Newton-Raphson Method

Find the root of .

Newton: "I wonder if I can represent this as a line, and I can always find the zero of a line."

What is ?

From IVT, since our function is continuous, there's a root somewhere between 2 and 3 (i.e the line can't teleport across the axis).
So the root is .

Then we take the point where the tangent line crosses the x-axis, find the point on curve that corresponds to this x coordinate, and repeat with a tangent line at this point. This is Newtons Method.

Where does cross the x-axis? Setting ,

so

Pretty good approximation, but lets try again:

so

Eventually, extra iterations are indistinguishable.

Compare this to the actual value: .

1. Start with a point that you know which is close to the desired root
2. Find the tangent line at this point. Make sure it is not zero.
3. Find the x-coordinate where the tangent line crosses the x-axis
4. Find the point on the curve which corresponds to this x-coordinate (i.e )
5. Repeat step 2 with this new point

In general, Newton's Method:

If we isolate , we get the recurrence relation:

The sequence will fail to converge if does not exist, is not continuous, or equals 0.

Consider the equation .

1. Show that there is a root in the interval and
2. Using Newton's method, approximate the root of the equation between and , to 6 decimal places
3. Using Newton's method, approximate the root of the equation between and , to 6 decimal places

solution
Let

Since the is continuous by inspection, we can use IVT.
We have:

So we can conclude that there is a root between and another between .

Next, we have . We will use the recurrence relation:

For the first interval, it seems is closer to the root, so we will use .

so we estimate the root to be .

For the second interval, we chose .

so we estimate the root to be .