# 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 CalculusPart I.

• Objective 01–03: I can explain the meaning of the Fundamental Theorem of CalculusPart 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 CalculusPart I.

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

• Objective 01–07: I can use the Net Change Theorem to identify the meaning of ${\int }_{a}^{b}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}\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.

## Net Change Theorem Revisited

### Definition: Net Change Theorem

Net Change Theorem: The definite integral of a rate of change is the net change: ${\int }_{a}^{b}{F}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=F\left(b\right)-F\left(a\right)$$\int_a^b{F^{\,\prime}(x)\,dx}=F(b)-F(a)$.

Example 1:

Suppose ${f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)$$f^{\,\prime}(t)$ is the speed (in $\frac{km}{s}$$\frac{km}{s}$) of a Falcon 9 rocket $t$$t$ seconds after liftoff, what does ${\int }_{120}^{180}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_{120}^{180}{f^{\,\prime}(t)\,dt}$ represent?

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, $t$$t$, is in seconds.

Thus, by the Net Change Theorem,

${\int }_{120}^{180}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_{120}^{180}{f^{\,\prime}(t)\,dt}$ represents a net change – in this case the net change is in terms of distance (because the rate of change is measured in $\frac{km}{s}$$\frac{km}{s}$ which is distance per time).

So, ${\int }_{120}^{180}{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_{120}^{180}{f^{\,\prime}(t)\,dt}$ represents the net distance (in $km$$km$) traveled by the Falcon 9 rocket between $120s$$120 s$ and $180s$$180 s$ after liftoff.

#### Investigation 20

1. If a population grows at a rate of ${p}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)$$p^{\,\prime}(t)$ thousands of people per year at time $t$$t$, what does ${\int }_{0}^{10}{p}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_0^{10}{p^{\,\prime}(t)\,dt}$ represent?

2. If an object moves along a straight line with velocity, ${s}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)$$s^{\,\prime}(t)$ (in feet/second), what does ${\int }_{3}^{7}{s}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$\displaystyle \int_3^7{s^{\,\prime}(t)\,dt}$ represent?

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

## Homework

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

• Section 5.4: # 59, 61, 63, 65.