Illustrating the Fundamental Theorem

Daniel Lotesto and Susan Thurmond

Introduction

The Fundamental Theorem of Calculus, which states

,

is one of the most difficult concepts for a student to grasp. Students do not always see the relationship between the area bounded by y = f(x), the x-axis, x=a, and x=b, and the difference between the antiderivative values F(b) and F(a). This spreadsheet activity illustrates the theorem by using Euler's Method to approximate the value for F(b) - F(a) and Riemann Sums to approximate the definite integral. The graphs offer a visual representation of the theorem.

Preliminary Discussion of Euler's Method

Given a function F(x), and its derivative = F'(x) = f(x) , we can approximate function values for F assuming that we know initial values (x₀, y₀) where y₀ = F(x₀). Choosing a value for x, we can generate values of x and y using the following recursive equations:

x_n = x_n-1 + x

F(x_n) y_n = y_n-1 + f(x_n-1) x

An example is shown in Figure 1 using two iterations.

Where P₀ = (x0 ,F(x₀)), P₁ = (x₁, y₀ + f(x₀) x), P₂ = (x₂, y₁ + f(x₁) x).

As x 0, the y_n values more closely approximate F(x_n).

A particularly useful function to use for the derivative is F'(x) = f(x) = since, in most courses, F(x) = ln x has not yet been defined as

.

Therefore, not only are students unburdened by preconceptions of the graph of y = F(x) while building the spreadsheet, they actually construct the graph of y = ln x in the process.

Activity

Working in pairs or small groups, have the students prepare spreadsheets to generate y values using Euler's Method and to find the areas of the corresponding rectangles using the left endpoint rule. The amount of instructor input will depend upon the ability level of the class. Sample spreadsheets using Microsoft Excel are shown in this article as well as a brief description of how to generate them. They are offered as a guide. The instructor is encouraged to work with her/his students to create their own version.

Spreadsheet #1 uses Euler's Method to find the approximate y values for y = F(x). The initial values for x₀ and y₀ are entered in cells [A12] and [D12], respectively. The value for x is entered in [C5]. The following formulas are used to complete the spreadsheet:

[A13] =A12+$C$5, fill down

[B13] =1/A12, fill down

[C13] =B13*$C$5, fill down

[D13] =D12+C13, fill down

[E12] =D12 - $D$12, fill down

Spreadsheet #2 is a continuation of spreadsheet #1. It uses the left endpoint rule to compute areas of rectangles. The formulas are as follows:

[F7] =A12

[F8] =F7+$C$5, fill down

[G7] =1/F7, fill down

[H7] =G7*$C$5, fill down

[I7] =H7

[I8] =I7+H8, fill down

[K6] =E12, fill down

SPREADSHEET #1

Figure 2

SPREADSHEET #2

Figure 3

Sample Questions to Explore:

1. How do the values in the columns labeled SArea and Approx F(x) - F(a) compare? What does this suggest to you?

2. Let a = 1.6. Without changing your spreadsheet, what is the sum of the areas from a to x_n = b = 2.7? [Hint: the left-endpoint of the last interval is 2.6, not 2.7]. What is the value of F(b) - F(a)? Show on your graphs what these two numbers represent.

3. Change Dx to 0.05. [C5]. Does SAreas = Approx F(x) - F(a)?

4. Change Dx to 0.01. [C5]. Result?

5. Change Dx back to 0.1. Change the interval from [1,3] to [0.5, 2.5] by making the following changes: x₀ = 0.5 [A12] ; y₀ = -.6931 [D12]. These two values must be given. How do the values in the last two columns compare? Write out what these two values represent. [Note: a rectangle lying under the x-axis indicates a "negative area."]

6. Extend your spreadsheet to include 40 intervals so that the domain of x is [0.5, 4.5]. What is the equation for y = F(x)? Using your calculator, test your "guess".

7. Explore the function f(x) = -cos x from x₀ = a = 0.5 to x_n = b = 2.5. Change row 4 to read f(x) = -cos x ; F(a) = F(0.5) = -.4794. Change the following cells: [C5] to 0.1; [A12] to 0.5; [D12] to -.4794; [B13] to = -cos(A12), fill down; and [G7] to = -cos(F7), fill down. What is the approximate value of F(2.5) - F(0.5)? What is the value of Areas? Look at your graphs. Are they what you expect? Is the graph of F(x) the graph of the antiderivative of f(x) = -cosx?

Note to the instructor: Below is an analytical verification of what the comparison of the two spreadsheets illustrates. [Kevin Bartkovich, NCSSM]

f(x) is the derivative of some function F(x). The y values refer to points on the graph of the curve that approximates F(x) using Euler's Method. The point (x₀,y₀) is the initial point.

y₁ = y_o + f(x_o) x

y₂ = y₁ + f(x₁) x

•

•

•

y_n = y_n-1 + f(x_n-1) x

y₁ + y₂ + ... + y_n = (y_o + f(x_o) x) + (y₁ + f(x₁) x) + ... + (y_n-1 + f(x_n-1) x)

y_n = y_o + [f(x_o) + f(x₁) + ... +f(x_n-1)] x

y_n = y_o +

The net change in y using Euler's Method is

y_n - y_o =

If x_o = a and x_n = b, then the net change in y as is

F(b) - F(a) =

This limit is the definite integral and we have F(b) - F(a) =

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