We talked about averages in this article (link the article that talks about averages)

I assumed that you’ve read my article on average. From our explorations of mean or averages, we learnt that if the mean consider all datas from a sample of data. Imagine a machine that computes the mean score of an exam and you have to feed student scores one after the other. For each score you feed, the machine compute the mean on its screen. When you feed the data of student 1 who score 7/10 on our exam. The mean is computed by (7/1) (here use latex and show the formula for mean.) We take student 2 result of 3/10 and feed into the machine? what do you expect to happen? Our machine will now show(7+3/2) as mean. The mean score decreased. Now what if we add a 10/10. The mean score increased. So the mean tells emcopansess everyone performance. If you compare yourself to the mean, you know how well you’re considering eveyrone performance.

A function is mapping from a set to another set

A function can be written as a relation between variable in other word that mapping can be represented as a relation between them. For every penny you make in a bet you want that result to be twice as that. That’s a relationship between the member of one set to the member of another set. The first set being your penny you have and the the second set the element you get after applying that relationsip. We sometimes use variables.

We can graph our example function here in the cartesian plane. See the cartesian plane as a way to visualize our algebraic relationship. If you start with 2 penny what do you have? If you go with 3 what do you have. an increase in one penny caused and 4 fold increase in the price you gained. The ratio of the change in the gained price and your original money is the average rate of change of your function

For one it’s very useful to see how things will change you can now predict that investing 10 more dollar makes will result in a x increase in your money received. Thing of it like the speed at which you gain increase.

If a car was coverying goes 20km every hour so for a hour increase in time the car cover a distance of 20km.. That 20km is said to be the rate of change of distance with respect to time. Kind of like our example. in fact the speed of that car is said to be 20km/h an hour.

Average rate isnt good when the distance the object travel vary in that interval. If the car slow down, just taking the average rate of change won’t account for that change in speed for example. Since average is end minus beginning

When rate of change or speed is no longer static like our 2 examples, it doesn’t make sense to talk of it in terms of interval. One more interesting idea is the instatanoues rate of change

The speed at which the bullet hits the target will be more interesting that the speed the bullet had in any interval. As close as we can get to that speed the better.

The derivative of a function tell you the instantanoes rate of change of that function

The notation is d(fx)/dx. Remember it was a ratio of change in one variable to the change in another. F(x) representdt he relation with talked about

If d(fx)/dx is the ratio of change d(fx)= must equal the dervative times a small change in dx so d(fx)=derivative times dx. This is very important to the concept of the integral. A product of the tiny change in our x variable that in our y variable. That’s very important because that product relates to the integral of a curve.

Such product can be intrepreted in mathematic as an area. SHow a diagram showing such product

This notation shows you that the change in a function can be seen as the product of the change in it’s derivative times the change in 14:53

Draw you oold poly Function

To find the area under this curve you can think for an intervale. You can think of approximateing it with a rectangle of height f(x) and change x. As x approaches 0 the rectangle becomes very thin. and it approximates the curve better and better.

The area of that approximating triangle is dx times f(x) . Taking the derivating under and interval is taking summing those rectangle of that size over and over again inside that place.

This thing represent a tiny change in the direction of x on a graph you can show it using the thing we know as tangent

It approximates the area well

We have seen this product before remember? How we have derivative of a function times a changes in x result in the change of that function itself.

The fundamental theorem of calculus tells us that if we take the derivative of a function times change in x then we get the original funtion back.

AI clean up

1. Functions as Mappings

A function (f\colon A\to B) assigns each element of (A) to exactly one element of (B).
Example: if you double every penny you bet, then (f(p)=2p).

Link to our article on derivatives

Sometimes we have things that are not functions

You have no clear relationship of one variable to another like the equation of a circle

How to find the rate of change here?

We take our differential as a variable and observe how each element of this expression will change when we change that

Function as a mapping that follow s 2 rules

Every input in the domain must be used but not everyone in the domain

When we say f:A->B, the codomain must not be all ———-

[Diagram showing that]

The part of the codomain that is actually hit is the range or image of the Function example x^

That image is always a subset of the codomain

Now the preimage operation is a mapping that takes a subset of the codomain and map it back to the domain. If the domain exists it maps back to the original. If there isn’t corresponding element it gives us the empty

Let’s explore this function right here. What if you give it the image?

Let’s take our x^2 defined for f:R->R

Sometimes preimage gives use something and sometimes if gives us the empty set. The preimage can’t be a function based on the properties we evoked earlier

The preimage operation gives us a set

It always exists since we haven’t even forced him to be a function

g

Starting with x^2

How does scaling you changes you. between 1 and positive infinity seems to make the parabola thinner. Why because scaling the. The parabolla becomes stipper and stipper and changes more rapidly. If you like you are streched vertically.

Imagine what will happen if you take someone and elongate them without changing their weight they will become more and more spagethi like an elastic

It’s the same in the negative direction just oriented down

Now a scale factor less between -1<x<1 makes you fatter by reveerse. you grow so slowly

Mathematical statement is a mapping from a set to a set of truth value

If you don’t specify the element from the domain, you get the open statement

We map the empty set to the truth always so any statement on the empty set result in the truth

We can define quanitifiers using and or from logic

Since it’s a function every statement from the domain can either be true or false but not both, Remember the property of functions

Another property to be a function is that you must have a corresponding element or in other words all statement have a truth value

A corollary is any statement is exactly true or false but never both

You can interpret if p then q a lot of ways. p was already a mapping

Position vs Direction Vector

Every vector has a position vector, even vectors at the origin. We omit the origin for vectors because it’s easier

Visually, a vector is a line segment with a tip and a tail in the 2D plane. The vector is a straight line, so if I give you the coordinate of the tip of the vector and the coordinate of the tail, you only have to draw a straight line between those two points to draw our vector or line segment.

What’s the curse of knowledge

Bad writing doesn’t always come from bad faith. The curse of knowledge poison the writing of well intentionned and smart people. When you write thinking that your audience will understand your writing because they know everything that you know of course. You don’t know what your audiences doesn’t know.

List of recommadations to guard yourself against the curse of knowledge

• Assume that your audience know less than you think they do rather than more. It keeps your writing accessible.
• Keep track of what your text requires from your audience as to clarify what they know
• If you see an abstract noun and ask yourself:

Is this a description of a feeling? if yes, replace with one example of how that feeligns manifest in people in way that we can observe concretely. If it’s you it’s a description of something abstract see if a concrete analogy or a simpler explanation can improve the writing.