solve problems that high-school students can? by Cornelius Goh

https://www.quora.com/How-likely-is-it-that-a-mathematics-student-cant-solve-IMO-problems-Is-there-a-fear-of-embarrassment-in-being-a-math-Ph-D-who-cant-solve-problems-that-high-school-students-can/answer/Cornelius-Goh?share=311c5a88&srid=oZzP

School System Video (Do not make a fish climb trees)

Singapore Maths Tuition

Singapore is being mentioned around 4:54. Very nice video. The truth is that the classroom of today is still nearly the same as the classroom of 150 years ago. There needs to be a “Educational Revolution” parallel to that of the Industrial Revolution. Many children cannot fit into the single classroom model, leading to growth in diagnosis of behavioral “problems” such as ADHD in developed nations.

Americans who are tired of “Common Core” may want to check out Singapore Math for their kids, which is highly acclaimed in the educational realm.

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Population Differential Equations and Laplace Transform

Singapore Maths Tuition

Malthus Model
$latex displaystyle frac{dN}{dt}=BN-DN=kN$

$latex N$: Total population

$latex B$: Birth-rate per capita

$latex D$: Death-rate per capita

$latex k=B-D$

Solution to D.E.:
$latex displaystyle boxed{N(t)=widehat{N}e^{kt}},$

where $latex widehat{N}=N(0)$.

Logistic Equation
$latex begin{aligned}
D&=sN
frac{dN}{dt}&=BN-sN^2
widehat{N}&=N(0)
N_infty&=B/s
end{aligned}$

Logistic Case 1: Increasing population ($latex widehat{N}<N_infty$)
$latex begin{aligned}
N(t)&=frac{B}{s+(frac{B}{widehat{N}}-s)e^{-Bt}}
&=frac{N_infty}{1+(frac{N_infty}{widehat{N}}-1)e^{-Bt}}
end{aligned}$

The second expression can be derived from the first: divide by $latex s$ in both the numerator and denominator.

Logistic Case 2: Decreasing population ($latex widehat{N}>N_infty$)
$latex begin{aligned}
N(t)&=frac{B}{s-(s-frac{B}{widehat{N}})e^{-Bt}}
&=frac{N_infty}{1-(1-frac{N_infty}{widehat{N}})e^{-Bt}}
end{aligned}$

Logistic Case 3: Constant population ($latex widehat{N}=N_infty$)
$latex displaystyle N(t)=N_infty$

Harvesting
Basic Harvesting Model: $latex displaystyle boxed{frac{dN}{dt}=(B-sN)N-E}.$

$latex E$: Harvest rate (Amount harvested per unit time)

Maximum harvest rate without causing extinction: $latex boxed{dfrac{B^2}{4s}}$.

$latex displaystyle boxed{beta_1,beta_2=frac{Bmpsqrt{B^2-4Es}}{2s}}.$

$latex beta_1$: Unstable equilibrium population

$latex beta_2$: Stable equilibrium population

Extinction Time: $latex displaystyle boxed{T=int_{widehat{N}}^0frac{dN}{N(B-sN)-E}}.$

Laplace transform of $latex f$
$latex displaystyle F(s)=L(f)=int_0^infty e^{-st}f(t),dt$

Tip: Use this equation when the questions…

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