1. NR
    • Aim:Have a function f(x), and want to find its roots, as in where f(x)=0. Given f(x) we can determine df/dx. Once we have this we can apply the NR algorithm to find the root:
      1. Pick an x.
      2. Is this a root? If not then using the value of the slope we estimate where f(x)=0:
        xnew = x - f(x)/m(x)
        If the slope was one and f(5)=5, then we could estimate that f(4) would be 4, f(3) would be 3 and f(0)=0.

        This would be the case if the function was linear which is not normally the case since if it was we can find the answer very easily anyway.

        We now use xnew as our x and see whether f(xnew) = 0. If it isn't we repeat the process and make a new estimate.

        We keep on doing this till we find the solution.

        NOTE:Add two points here:

        • In general we would stop if f(x) was very close to zero since that is good enough.
        • It does not always converge, so if you find that it doesn't seem to be working then mayb start again with a different initial guess. If, for example, we reach a point with a slope of zero then we have a problem.

  2. Max/Min
    • Aim:Have f(x), want to find it's max or min; its highest or lowest point.

    • Again from f(x) we can determing df/dx. If a point is a max or a min then df/dx at the point equals zero. We find where df/dx=0 and then check these points to see whether they a

    • The algorithm:
      1. From f(x) we can determing df/dx.
      2. If a point is a max or a min then df/dx at the point equals zero.
      3. We find where df/dx=0 and then check these points. If a point is a max then the second derivative, at that point,will be negative. If it's a min then the second derivative, at that point, will be positive.

    • What is the max value the function -4x2+2x+1 takes?
      1. From f can get its derivative:df/dx = -8x+2 = 0.
      2. Find where this equals zero, at x=1/4 here.
      3. What is the sign of the 2nd derivative at this point:
        d2f
        dx2
        		 = -8 < 0,
        so is where the function takes its max value.
      4. The max value is thus
        -4(1/4)2+2(1/4)+1 = -1/4+2/4+1 = 1 1
        4
        		 .

  3. For the mathematica problem, I used TrigReduce. Standard syntax, so
    TrigReduce[Cos[x]Sin[x]-Sin[x]]
    which produces the output
    1
    2
    		 (-2 Sin[x]+Sin[2x]).

File translated from TEX by TTH, version 2.70.
On 4 Dec 2002, 11:13.