Example

> f:=x->ln(cos(x));

> df:=D(f):ddf:=D(df);

> plot(ddf,-1..1);

So the second derivative is bounded by 3.5. Let us now apply the forward-difference formula to estimate the derivative at :

> for i from 1 to 5 do h:=10.^(-i): df1:=(f(0.4+h)-f(0.4))/h: maxerr:=h*3.5/2: acterr:=df1-df(0.4): print(h,df1,maxerr,acterr); od:

A good estimate is obtained for sufficiently small values of . But the trend does not continue much longer:

> for i from 5 to 10 do h:=10.^(-i): df1:=(f(0.4+h)-f(0.4))/h: maxerr:=h*3.5/2: acterr:=df1-df(0.4): print(h,df1,maxerr,acterr); od:

This is caused by round-off error. Generally, there will be some optimal value for .