
(a) When calculating the derivative of a function as an approximation numercially, it is convenient to express the derivative in a symmetric way:
The derivative is defined as the limit of the right hand side above as ε approaches 0. Demonstrate that this is correct to O(ε^{2}) using the Taylor series and show what it means graphically in terms of the slope of the tangent to the curve.
(b) The integral of the function f is defined as the limit of the Riemann sum:
where Δ = (ba)/N. The integral is defined as the limit of the right hand side above as N approaches infinity. Give a graphical interpretation of this result explaining why the argument of the g on the right is what it is.
(c) Combine these two results by considering the integral of the function
and using the approximation for the derivative with ε = Δ. Demonstrate the result (the fundamental theorem of calculus)
.
Last revision 28. November, 2005.