In this activity, students explore the fundamental theorem of calculus. They apply this theorem to understand the number "e" - the base of the natural logarithm. They also use integration and natural logarithmic function to solve a typical problem in exponential growth.

Before the Activity

See the attached PDF file for detailed instructions for this activity

Print pages 89 - 99 from the attached PDF file for your class

During the Activity

Distribute the pages to the class.

Follow the Activity procedures:

Use the calculator to graph and calculate the area under the curve

Find the definite integral of a function

Find a value of b such that integral b to1 of (1/x) dx is equal to 1

Observe that changing the value of b changes the area under the curve

Graph the function myst(x) = (1 / t )dt

Find where myst(u) intersects the lines y = 0,1,2,3 to solve integral u to 1 (1/t ) dt = 1, integral from u to2 (1 / t) dt = 2 and so on

Observe that the point of intersection is (1, 0)

Find 3 other points of intersection

Observe that the values of y increase linearly and values of x increase rapidly

Find the common ratios of the x values and observe this value defines the number e

Enter the differential equation to solve a problem in exponential growth

Integrate each side of the equation separately with a constant k of integration on the right

Find the value of constant k

Enter the value of k in the exponential equation and find the result

After the Activity

Review student results:

As a class, discuss questions that appeared to be more challenging