# 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

### Trigonometry

#### Evaluating Trigonometric Ratios

There is NO reason to memorize the unit circle. Look at the Table below and consider the patterns of the numbers. Use these patterns to recall your trig ratios1.

NOTE: On all Assessments, exact trigonometric values are expected, except where instructed otherwise. There are two options available for evaluating trigonometric ratios, e.g., $\mathrm{sin}\left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$:

1. Leave the answer as $\mathrm{sin}\left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$; or
2. Rewrite $\mathrm{sin}\left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$ as $-\frac{\sqrt{3}}{2}$$-\frac{\sqrt{3}}{2}$

#### Practice 02

Evaluate the following:

1. $\mathrm{sin}\left(\frac{4\pi }{3}\right)$$\sin{\left(\frac{4\pi}{3}\right)}$
2. $\mathrm{sec}\left(\frac{5\pi }{6}\right)$$\sec{\left(\frac{5\pi}{6}\right)}$
3. $\mathrm{tan}\left(\frac{3\pi }{2}\right)$$\tan{\left(\frac{3\pi}{2}\right)}$
4. $\mathrm{csc}\left(\frac{\pi }{6}\right)$$\csc{\left(\frac{\pi}{6}\right)}$
5. $\mathrm{cos}\left(\frac{11\pi }{6}\right)$$\cos{\left(\frac{11\pi}{6}\right)}$
6. $\mathrm{cot}\left(\frac{3\pi }{4}\right)$$\cot{\left(\frac{3\pi}{4}\right)}$
7. $\mathrm{cos}\left(\frac{5\pi }{3}\right)$$\cos{\left(\frac{5\pi}{3}\right)}$
8. $\mathrm{sin}\left(\frac{2\pi }{3}\right)$$\sin{\left(\frac{2\pi}{3}\right)}$
9. $\mathrm{cos}\left(\pi \right)$$\cos{\left(\pi\right)}$
10. $\mathrm{tan}\left(\frac{7\pi }{6}\right)$$\tan{\left(\frac{7\pi}{6}\right)}$
##### Use Technology to Rewrite Using Algebra

Mathematica

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

1(* Evaluate: sin(4 pi/3) *)2Sin[4Pi/3]

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. Remember, correct Mathematica code will be all black except for variables.
3. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Python

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., $4\pi$$4\pi$ is $4\text{*pi}$$4\text{*pi}$.
3. To execute code, press the “View the result” button:

#### Rewriting Trigonometric Expressions

Half-Angle Formulas

${\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)=\frac{1}{2}\left(1+\mathrm{cos}\left(2x\right)\right)$$\displaystyle \cos^2{\!(x)}=\frac{1}{2}\left(1+\cos{(2x)}\right)$

${\mathrm{sin}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)=\frac{1}{2}\left(1-\mathrm{cos}\left(2x\right)\right)$$\displaystyle \sin^2{\!(x)}=\frac{1}{2}\left(1-\cos{(2x)}\right)$

Double Angle Formulas

$\mathrm{sin}\left(2x\right)=2\mathrm{sin}\left(x\right)\mathrm{cos}\left(x\right)$$\displaystyle \sin{(2x)}=2\sin{(x)}\cos{(x)}$

$\mathrm{cos}\left(2x\right)={\mathrm{cos}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)-{\mathrm{sin}}^{2}\phantom{\rule{-0.167em}{0ex}}\left(x\right)$$\displaystyle \cos{(2x)}=\cos^2{\!(x)}-\sin^2{\!(x)}$

##### Use Technology to Rewrite Using Trigonometric Identities

Mathematica

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

​x1(* Rewrite: sin(x)^2 *)2TrigReduce[Sin[x]^2]3
4(* Rewrite: sin(2x) *)5TrigExpand[Sin[2x]]

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. Remember, correct Mathematica code will be all black except for variables.
3. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.