Fundamental Theorem of Calculus (FTC)

Author: John J Weber III, PhD Corresponding Textbook Sections:

Expected Educational Results

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Area Function

Definition: Area Function

According to the definition of a Definite Integral [Section 5.2 of the textbook and discussed in Math 2211 – Calculus of a Single Variable - I], abf(t)dt is the Net Area under the curve f(t), where

net area=area under f(x) above x-axisarea under f(x) below x-axis

The following DESMOS activity will help you graphically understand the Fundamental Theorem of Calculus - Part I:

Recall,

Fundamental Theorem of Calculus - Part I [FTC-I]: Let f(t) be continuous on [a,b] and for any x in [a,b], let F(x)=axf(t)dt. Then F(x) is differentiable on [a,b] and F(x)=f(x), for any x in [a,b].

Check Conditions of FTC-I

Procedure

Let g(x)=0af(t)dt.

NOTE: I used g(x) rather than F(x), the letter used to represent functions is arbitrary.

In the embedded DESMOS page above,

  1. Move the slider for the parameter a [Box 3] to values a=0,0.5,1,1.5,,11,11.5,12.

  2. Identify the value of 0af(t)dt [Box 2] and compare to the area [shaded blue and red and the value listed in the Table [Box 5] under f(t) [black curve].

  3. What happens to g(x) from a=7 to a=7.5? Explain.

Activity 05

  1. What does the y-value, i.e., for each green point refer to? Explain.

  2. Explain what happens to the green curve, i.e., g(x), when a increases.

Activity 06

Using Calculus I, determine which of the following is true:

Activity 07

Relate your answer to Activity 06 to the FTC-I.

Homework

At this time, you should be able to complete the following assignments:

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Created: Tuesday, 25 August 2020 02:28 EDT Last Modified: Monday, 15 August 2022 - 15:36 (EDT)