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Given an IVP, we can use Euler's (pronounced "Oiler's") method to find approximate values of the solution to the IVP (near the initial point). Before we fully explain what Euler's method is, we need to remember some things about slopes and tangent lines.

### Slopes (Again)

We know that the slope of a line is given by

or by

Since

*y*is usually the dependent variable and*x*is usually the independent variable, you may also seeor

The symbol Δ is the Greek capital letter "Delta", which mathematicians use to mean "change."

We usually use the slope formula to calculate the slope of a line given Δ

*y*and Δ*x*. If we know the slope and Δ*x*we can instead use the slope formula to find Δ*y*.Using algebra, if

then multiplying both sides by Δ

*x*we getTherefore if we know the slope of a line and we move over by Δ

*,*then we know*y*is needs to to change by (slope × Δ*x*).### Tangent Line Approximations (Again)

Tangent line approximation can also be called

**local linearization**,**linear approximation**, and probably a bunch of other names. The important thing is that you're*using a line to approximate a curve*.Here's a reminder of how tangent line approximation works:

Suppose we know the value of

*f*at a particular value of*x*:and we want to know the value of

*f*at a nearby value of*x*:We draw the tangent line to

*f*at the point we know:Then we find the value on the tangent line at the nearby value of

*x*:This tangent line value is close to the value we actually wanted.

Using the language from the last section, let's talk this through again.

We know

*y*_{old}:We want to know the value of

*f*if we change*x*by Δ*x*:We draw the tangent line to

*f*at the point we know:and we use the tangent line to find

*y*_{new}, which is close to the value we actually wanted.### Sample Problem

Another kind of problem you may run into is the kind where you're given a formula for the function

*f*and asked to use a tangent line to approximate*f*at some particular value*x*^{*}.To do this, pick some

*x*=*a*close to*x*^{*}. Choose*a*so that*f*(*a*) and*f*'(*a*) are easy to calculate. Then we have*y*_{old}=*f*(*a*), Δ*x*=*x*^{*}–*a*, and slope =*f*'(*a*).From there we know how to calculate y

_{new}, which is the approximation we want.### The Scoop on Euler

**Euler's Method**is a bunch of tangent line approximations stuck together. The basic idea is that you start with a differential equation and a point. You do a tangent line approximation to get a new point.Then you use the new point to do another tangent line approximation.

You do this over and over until you get to the end (which will be specified in the problem). The catch is that, after the first tangent line, instead of drawing real tangent lines you'll be drawing pretend tangent lines. This will make more sense after a couple of examples.