# 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.

## Using FTC-I

#### Investigation 12

Use what you learned from Activity 11 to answer the following:

Let $g\left(x\right)={\int }_{a}^{x}f\left(t\right)\phantom{\rule{0.167em}{0ex}}dt$$g(x) = \int_a^x{f(t)\,dt}$, where the graph of $f\left(t\right)$$f(t)$ is shown below:

1. Identify the graph of ${g}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$g^{\,\prime}(x)$. Explain.

2. For what value(s) of $x$$x$, does $g\left(x\right)$$g(x)$ have relative (local) minimum(s)? Explain.

3. For what value(s) of $x$$x$, does $g\left(x\right)$$g(x)$ have relative (local) maximum(s)? Explain.

4. For what value(s) of $x$$x$, does $g\left(x\right)$$g(x)$ have absolute (global) minimum(s)? Explain.

5. For what value(s) of $x$$x$, does $g\left(x\right)$$g(x)$ have absolute (global) maximum(s)? Explain.

6. On what interval(s) of $x$$x$, is $g\left(x\right)$$g(x)$ increasing? Explain.

7. On what interval(s) of $x$$x$, is $g\left(x\right)$$g(x)$ decreasing? Explain.

8. On what interval(s) of $x$$x$, is $g\left(x\right)$$g(x)$ concave up? Explain.

9. On what interval(s) of $x$$x$, is $g\left(x\right)$$g(x)$ concave down? Explain.

10. For what value(s) of $x$$x$, does $g\left(x\right)$$g(x)$ have inflection point(s)? Explain.

### Homework

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

• Section 5.3: # 3.