Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyNet Change Theorem RevisitedDefinition: Net Change TheoremInvestigation 20HomeworkCC 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.

**Net Change Theorem**: The definite integral of a rate of change is the **net change**:

**Example 1**:

Suppose *Falcon 9* rocket

**Solution**:

Let's check the conditions of the **Net Change Theorem**:

The integral is a definite integral, i.e., there is a lower- and upper-limit of integration;

The integrand, i.e., the function to be integrated, is a

*rate of change*.

**NOTE**: The independent variable,

Thus, by the **Net Change Theorem**,

*net change* – in this case the net change is in terms of distance (because the rate of change is measured in

So,

If a population grows at a rate of

thousands of people per year at time${p}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t)$ , what does$t$ represent?${\int}_{0}^{10}{p}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ If an object moves along a straight line with velocity,

(in feet/second), what does${s}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t)$ represent?${\int}_{3}^{7}{s}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(t){\textstyle \phantom{\rule{0.167em}{0ex}}}dt$ In economics, the marginal cost of production is defined to be the cost of producing one more unit, that is, the rate of increase of cost, where cost is a function of the number of units produced. If

is the cost of producing$C(x)$ units of a commodity, then$x$ is the marginal cost at production of${C}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x)$ units. What does$x$ represent?${\int}_{3000}^{4000}{C}^{{\textstyle \phantom{\rule{0.167em}{0ex}}}\prime}(x){\textstyle \phantom{\rule{0.167em}{0ex}}}dx$

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

**Section 5.4**: # 59, 61, 63, 65.

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 04:01 EDT
**Last Modified**: Tuesday, 30 May 2023 – 20:08 (EDT)