What is calculus?
This isn't one of those things we inherited from the ancient Greeks like geometry.
This subject was created more recently in the late 1600s by Isaac Newton and Gottfried
Leibniz.
They didn't work together.
They each created calculus on their own, and as a result there was a huge argument over
who should receive credit for its discovery.
But we'll save that story for another day.
Today, let's talk about what they discovered.
In Calculus, you start with two big questions about functions.
First, how steep is a function at a point?
Second, what is the area underneath the graph over some region?
The first question is answered using a tool called the DERIVATIVE.
And to answer the second question, we use INTEGRALS.
Let's take a look at the derivative - the tool that tells us how steep a function is
at a point.
Another way to think about the derivative is it measures the rate of change of a function
at a point.
As an example, let's use the function f(x) = x^3 -x^2 - 4x +4.
Suppose we want to find the steepness of the graph at the point (-1,6).
How would we do it?
And what do we even mean by steepness?
In Algebra, you find the rate of change of a line by computing the slope (the change
in y divided by the change in x).
But this is a CURVE, not a line.
So we get a good look, let's zoom in a bit.
Here's the idea: Pick a second point nearby.
How about the point (-0.8, 6.048).
Next, draw a line through these two points.
The slope of this line is a good approximation for the steepness of the curve at the point
(-1,6).
If you compute the slope, you get 0.24.
This is a good approximation, but we can do better.
What if we pick a different point that's even closer?
How about (-0.9, 6.061).
If you compute the slope of the line between this point and (-1,6), you get 0.61.
If you keep picking closer and closer points, and computing the slopes of the lines, you'll
get a sequence of slopes which are getting closer and closer to some number.
The lines are getting closer and closer to the TANGENT LINE.
And the slopes are approaching 1.
So we say the "slope" of the curve at the point (-1,6) is 1.
We call this number the DERIVATIVE of f(x) at the point where x = -1.
This is the SLOPE of the TANGENT LINE through the point (-1,6).
Luckily, you won't have to do this every time you want to measure the rate of change at
a point.
In Calculus, you'll learn how to find a function that will give you the slope of any tangent
line to the graph.
This function is also called the DERIVATIVE.
Next, let's take a look at the INTEGRAL.
This is the tool that lets you find areas under curves.
As an example, let's look at the function g(x)= sin x.
What if we wanted to find the area under this curve between x=0 and x=pi?
How would we do it?
We know how to find the area of simple shapes, like rectangles and circles, but this is much
more curvy and complicated.
Let's zoom in to get a closer look.
Here's the idea: Slice the region into a bunch of very thin
sections.
Let's start with 10 slices.
For each section, find the area of the tallest rectangle you can fit inside.
There are 10 thin rectangles.
The width of each rectangle is pi/10.
And we can find the height using the function g(x).
Next, add up the areas of all 10 rectangles.
We get a combined area of 1.66936.
This is a pretty good approximation to the area under the curve, but we can do better.
What if we do this again, but we use 25 slices instead?
This time, we'll get an approximate area of 1.87195.
Let's do this again, and again, using thinner and thinner slices.
50 slices...100 slices...1000 slices.
You get a sequence of areas that are getting closer and closer to some number.
It looks like the area is approaching 2.
We call this area the INTEGRAL of g(x) from x=0 to x=pi.
So we have these two tools: the derivative and the integral.
The derivative tells us about the function at a specific point... while the integral
combines the values of the function over a range of numbers.
But notice there was something similar to how we found the derivative and the integral.
In the case of the derivative, we found two points that were close to each other.
Then we let one point get closer and closer and closer to the point that we're interested
in.
In the case of the integral, we took the curve, and we chopped it up into a bunch of rectangles
to approximate the area under the curve.
Then we took thinner and thinner rectangles to get better and better approximations.
In both cases, we're using the same technique.
In the case of the derivative, we're letting the points get closer to each other.
In the case of the integral, we're letting the rectangles get thinner.
In both instances we're getting better and better approximations, and we're looking at
what number the approximations are approaching.
The number they are approaching is called the LIMIT.
And because limits are key for computing both the derivative and the integral, when you
learn calculus you usually start by learning about limits.
A lot of your time in Calculus will be spent computing derivatives and integrals.
You'll start with the essential functions: Polynomials
Trig Functions (sine, cosine, and tangent) Exponential functions
And Logarithmic functions.
These are the building blocks for most of the functions you'll work with.
Next, you'll make up more complex functions by adding, subtracting, multiplying, and dividing
these functions together.
You'll even combine them using function composition.
In Calculus, there are a lot of rules to help you find derivatives and integrals of these
more complex functions.
The derivative rules have names like the Product Rule, Quotient Rule, and Chain Rule.
The integral rules include U-substitution, Integration by Parts, and Partial Fraction
Decomposition.
When you first start Calculus, your focus will be on basic functions.
Functions with one input and one output.
But we don't live in a one-dimensional world.
Our universe is much more complicated.
So once you've mastered Calculus for basic functions, you'll then move up to higher dimensions.
For example, consider a function with two inputs and one output.
Like f(x,y) = e^-(x^2+ y^2).
Earlier, we computed the derivative by computing slopes of tangent lines.
But in higher dimensions, things are a bit more complex.
This is because on a surface, instead of a tangent line, you'll have a tangent PLANE.
To handle this, you'll compute the derivative both in the x direction and in the y direction.
We call these partial derivatives.
These two partial derivatives are what you need to describe the tangent plane.
We'll also need to generalize the integral.
The region below a surface is 3-dimensional.
It has a volume, not an area.
To compute the volume, we'll approximate it using a bunch of skinny boxes.
To sum up all the volumes, you'll need to use a DOUBLE INTEGRAL, because the boxes are
spread out in 2 dimensions.
But don't forget we live in 3 spatial dimensions.
So you'll also need to learn calculus for functions with three inputs (x, y, and z).
If a function has three inputs and one output, we call it a SCALAR FIELD.
An example would be a function returning the temperature at any point in space.
And the outputs of functions don't have to be simply numbers.
They can also be vectors.
Functions with three inputs and a vector output are called VECTOR FIELDS.
An example would be a function that gives the force vector due to gravity at every point
in space.
To recap, the two main tools you'll learn in calculus are the derivative and the integral.
The derivative tells you about a function at a specific point.
Namely, it tells you how quickly the function is changing at that point.
The integral combines the values of a function over a region.
You'll start your study of calculus by learning how to compute the derivatives and integrals
of a wide variety of different functions, and you'll learn a lot of rules to help you.
Next you're going to take these tools and apply them to higher dimensions by using things
called partial derivatives and multiple integrals.
And along the way, you'll learn how to apply derivatives and integrals to solve real-world
problems.
Now that you've seen the big picture, it's time to start learning the details, so let's
get to work!
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