Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyArea FunctionDefinition: Area FunctionCheck Conditions of FTC-IProcedureActivity 05Activity 06Activity 07HomeworkCC BY-NC-SA 4.0

**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

**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 .${\int}_{a}^{b}{f}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}\text{d}x$ **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.

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.

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*], **Net Area** *under* the curve

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

is the black graph.$f(t)$ is$f(t)$ **continuous**on .$[0,12]$ **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.

Let

**NOTE**: I used

In the embedded DESMOS page above,

Move the slider for the parameter

[Box 3] to values$a$ .$a=0,0.5,1,1.5,\dots ,11,11.5,12$ Identify the value of

[Box 2] and compare to the area [shaded blue and red and the value listed in the Table [Box 5]${\int}_{0}^{a}f(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ *under* [black curve].$f(t)$ What happens to

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

What does the

-value, i.e., for each green point refer to? Explain.$y$ Explain what happens to the green curve, i.e.,

, when$g(x)$ increases.$a$

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

The black curve,

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

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

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

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

**Section 5.3**: # 5.

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)