Why Are Greek Letters Used In Math?
Waterloo Standard
by James Lowman
4y ago
Sometimes, maths literally is written in a foreign language – Greek. So, why are Greek letters Used in Mathematics? The next time you’re at a party (if those even exist anymore?), mention the word “math” out loud. Aside from the host immediately kicking you out for daring to suggest such a topic at any kind of social gathering, you’ll most likely send a chill down the spine of the unfortunate people within earshot. You don’t need a degree in Psychology from Harvard to know that when it comes to suppressed childhood memories, the enigmatic subject of math is a stock villain. Daunting equations ..read more
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Taylor Series: Formula, Examples, Real-World Uses
Waterloo Standard
by James Lowman
4y ago
What is the Taylor Series? The Taylor series is a way to write a function as an infinite sum of terms that are constructed from the function’s derivatives at a single point. A Taylor series is composed of individual terms called Taylor polynomials. Summation of Taylor polynomials will approximate a function with increasing accuracy as the number of polynomials is increased. If the point where derivatives are calculated is zero, the Taylor series is called a Maclaurin series. The Taylor Series, named after famed mathematician Brook Taylor, is a way to represent a function as a sum of its deriv ..read more
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How to Find the Magnitude of a Vector
Waterloo Standard
by James Lowman
4y ago
Magnitude of a Vector Formula The Formula for Magnitude of a Vector is: $$\left| \vec{v} \right| = \sqrt{v_1^2 + v_2^2}$$ Where the vector \(\vec{v}\) is defined as \(\vec{v} = \left< v_1,v_2 \right>\) What is the Magnitude of a Vector? The magnitude is another way of saying length. The magnitude of a vector is the length of that vector. Length, in this case, is described as measuring from the tip to the tail of the vector. The magnitude of a vector is always positive. This is intentional. If the magnitude is length, then it makes sense that there are no negative lengths. When we ..read more
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Intro to Speech Recognition
Waterloo Standard
by Grigoriy Kimaev
4y ago
In this post, we will look at some basic code (implemented in Python 3) for speech recognition. We will use a readily available Python speech recognition library (called “SpeechRecognition”) and apply it for our purposes. First, you have to install the library on your computer. The installation command (specified here) that you should type in your Command Prompt (on Windows) or Terminal (on Mac or Linux) is: pip install SpeechRecognition If it doesn’t work, maybe you don’t have the “pip” installer. Consider installing Anaconda on your laptop. If you want to use your laptop’s microphone for s ..read more
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Mean Value Theorem and Rolle’s Theorem
Waterloo Standard
by James Lowman
4y ago
The Mean Value Theorem states that the rate of change at some point in a domain is equal to the average rate of change of that domain. This is what is known as an existence theorem. In order to utilize the Mean Value Theorem in examples, we need first to understand another called Rolle’s Theorem. The proof of the Mean Value Theorem and the proof of Rolle’s Theorem are shown here so that we may fully understand some examples of both. Rolle’s Theorem Rolle’s Theorem states the rate of change of a function at some point in a domain is equal to zero when the endpoints of the function are equal ..read more
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The Fundamental Theorem of Calculus, Part 1 & 2
Waterloo Standard
by James Lowman
4y ago
The Fundamental Theorem of Calculus (FTC) is the connective tissue between Differential Calculus and Integral Calculus. Differential Calculus is the study of derivatives (rates of change) while Integral Calculus was the study of the area under a function. Part 1 and Part 2 of the FTC intrinsically link these previously unrelated fields into the subject we know today as Calculus. First Fundamental Theorem of Calculus The First Fundamental Theorem of Calculus links the two by defining the integral as being the antiderivative. Fundamental Theorem of Calculus: Part 1 Let \(f(x)\) be continu ..read more
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If Statements
Waterloo Standard
by Grigoriy Kimaev
4y ago
Conditional statements are used to control the flow of your program. Depending on some set of conditions, we might want our program to do one thing or another. We accomplish the “choices” between options with conditional statements: “if”, “else if”, “else”. Sometimes they are referred to as “if-then-else” statements. Pseudocode Consider the following example written in pseudocode (i.e. code that is easy for humans to read but no computer can implement – at least not as of this writing). Alarm clock Consider the program that controls a light alarm clock. You can set the days of the week and tim ..read more
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Solving Trig Equations using Identities and Factoring
Waterloo Standard
by James Lowman
4y ago
Solving trigonometric equations using identities is no different from solving basic trigonometric equations. The only difference is now we have learned all about the trigonometric identities. We can leverage the knowledge of converting trig functions using these identities to solve more complex equations. Example 01: Solving Trigonometric Equations Using Identities and Factoring Solve the following trigonometric equation: \(2\cos^2(x) + \sin(x) = 1\), for all possible values of \(x\). Solution 01: Step one to solving this trig equation problem is to identify that \(\cos^2(x)\) is part of ..read more
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Quotient Rule for Derivatives: Proof
Waterloo Standard
by Alex Vasile
4y ago
The quotient rule is another useful tool to have. It is similar to the product rule, except it focus on the quotient of two functions rather than their product. Formula for the Quotient Rule Leibniz Notation $$ \frac{d}{dx} \left( \frac{u(x)}{v(x)} \right)=\frac{v(x)\frac{du}{dx}- u(x)\frac{dv}{dx}}{[v(x)]^2}$$ Prime Notation $$ \left( \frac{u(x)}{v(x)} \right)’= \frac{v(x)u'(x)- u(x)v'(x)}{[v(x)]^2}$$ The final result above may seem confusing, but it will make a lot more sense after we go through it step-by-step. And since not everybody learns ..read more
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While loops
Waterloo Standard
by Grigoriy Kimaev
4y ago
While loops are another fundamental programming concept. Let’s revisit the summation example from the for loop post. Even though the for loop is best-suited for that example, we can achieve the same goal with a while loop. $$ \sum_{i=1}^{n=5} = 1 + 2 + 3 + 4 + 5 = 15 $$ Let’s again consider what we know about this problem. To accomplish this summation, we must start counting the digits at 1. We keep counting until we reach 5 (or, while the digit is less than or equal to 5). We repeat the operation of adding each new digit to the sum. Since we have repetition, we can use ..read more
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