Start with any continuous function [math]f(x)[/math]. The [b]Accumulating Function for f[/b], or the "Area-Eating Function", is defined by:[br] [math]A(x) = \int_0^x f(t)dt[/math].[br][br]Directions: Use the slider below to calculate A(x) at various x-values.

Questions:[br][br]1a) Set the slider to x =0.8 . [i]Geometrically[/i], how does A(0.8) relate to the graph of f(x)? [br]1b) Which is greater: A(0.5) or A(0.8)? Use the graph of f(x) to explain why.[br]1c) Which is greater: A(0.8) or A(1.6)? Use the graph of f(x) to explain why.[br]1c) Why did we call A(x) an "Area-Eating function"?[br]1d) Give a geometric argument for why A(0) = 0.[br][br]2.) Select the box "Show Graph of A(x)". [br] a.) Roughly on which intervals is the graph of A(x) (Purple graph) increasing?[br] b.) Give an explanation why A(x) is decreasing on [1.1, 2.1][br] c.) The [b]First Fundamental Theorem of Calculus[/b] says that [math]\frac{d}{dx}A(x) =f(x)[/math]. Using the features of both graphs above as evidence, can you find three reasons to support the First Fundamental Theorem?