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 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}^{\prime }(x)\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.

Some Additional Antiderivative Rules

Power Rule for Antiderivatives

Investigation 13

1. Compute the derivative ${\displaystyle \frac{d}{dx}{x}^n}$$\displaystyle \frac{d}{dx} x^n$

2. Then what is the antiderivative of $n{x}^{n-1}$$nx^{n-1}$, i.e., evaluate ${\displaystyle \int n{x}^{n-1}dx}$$\displaystyle \int{nx^{n-1}\,dx}$? Explain.

Definition: Power Rule for Antiderivatives

$\int x^{n}dx=\frac{x^{n+1}}{n+1},n\ne -1$$\int{x^n\,dx}=\frac{x^{n+1}}{n+1},\,n\ne-1$

Derivation of Power Rule

NOTE: You will not be assessed on the derivation of the Power Rule for Antiderivatives.

So, we know how to evaluate $\int n{x}^{n-1}dx$$\int{nx^{n-1}\,dx}$ by using a known derivative formula.

Now, let's consider the following integral: $\int x^{n}dx$$\int{x^{n}\,dx}$.

We know $\frac{d}{dx}{x}^{n+1}=(n+1)x^n$$\frac{d}{dx} x^{n+1}=(n+1)x^{n}$.

Rewrite this equation by dividing both sides of the the equation by $(n+1)$$(n+1)$:

$\frac{d}{dx}{x}^{n+1}=(n+1)x^n$$\frac{d}{dx} x^{n+1}=(n+1)x^{n}$

$\Rightarrow \left(\frac{1}{n+1}\right)\frac{d}{dx}{x}^{n+1}=\frac{(n+1)x^n}{n+1}$$\Rightarrow \left(\frac{1}{n+1}\right)\frac{d}{dx} x^{n+1}=\frac{(n+1)x^{n}}{n+1}$

$\Rightarrow \frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right)=x^n$$\Rightarrow \frac{d}{dx} \left(\frac{x^{n+1}}{n+1}\right)=x^{n}$

$\Rightarrow \frac{x^{n+1}}{n+1}+C=\int x^n\text{d}x$$\Rightarrow \frac{x^{n+1}}{n+1}+C=\int{x^{n}\,\text{d}x}$

Investigation 14

1. For the integral ${\displaystyle \int x^{n}dx}$$\displaystyle\int{x^n\,dx}$, explain why $n\ne -1$$n\ne -1$.

2. Evaluate: ${\displaystyle \int x^{-1}dx}$$\displaystyle\int{x^{-1}\,dx}$. Explain.

Investigation 15

Using your Calculus I knowledge, explain why ${\displaystyle \left(\frac{1}{n+1}\right)\frac{d}{dx}{x}^{n+1}=\frac{d}{dx}\left(\frac{x^{n+1}}{n+1}\right)}$$\displaystyle\left(\frac{1}{n+1}\right)\frac{d}{dx} x^{n+1}= \frac{d}{dx} \left(\frac{x^{n+1}}{n+1}\right)$

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Created: Tuesday, 25 August 2020 03:20 EDT Last Modified: Tuesday, 30 May 2023 – 20:13 (EDT)