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Differential Equations

Differential Equations

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 yold:

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 ynew, 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

yold = f (a), Δ x = x*a, and slope = f '(a).

From there we know how to calculate ynew, which is the approximation we want.

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