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

Prerequisite Knowledge

Algebra

Factoring Expressions

• Difference of Two Squares: $\left({a}^{2}{x}^{2}-{b}^{2}\right)=\left(ax+b\right)\left(ax-b\right)$$\displaystyle (a^2x^2-b^2)=(ax+b)(ax-b)$
• Sum of Two Squares: [NOT factorable using Real numbers, $\mathbb{R}$$\displaystyle \mathbb{R}$]
• Difference of Two Cubes: $\left({a}^{3}{x}^{3}-{b}^{3}\right)=\left(ax-b\right)\left({a}^{2}{x}^{2}+abx+{b}^{2}\right)$$\displaystyle (a^3x^3-b^3)=(ax-b)(a^2x^2+abx+b^2)$
• Sum of Two Cubes: $\left({a}^{3}{x}^{3}+{b}^{3}\right)=\left(ax+b\right)\left({a}^{2}{x}^{2}-abx+{b}^{2}\right)$$\displaystyle (a^3x^3+b^3)=(ax+b)(a^2x^2-abx+b^2)$
• Factor Using Distributive Property: ${x}^{2}-3x=x\left(x-3\right)$$\displaystyle x^2-3x=x(x-3)$
• Factor Quadratic Expressions: $a{x}^{2}+bx+c=\left(x+s\right)\left(x+t\right)$$\displaystyle ax^2+bx+c=(x+s)(x+t)$, where $\left(s\right)\left(t\right)=c$$\displaystyle (s)(t)=c$ and $s+t=b$$\displaystyle s+t=b$
• Factor Quadratic-like Expressions: $a{x}^{2n}+b{x}^{n}+c$$\displaystyle ax^{2n}+bx^n+c$. Let $u={x}^{n}$$\displaystyle u=x^n$. Then the expression becomes $a{u}^{2}+bu+c=\left(u+s\right)\left(u+t\right)$$\displaystyle au^2+bu+c=(u+s)(u+t)$, where $\left(s\right)\left(t\right)=c$$\displaystyle (s)(t)=c$ and $s+t=b$$\displaystyle s+t=b$.
Finally, use the let statement to convert from $u$$u$ to $x$$x$.
• Factor by Grouping: $a{x}^{3}+b{x}^{2}+cx+d$$\displaystyle ax^3+bx^2+cx+d$, where $c=sa$$c=sa$ and $d=sb$$d=sb$ for some $s$$s$. Then $\left(a{x}^{3}+b{x}^{2}\right)+\left(cx+d\right)={x}^{2}\left(ax+b\right)+s\left(ax+b\right)=\left(ax+b\right)\left({x}^{2}+s\right)$$\displaystyle (ax^3+bx^2)+(cx+d)=x^2(ax+b)+s(ax+b)=(ax+b)(x^2+s)$
Use Technology to Rewrite Using Algebra

Mathematica

NOTE: On all Assessments, Mathematica may be used to rewrite any expression without showing any work.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. Factor[ ] has at least one argument:

1. The expression to be factored.
3. For help on using the Factor[ ] function:

1. In Mathematica, execute the code: $\text{?Factor}$$\text{?Factor}$
2. Click on $\vee$$\or$ near the bottom-left of output
3. Click on local
4. Read how to use the Factor[ ] function – you will be able to copy-paste code.
4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

NOTE: On all Assessments, Python may be used to rewrite any expression without showing any work.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.
2. Python requires ∗ for explicit multiplication, e.g., $8x$$8x$ is $8\text{*}x$$8\text{*}x$.
3. Python requires ∗∗ for exponentiation, e.g., ${x}^{3}$$x^3$ is $x\text{**}3$$x\text{**}3$.
4. Remember to use parens ( and ) to group factors consisting of multiple terms.
5. To execute code, press the “View the result” button:

Distributive Property

${x}^{d}\left({a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\cdots +{a}_{n}{x}^{n}\right)={a}_{0}{x}^{d}+{a}_{1}{x}^{1+d}+{a}_{2}{x}^{2+d}+\cdots +{a}_{n}{x}^{n+d}$$\displaystyle x^d\left(a_0+a_1 x+a_2x^2+\cdots +a_nx^n\right)=a_0x^d+a_1 x^{1+d}+a_2x^{2+d}+\cdots +a_nx^{n+d}$

and

$\frac{{a}_{0}+{a}_{1}x+{a}_{2}{x}^{2}+\cdots +{a}_{n}{x}^{n}}{{x}^{c}}={a}_{0}{x}^{-c}+{a}_{1}{x}^{1-c}+{a}_{2}{x}^{2-c}+\cdots +{a}_{n}{x}^{n-c}$$\displaystyle \frac{a_0+a_1 x+a_2x^2+\cdots +a_nx^n}{x^c}=a_0x^{-c}+a_1 x^{1-c}+a_2x^{2-c}+\cdots +a_nx^{n-c}$

Distributive Property – Table Form

$\left({a}_{0}+{a}_{1}+\cdots +{a}_{n}\right)\left({b}_{0}+{b}_{1}+\cdots +{b}_{n}\right)={b}_{0}{a}_{0}+{b}_{0}{a}_{1}+\cdots$$\displaystyle (a_0+a_1+\cdots +a_n)(b_0+b_1+\cdots +b_n)=b_0 a_0 + b_0 a_1 + \cdots$

Distributive Property – Method of Detached Coefficients

Example 01: $\left({x}^{3}-2{x}^{2}-1\right)\left(2{x}^{2}+3x+5\right)$$\displaystyle (x^3-2x^2-1)(2x^2+3x+5)$

which corresponds to the polynomial $2{x}^{5}-1{x}^{4}-1{x}^{3}-12{x}^{2}-3x-5$$2x^5-1x^4-1x^3-12x^2-3x-5$.

Use Technology to Rewrite Using Algebra

Mathematica

NOTE: On all Assessments, Mathematica may be used to rewrite any expression without showing any work.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. Factor[ ] has at least one argument:

1. The expression to be factored.
3. For help on using the Expand[ ] function:

1. In Mathematica, execute the code: $\text{?Expand}$$\text{?Expand}$
2. Click on $\vee$$\or$ near the bottom-left of output
3. Click on local
4. Read how to use the Expand[ ] function – you will be able to copy-paste code.
4. You need to use parens, $\left($$\left(\right.$ and $\right)$$\left.\right)$, to group multiple terms1.

5. Remember, correct Mathematica code will be all black except for variables.

6. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

NOTE: On all Assessments, Python may be used to rewrite any expression without showing any work.

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.
2. Python requires $\ast$$*$ for explicit multiplication.
3. Python requires $\ast \ast$$**$ for exponentiation, e.g., ${x}^{3}$$x^3$ is $\text{x**3}$$\text{x**3}$.
4. Remember to use parens $\left($$\left(\right.$ and $\right)$$\left.\right)$​ to group factors consisting of multiple terms.
5. You can use pprint( ) for pretty print.
6. To execute code, press the “View the result” button:

Even and Odd Functions

Even Function: $f\left(-x\right)=f\left(x\right)$$\displaystyle f(-x)=f(x)$.

Even functions have reflectional symmetry over the $y$$y$-axis.

Odd Function: $f\left(-x\right)=-f\left(x\right)$$\displaystyle f(-x)=-f(x)$.

Odd functions have rotational symmetry about the origin.

Use the embedded DESMOS file below:

• If $g\left(x\right)$$g(x)$ is the same graph as $f\left(x\right)$$f(x)$, then $f\left(x\right)$$f(x)$ is an odd function;
• If $h\left(x\right)$$h(x)$ is the same graph as $f\left(x\right)$$f(x)$, then $f\left(x\right)$$f(x)$ is an even function.

Rational Exponents

$\sqrt[b]{{x}^{a}}={x}^{\frac{a}{b}}$$\displaystyle\sqrt[b]{x^a}=x^{\frac{a}{b}}$

Properties of Exponents

• ${a}^{b+c}={a}^{b}{a}^{c}$$\displaystyle a^{b+c}=a^ba^c$
• ${a}^{m-n}=\frac{{a}^{m}}{{a}^{n}}$$\displaystyle a^{m-n}=\frac{a^m}{a^n}$

Absolute Value Functions

The absolute value function, $|f\left(x\right)|$$\left|f(x)\right|$, reflects any point below the $x$$x$-axis over the $x$$x$-axis, i.e., the absolute value function makes any negative $y$$y$-value into a positive value.

Practice 01

1. Rewrite using rational exponents: $\sqrt[3]{{x}^{2}}$$\displaystyle\sqrt[3]{x^2}$.

2. Rewrite using rational exponents: $\sqrt[5]{\left(x-2{\right)}^{7}}$$\displaystyle\sqrt[5]{(x-2)^7}$.

3. Rewrite as a product of rational functions: ${e}^{x+3}$$\displaystyle e^{x+3}$.

4. Rewrite as a product of rational functions: ${e}^{2-x}$$\displaystyle e^{2-x}$.

5. Factor completely over $\mathbb{R}$$\mathbb{R}$2, if possible3: ${x}^{2}-x-6$$\displaystyle x^2-x-6$.

6. Factor completely over $\mathbb{R}$$\mathbb{R}$, if possible: ${x}^{3}+5{x}^{2}+6x$$\displaystyle x^3+5x^2+6x$.

7. Factor completely over $\mathbb{R}$$\mathbb{R}$, if possible: ${x}^{4}-1$$\displaystyle x^4-1$.

8. Factor completely over $\mathbb{R}$$\mathbb{R}$, if possible: ${x}^{3}-3{x}^{2}+2x-6$$\displaystyle x^3-3x^2+2x-6$.

9. Factor completely over $\mathbb{R}$$\mathbb{R}$, if possible: ${x}^{2}+9$$\displaystyle x^2+9$.

10. Expand: $\left({x}^{2}+x-1\right)\left(x-2\right)$$\displaystyle (x^2+x-1)(x-2)$.

11. Expand: $\left({x}^{2}-2x-1\right)\left({x}^{2}+x+4\right)$$\displaystyle (x^2-2x-1)(x^2+x+4)$.

12. Expand: $\left({x}^{2}+3x-2{\right)}^{2}$$\displaystyle (x^2+3x-2)^2$.

13. Expand: $\left(x-2{\right)}^{3}$$\displaystyle (x-2)^3$.

14. Distribute: $\frac{{x}^{2}-3x+2}{x}$$\displaystyle \frac{x^2-3x+2}{x}$.

15. Determine if the function is even, odd, or neither even nor odd: $f\left(x\right)={x}^{2}-3x+2$$\displaystyle f(x)=x^2-3x+2$. Explain.

16. Determine if the function is even, odd, or neither even nor odd: $g\left(x\right)={\mathrm{tan}}^{-1}\left(x\right)$$\displaystyle g(x)=\tan^{-1}(x)$. Explain.

17. Determine if the function is even, odd, or neither even nor odd: $h\left(x\right)={x}^{3}-2x$$\displaystyle h(x)=x^3-2x$. Explain.

18. Determine if the function is even, odd, or neither even nor odd: $j\left(x\right)={x}^{2}+5$$\displaystyle j(x)=x^2+5$. Explain.

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2 “over $\mathbb{R}$$\mathbb{R}$” means using Real numbers only.
3 Expressions not factorable over $\mathbb{R}$$\mathbb{R}$ are called irreducible.