Solving Math Questions Essay (Essay Sample)

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Calculus: Question 1
* From probability theory comparing the cumulative distribution function Fxand the probability distribution functionfx is given by the relationship
fx=dydxFx,
hence from the above equation we can be written in this manner
abdydxFxdx=Fb-Fa, but from probability theory perspective,
Fb=PrX≤b=0bfxdx, and Fa=PrX≤a=0afxdx ,
this is obviously with the assumption that the lower limit is 0 and not-„.
Now lets say, as an example
fx=2x, then 2x dx=x2, lets substitute and see,
Fb=0b2x dx=b2 , same for Fa=a2, hence over the interval (a,b),
=>ab2x dx=b2-a2=Fb-Fa,
so the statement only holds true for intervals of a,b that are discrete and
„>b>a<-„.
But if b→„, and a→-„, then the above statement doesn't hold true ,since
Fb=-„„fxdx,hence if b→„ and a→-„, then the statement
-„„fx=F„-F-„ does not hold true, this is because F-„ is already defined in the function F„.
* We prove this statement is false by using an inverse exponential function.
Let us consider a curve with the equation fx=1x ,where x≥1
this implies the interval is 1,„ and then rotate it on the x-axis.
We will compare the length implication, surface area and volume of this equation upon rotation on the x-axis.
* For one Dimension – Obviously this structure is infinitely long. Getting its Surface area and volume,
(ii) For two Dimensions - surface area=2π1„1xdx=2πln„-ln1=„
(iii) Three Dimensions - volume=π1„x-2 dx=π-1„+11=π
This structure has an infinite length=„ from a one dimension view, infinite surface area=„from a two dimension view, but a finite volume=π from a thre...