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

Some Additional Antiderivative Rules

Power Rule for Antiderivatives

Investigation 13

  1. Compute the derivative ddxxn

  2. Then what is the antiderivative of nxn1, i.e., evaluate nxn1dx? Explain.

Definition: Power Rule for Antiderivatives

xndx=xn+1n+1,n1

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 nxn1dx by using a known derivative formula.

Now, let's consider the following integral: xndx.

We know ddxxn+1=(n+1)xn.

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

ddxxn+1=(n+1)xn

(1n+1)ddxxn+1=(n+1)xnn+1

ddx(xn+1n+1)=xn

xn+1n+1+C=xndx

Investigation 14

  1. For the integral xndx, explain why n1.

  2. Evaluate: x1dx. Explain.

Investigation 15

Using your Calculus I knowledge, explain why (1n+1)ddxxn+1=ddx(xn+1n+1)

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

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