Fundamental Theorem of Calculus (FTC)Expected Educational ResultsBloom’s TaxonomyApplication of FTCDefinition: VelocityDefinition: AccelerationDefinition: DisplacementDefinition: Total Distance TraveledArea *Under* the CurveInvestigation 21HomeworkCC 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.

Let **velocity** at time

Let **acceleration** at time

Let **displacement** (i.e., Net Change of Position) over the time interval

Let **total distance traveled** over the time interval

**Example 01**:

Let **displacement** over the time interval

**Solution**:

The object is

The **displacement** is the **net** area *under* the curve. This is shown in the Figure below:

**NOTE**: The area of the blue triangle is **Net Change Theorem** suggests we can “*add*” these areas. However, since the red triangle is below the **displacement** of the object is

**Example 02**:

Let **total distance traveled** over the time interval

**Solution**:

The **total distance traveled** is the **net** area *under* the absolute value of the curve. This is shown in the Figure below:

**NOTE**: The area of the blue triangle is **Net Change Theorem** suggests we can “*add*” these areas. Applying the *absolute value* to **total distance traveled** of the object is

The object traveled a total distance of

Suppose an object moves along a straight line with velocity

(in meters/second)$v(t)=6t-3$ Find the total distance traveled on

$0\le t\le 2$ Find the displacement on

$0\le t\le 2$

Suppose an object moves along a straight line with acceleration

(in$a(t)=3t+4$ ),$\text{m}/{\text{s}}^{2}$ ,$v(0)=3$ $0\le t\le 6$ Find the velocity at time

,$t$ $v(t)$ Find the total distance traveled

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

**Section 5.4**: # 69, 71.

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**Created**: Tuesday, 25 August 2020 04:32 EDT
**Last Modified**: Monday, 23 August 2021 - 06:06 (EDT)