Numerical Analysis Estimate the derivative of sinx at x1 usi

Numerical Analysis: Estimate the derivative of sin(x) at x=1 using the divided difference formula (sin(1+h)

Solution

(sin(1 + h) - sin(1)) / h

With h = 0.1 :
(sin(1 + 0.1) - sin(1)) / 0.1 ---> 0.49736375253

With h = 0.01 :
(sin(1 + 0.01) - sin(1)) / 0.01 --> 0.536085981

With h = 0.001 :
(sin(1 + 0.001) - sin(1)) / 0.001 --> 0.53988148

With h = 0.0001 :
(sin(1 + 0.0001) - sin(1)) / 0.0001 --> 0.54026023

So, as the value of h keeps reducing, it appears that the value of the difference quotient is roundabout 0.5403

A good estimate would be : 0.5403

The exact answer = 0.540302305868

Explanation :
The divided difference formula is only an estimate of the derivative. It does not give the EXACT answer. But if the value of h were continually reduced and the divided difference evaluated at each step, one observes that we approach closer and closer to the right answer. Example in point is the problem that was just solved above