Discovery project

Page 1

A DISCOVERY PROJECT AREA FUNCTIONS In this section, we will discuss four main problems; the fourth problem will ask us to make a conclusion based on the results of the first three problems. Study each problem, its solution and explanations carefully. The results will be the subject of discussion in the next section. Once again, that this section is aimed at providing a “sneak-peek� into the idea of the fundamental theorem. Here we go:

PROBLEM 1 a)

Draw the line y = 2t + 1 and use geometry to find the area under this line, above the t-axis, and between the vertical lines t = 1 and t = 3.

b)

If x > 1, let A(x) be the area under the region that lies under the line y = 2t + 1 between t = 1 and t = x. Sketch this region and use geometry to find an expression for A(x).

c)

Differentiate the area function A(x). What do you notice?

Solution a).

To start with, we graph the function f(t) = 2t + 1 on the interval [1, 3]:


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