# The "Area-Eating Function"

- Author:
- DNghiem

- Topic:
- Area

Start with any continuous function . The .
Directions: Use the slider below to calculate A(x) at various x-values.

**Accumulating Function for f**, or the "Area-Eating Function", is defined by:Questions:
1a) Set the slider to x =0.8 . . Using the features of both graphs above as evidence, can you find three reasons to support the First Fundamental Theorem?

*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