## 17Calculus - Calculus Notation

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Using correct notation is extremely important in calculus. If you truly understand calculus, you will use correct notation. Take a few extra minutes to notice and understand notation whenever you run across a new concept. Start using correct notation from the very first.

You may not think this is important. However, if your teacher doesn't require correct notation now, you may (and probably will) get a calculus teacher in the future that WILL require correct notation. And it is easier to learn it correctly from the first than have to correct your notation later, after you have been doing it incorrectly for a while.

This brings up another point. Many teachers don't encourage you to learn math that you can use. It is easier as a teacher to get you to parrot back what they say than it is to teach you so that you learn. So, no matter what kind of teacher you get (good or bad), it is up to you to learn math. Take that responsibility yourself. If you do, you will be able to learn on your own. When you are in college, this is the time to start learning without needing a teacher to teach you. Doing so will free you to enjoy learning.

Simplifying

Every teacher has their own idea on what they think simplifying is and usually their idea is based on what they are teaching at the time. Sometimes, simplifying means multiplying out. Sometimes it means factoring. If your teacher asks you to simplify your answer, it is good to ask them to explain what they mean by simplifying. You will find in calculus, most teachers want you to simplify your answer by factoring and canceling common terms in fractions. That is the standard this site follows.

Use of The Greek Alphabet in Mathematics

The use of greek letters is widespread in calculus. You probably saw it a lot in trig to represent angles. Greek letters are also used in limits and all throughout calculus. When you see greek letters or any other kind of unusual use of notation, it is best not to change the variables to something you are familiar with. You are probably used to using $$x$$ as a variable from algebra. However, you need to get used to using greek letters. A good teacher will encourage this by taking off points if you change notation to something you are familiar with. You need to learn the new notation to succeed in calculus. And it's not that hard.

As you continue on in calculus and higher math, you will find that most mathematicians use the same or similar variables in similar contexts. This means variables are not just chosen randomly. They usually carry some meaning along with them. Here is a rundown on what some of them usually mean. However, this list is not cast in stone. You may occasionally find mathematicians or contexts that depart from this list.

Letter(s)

Usual Meaning (depending on the context)

θ, α, γ

Angles

δ, ε

Very small (usually positive) numbers

Δ

Indicates change in a variable; often written as Δx; this is not Δ times x but is one variable and written this way to indicate a change in another variable x.

λ, μ

Parameters in parametric equations

 Hover your cursor over a letter in the first column to reveal it's name and case.

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### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia] Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

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