Name: 什麼是微積分？ (數學) (What is Calculus? (Mathematics))
Uploaded: 2021-01-14T08:51:12.000Z
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This isn't one of those things we inherited from the ancient Greeks like geometry.

基礎被動語態

This subject was created more recently in the late 1600s by Isaac Newton and Gottfried

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

Another way to think about the derivative is it measures the rate of change of a function

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).

In Algebra, you find the rate of change of a line by computing the slope (the change

So we get a good look, let's zoom in a bit.

Here's the idea: Pick a second point nearby.

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

This is a good approximation, but we can do better.

What if we pick a different point that's even closer?

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.

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

In Calculus, you'll learn how to find a function that will give you the slope of any tangent

This function is also called the DERIVATIVE.

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?

We know how to find the area of simple shapes, like rectangles and circles, but this is much

Here's the idea: Slice the region into a bunch of very thin

For each section, find the area of the tallest rectangle you can fit inside.

And we can find the height using the function g(x).

Next, add up the areas of all 10 rectangles.

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.

You get a sequence of areas that are getting closer and closer to some number.

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 the case of the integral, we took the curve, and we chopped it up into a bunch of rectangles

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

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

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