Fundamental Theorem of Calculus (FTC)

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

• Section 4.9 – Antiderivatives

• Section 5.3 – The Fundamental Theorem of Calculus

• Section 5.4 – Indefinite Integrals and the Net Change Theorem

Expected Educational Results

• Objective 01–01: I can state the Fundamental Theorem of Calculus.

• Objective 01–02: I can explain the meaning of the Fundamental Theorem of Calculus – Part I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of Calculus – Part II.

• Objective 01–04: I can determine the properties of an area function using FTC-I and Calculus I.

• Objective 01–05: I can evaluate indefinite integrals using the Fundamental Theorem of Calculus – Part I.

• Objective 01–06: I can evaluate definite integrals using the Fundamental Theorem of Calculus – Part II.

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_a^b{f}^{\prime }(x)\text{d}x$$\int_a^b{f^{\,\prime}(x)\,\text{d}x}$.

• Objective 01–08: Given velocity of an object in motion along a line, I can find the object’s displacement and the total distance traveled.

• Objective 01–09: Given acceleration of an object in motion along a line, I can find the object’s velocity, displacement, and the total distance traveled.

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.

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], ${\int }_a^{b}f(t)dt$$\int_a^b{f(t)\,dt}$ is the Net Area under the curve $f(t)$$f(t)$, where

${\displaystyle \text{net area}=\text{area under }f(x)\text{ above }x\text{-axis}-\text{area under }f(x)\text{ below }x\text{-axis}}$$\displaystyle \text{net area}={\color{green}\text{area under } f(x) \text{ above }x\text{-axis}}-{\color{red}\text{area under } f(x) \text{ below }x\text{-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)$$f(t)$ be continuous on $[a,b]$$[a,b]$ and for any $x$$x$ in $[a,b]$$[a,b]$, let ${\displaystyle F(x)={\int }_a^{x}f(t)dt}$$\displaystyle F(x)=\int_a^x{f(t)\,dt}$. Then $F(x)$$F(x)$ is differentiable on $[a,b]$$[a,b]$ and $F^{\prime }(x)=f(x)$$F^{\,\prime}(x)=f(x)$, for any $x$$x$ in $[a,b]$$[a,b]$.

Check Conditions of FTC-I

• $f(t)$$f(t)$ is the black graph. $f(t)$$f(t)$ is continuous on $[0,12]$$[0,12]$. NOTE: A function is continuous on an interval IF:

  • there are NO holes in the graph;

  • there are NO jump discontinuities in the graph; and

  • there are NO infinite discontinuities in the graph.

Procedure

Let $g(x)={\int }_0^{a}f(t)dt$$g(x)=\int_0^a{f(t)\,dt}$.

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

In the embedded DESMOS page above,

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

2. Identify the value of ${\int }_0^{a}f(t)dt$$\int_0^a{f(t)\,dt}$ [Box 2] and compare to the area [shaded blue and red and the value listed in the Table [Box 5] under $f(t)$$f(t)$ [black curve].

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

Activity 05

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

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

Activity 06

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

• The black curve, $f(x)$$f(x)$, is the derivative of the green curve, $g(x)$$g(x)$. Explain.

• The green curve, $g(x)$$g(x)$, is the derivative of the black curve, $f(x)$$f(x)$. Explain.

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:

• Section 5.3: # 5.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

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