Calculate the determinant of the square Vandermonde matrix     (A is alpha)

     1   Α₁    A₁²   A₁³       

     1   A₂    A₂²   A₂³ 

V= 1   A₃    A

Use Newton's Method to find a local extrema for f(x)=sin(x^2)+x

with start point x=(1,0)

take derivative of f(x) then apply Newton's Method

My teacher started us off with this but I can't seem to get it to work the way she did, any help would be appreciated!

prcNewton:=proc( ) 

  local ftn,strpt,epsilon,maxlps,i,xn,dftn; 

  if nargs>4 then     

Investigate the limit of (1-cos(x))/x as x->0.  

More specifically, define f(x)=(1-cos(x))/x  and demonstrate that the limit of f(x) going to 0 is equal to 0 as follows:

Write a function that takes epsilon as input, and calculate an integer N as the output so that |f(1/N)|<epsilon, using a while loop.

Specifically, compute f(1);f(1/2);f(1/3);;;;until |f(1/N)<epsilon and return N.

I'm not sure how to even start this, any help would be appreciated....

Find the derivative of f(x)=|(x^3)-8*(x^2)+5*x+4|-0.5*x;x in [-1,7]

Find critical points of f(x) and dertimine the local maxima and local minima.

Output: Two lists of points (x,y), a list of local minima and a list of local maxima.

Hint: you may use Maple package Student[Calculus1]]

     use first derivative test to avoid 'kink' point i.e. undifferentiable point

     set delta=0.0001, test derivative around critical point x+delta and x-delta...

Define a function (a implies b) is equivlent to (b or not a)

implies(a,b) truth table: (a) is in left column and (b) is in the top row

Implies  |  true  false

-------------------------

true      |  true  false

false     |  true  false

such that : implies(true,false)=flase ,implies(false,true)...

you need to show results that meet the truth table