Fundamental Theorem of Calculus (FTC)

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

Expected Educational Results

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.

Bloom’s Taxonomy for different levels of understanding
Figure 1.1: Bloom's Taxonomy

Net Change Theorem Revisited

Definition: Net Change Theorem

Net Change Theorem: The definite integral of a rate of change is the net change: abF(x)dx=F(b)F(a).

Example 1:

Suppose f(t) is the speed (in kms) of a Falcon 9 rocket t seconds after liftoff, what does 120180f(t)dt represent?

Solution:

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

NOTE: The independent variable, t, is in seconds.

Thus, by the Net Change Theorem,

120180f(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 kms which is distance per time).

So, 120180f(t)dt represents the net distance (in km) traveled by the Falcon 9 rocket between 120s and 180s after liftoff.

Investigation 20

  1. If a population grows at a rate of p(t) thousands of people per year at time t, what does 010p(t)dt represent?

  2. If an object moves along a straight line with velocity, s(t) (in feet/second), what does 37s(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(x) is the cost of producing x units of a commodity, then C(x) is the marginal cost at production of x units. What does 30004000C(x)dx represent?

Homework

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

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Created: Tuesday, 25 August 2020 04:01 EDT Last Modified: Tuesday, 30 May 2023 – 20:08 (EDT)