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
This gives us
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
So what? Well, basic arithmetic tells us
Thus the difference between this and is . So the approximation appears to be decent to a couple of digits.
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?
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
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.