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

FTC-I

Recall, from CPT_01c_FTC_I.html,

Theorem: Fundamental Theorem of Calculus - Part I

Fundamental Theorem of Calculus - Part I: Let f(t) be continuous on [a,b] and for any x in [a,b], let F(x)=axf(t)dt. Then F(x) is differentiable on [a,b] and F(x)=f(x), for any x in [a,b].

Note: The variable x is the independent variable, and the variable t is called a dummy variable.

Conditions to be met:

  1. f(t) is continuous for all values on [a,b];

  2. F(x) is defined as the integral axf(t)dt, where

    • a constant, a, is the lower limit;

    • the variable, x, and not some function of x, is the upper limit;

    • the integral is in terms of a dummy variable, e.g., t, and not x.

Conclusion of FTC-I:

If the above conditions are met, then the conclusion of the FTC-I is

F(x)=ddxaxf(t)dt=f(x).

Activity 11

Use what you learned from the previous Activity page (ACT_01e_Derivative_of_Integral_Function.html). Without using any technology, differentiate the following:

  1. g(x)=2xt2+t1dt. Explain. Check your answer with the embedded quiz below.

  2. h(x)=x1t+1t2+1dt. Explain. Check your answer with the embedded quiz below.

  3. k(x)=1x2tan(t+1)dt. Explain. Check your answer with the embedded quiz below.

  4. m(x)=1/x1cos(2t)dt. Explain.

Homework

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

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Created: Tuesday, 25 August 2020 03:49 EDT Last Modified: Monday, 30 May 2022 - 13:48 (EDT)