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# 5.3: FTC Changed History

Difficulty Level: At Grade Created by: CK-12

ID: 9778

Time required: 45 minutes

Activity Overview

This activity builds student comprehension of functions defined by a definite integral, where the independent variable is an upper limit of integration. Students are led to the brink of a discovery of a discovery of the Fundamental Theorem of Calculus, that .

Topic: Fundamental Theorem of Calculus

• Graph a function and use Measurement > Integral to estimate the area under the curve in a given interval.
• Use Integral (in the Calculus menu) to obtain the exact value of a definite integral.

Teacher Preparation and Notes

• This investigation should follow coverage of the definition of a definite integral, and the relationship between the integral of a function and the area of a region bounded by the graph of a function and the axis.
• Before doing this activity, students should understand that if and , then:
• Before starting this activity, students should go to the HOME screen and select :Clean Up > 2:NewProb, then press ENTER. This will clear any stored variables.

Associated Materials

## Problem 1 – Constant integrand

Students explore the function . They should notice that there is a constant rate of change in the graph of This rate of change is 1.5.

1. The table looks like the one below.

1 1.5
2 3
3 4.5
4 6
5 7.5

2. ; There is zero area under the graph of from to .

3. 1.5 units

4. The graph will be a line through the origin with slope 1.5.

Students will enter their data into the lists and then view the graph of .

5. A line; yes (student answers may vary)

6. The same as before, except the slope would be 0.5 instead of 1.5.

## Problem 2 – Non-Constant Integrand

Students investigate the behavior of . Students should note that this function changes at a non-constant rate and are asked to explain why this is so (from a geometric point of view).

1 0.25
2 1
3 2.25
4 4
5 6.25

8. The height and the length of the triangle are 0 so the area is 0.

9. The area changes by a different amount each time because both the height and width are increasing.

10. The graph is not linear. It is a parabola as seen by the formula in the above screen shot.

## Problem 3 – An Integrand That Changes Sign

1
2
3
4
5
6
7
8
9
10
11
12
13
14

The fractions are approximated to the nearest hundredth.

11. After , the integral value begins to decrease.

12. The values for in which the integral decreases are ; the function is negative.

13. The values for which the integral is increasing are ; the function is positive.

14. The table seems to indicate . To find out for sure, use fMin on the integral. You have to restrict the domain to , to get the answer.

15. Yes, we have seen a similar situation. The minimum occurs on a function where the function stops decreasing and starts increasing.

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