**1. Motivation.** So we have some procedure to construct a line tangent to a curve at a given point. What good is it?

Well, suppose we want to compute . What is it? I don’t know!

Lets introduce the function . Then we want to evaluate . What to do?

Why not construct the line tangent to at , then make an approximation ?

Lets see if this really works. We calculate

(1)

This gives us

(2).

Is this a good approximation? I don’t know, how can we find out?

Why not square it, and see how far off we are from the original quantity ? Let us try! We see

(3).

So what? Well, basic arithmetic tells us

(3′).

Thus the difference between this and is . So the approximation appears to be decent to a couple of digits.

*Reflection.* What did we just do? We observed we wanted to calculate some quantity, and expressed it as a function

. We know what

is, and we wanted to find some approximate solution for small

.

Since is “small enough”, we just used the line tangent to a given point (for us ) as the approximation. The key moment is using a “small enough” . Right now we do not have the proper tools to determine what is “small enough”, but we will develop the tools later on.

**2. Approximations.** If we don’t have a calculator and want to calculate stuff, we may use calculus to help us.

We describe the quantity as a point on the curve . So we want to find some approximate value “close” to .

We do this by finding some point on the curve that’s “close enough” to . (That’s why in our motivating example we chose instead of : we have be dealing with instead of something 300 times bigger when calculating !)

Then we construct the line tangent to the curve at . We’ll describe this line as the points .

Now our approximation takes .

So what’s the algorithm?

**Algorithm A.**

Step 0: we want to calculate some quantity .

Step 1: construct some function such that at some point we have .

Step 2: Find a point “close enough” to .

Step 3: Construct the tangent line .

Step 4: Make the approximation .

*Remark.* We call this a linear approximation because the tangent line is a linear function.

**3. Reflection.** What if we work with when attempting an approximation of ? What happens? Lets try!

We see the tangent line to is described by

(4)

Lets now try to approximate . Is it any good?

Well, . Is this a good approximation? No! We see that , which tells us: a **horrible approximation!**

*Remark.* When we have a linear approximation , it generally gets worse as gets bigger.

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## About Alex Nelson

I like math. I like programming. Most of all, I love puzzles.

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