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.

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)$$f(t)$ be continuous on $[a,b]$$[a,b]$ and for any $x$$x$ in $[a,b]$$[a,b]$, let ${\displaystyle F(x)={\int }_a^{x}f(t)dt}$$\displaystyle F(x)=\int_a^x{f(t)\,dt}$. Then $F(x)$$F(x)$ is differentiable on $[a,b]$$[a,b]$ and $F^{\prime }(x)=f(x)$$F^{\,\prime}(x)=f(x)$, for any $x$$x$ in $[a,b]$$[a,b]$.

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

Conditions to be met:

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

2. ${\displaystyle F(x)}$$\displaystyle F(x)$ is defined as the integral ${\displaystyle {\int }_a^{x}f(t)dt}$$\displaystyle \int_a^x{f(t)\,dt}$, where

   • a constant, $a$$a$, is the lower limit;

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

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

Conclusion of FTC-I:

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

${\displaystyle F^{\prime }(x)=\frac{d}{dx}{\int }_a^{x}f(t)dt=f(x)}$$\displaystyle F^{\>\prime}(x)=\frac{d}{dx}\int_a^x{f(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. ${\displaystyle g(x)={\int }_2^x\sqrt{t^2+t-1}dt}$$\displaystyle g(x) = \int_2^x{\sqrt{t^2 + t - 1}\, dt}$. Explain. Check your answer with the embedded quiz below.

2. ${\displaystyle h(x)={\int }_x^1\frac{t+1}{t^2+1}dt}$$\displaystyle h(x) = \int_x^1{\frac{t+1}{t^2+1}\, dt}$. Explain. Check your answer with the embedded quiz below.

3. ${\displaystyle k(x)={\int }_1^{x^2}\tan (t+1)dt}$$\displaystyle k(x) = \int_1^{x^2}{\tan{(t+1)}\, dt}$. Explain. Check your answer with the embedded quiz below.

4. ${\displaystyle m(x)={\int }_{1/x}^{-1}\cos (2-t)dt}$$\displaystyle m(x) = \int_{1/x}^{-1}{\cos{(2-t)}\, dt}$. Explain.

Homework

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

• Section 5.3: # 7, 9, 11, 13, 15, 17, 19.

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 03:49 EDT Last Modified: Monday, 30 May 2022 - 13:48 (EDT)