The "Area-Eating Function"

Start with any continuous function . The Accumulating Function for f, or the "Area-Eating Function", is defined by: . Directions: Use the slider below to calculate A(x) at various x-values.
Questions: 1a) Set the slider to x =0.8 . Geometrically, how does A(0.8) relate to the graph of f(x)? 1b) Which is greater: A(0.5) or A(0.8)? Use the graph of f(x) to explain why. 1c) Which is greater: A(0.8) or A(1.6)? Use the graph of f(x) to explain why. 1c) Why did we call A(x) an "Area-Eating function"? 1d) Give a geometric argument for why A(0) = 0. 2.) Select the box "Show Graph of A(x)". a.) Roughly on which intervals is the graph of A(x) (Purple graph) increasing? b.) Give an explanation why A(x) is decreasing on [1.1, 2.1] c.) The First Fundamental Theorem of Calculus says that . Using the features of both graphs above as evidence, can you find three reasons to support the First Fundamental Theorem?