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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

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Lesson Transcript

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

This lesson will demonstrate how to use linear approximation with differentials to approximate function values near a given point. We will go over the steps and formulas involved in linear approximation.

Suppose you're at a carnival, and the announcer says that they are having a contest to see who can guess closest to the value of âˆš(9.24). You know that âˆš(9) = 3, so it must be that âˆš(9.24) is a little over 3, but you want to be as close as possible to win the contest.

Good news! We've got a way of approximating this value without a calculator, and it's called using linear approximation. **Linear approximation** involves finding the equation of a line tangent to the function at a given value of *x*, and using it to approximate the function value for points nearby.

To put this into perspective, let's consider the function *f*(*x*) = âˆš(*x*). If we could find a line that's close to the function at *x* = 9.24, we can use it to approximate *f*(9.24), or âˆš(9.24), which is exactly what we want to do.

Notice that the tangent line of *f*(*x*) = âˆš(*x*) at *x* = 9 has values that are very close to the function values of *f*(*x*) when *x* is near 9, so if we could find the equation of this line, we could use it to estimate the value of *f*(*x*) at 9.24, and find an approximate value for âˆš(9.24).

In general, if we're finding a line to use in linear approximation that passes through a point (*a*, *f*(*a*)), we use the following formula:

*L*(*x*) = *f*(*a*) + *m*(*x* - *a*), where *m* is the slope of the line at *x* = *a*.

In this case, our line is tangent to *f*(*x*) = âˆš(*x*), and passes through the point (9, *f*(9)) = (9, 3). Plugging these values in gives:

*L*(*x*) = *f*(9) + *m*(*x* - 9) = 3 + *m*(*x* - 9)

Wait a second, what about *m*? How do we find the slope at *x* = 9? This is where differentials come into play. Let's explore!

When finding the linear approximation for a function *f*(*x*) at the point (*a*, *f*(*a*)), we just saw that the general formula is *L*(*x*) = *f*(*a*) + *m*(*x* - *a*), where *m* is the slope of the line at *x* = *a*. To find the *m*, we use the fact that the derivative of a function at a point is equal to the slope of the tangent line at that point. Therefore, this general formula for the linear approximation can actually be rewritten as follows:

*L*(*x*) = *f*(*a*) + *f* ' (*a*)(*x* - *a*)

Great! So, all we have to do is find the derivative of *f*(*x*) and plug 9 into it, and we'll have all of the parts of our formula!

To find the derivative of *f*(*x*) = âˆš(*x*), let's rewrite it as *f*(*x*) = *x*(1/2). Now we can use the following rule: The derivative of *x**n* is *n**x*(*n*-1)

Thus, we have the following:

*f* ' (*x*) = (1/2)*x*(-1/2) and *f* ' (9) = (1/2)(9)(-1/2) = (1/2)(1/3) = 1/6

All together, we have the following:

*L*(*x*) = *f*(9) + *f* ' (9)(*x* - 9)

= 3 + (1/6)(*x* - 9)

= 3 + (1/6)*x* - (9/6)

= 1.5 + (1/6)*x*

*L*(*x*) = 1.5 + (1/6)*x*.

All we have to do is plug 9.24 into this formula for *x* and we have our approximation!

*L*(9.24) = 1.5 + 1/6(9.24) = 3.04

You submit your answer into the contest, and the announcer goes on to check the value of âˆš(9.24) on a calculator and gets 3.0397368307. We see that 3.4 is a very close approximation, and congrats! You won the contest!

**Linear approximation** involves finding the equation of a line tangent to the function at a given value of *x*, and using it to approximate the function value for points nearby. This process involves differentials in that the formula for a linear function that is a linear approximation of the function *f*(*x*) at the point (*a*, *f*(*a*)) includes the derivative of *f*(*x*). That is:

*L*(*x*) = *f*(*a*) + *f* ' (*a*)(*x* - *a*)

To perform linear approximation using differentials, we use the following steps:

- Identify the function,
*f*(*x*), and the point, (*a*,*f*(*a*)), that you're using to perform the linear approximation. - Find
*f*' (*x*) and use it to find*f*' (*a*). - Plug
*a*,*f*(*a*) and*f*' (*a*) into the linear approximation formula*L*(*x*) =*f*(*a*) +*f*' (*a*)(*x*-*a*). - Use this formula to approximate values of the function
*f*(*x*) for*x*-values near*a*.

This process comes in extremely handy when we need a close approximation of a function value, so let's make sure to keep it in mind for future use!

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GRE Math: Study Guide & Test Prep27 chapters | 182 lessons | 16 flashcard sets

- Using Limits to Calculate the Derivative 8:11
- The Linear Properties of a Derivative 8:31
- Calculating Derivatives of Trigonometric Functions 7:20
- Linear Approximations Using Differentials: Definition & Examples 5:09
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- Understanding Higher Order Derivatives Using Graphs 7:25
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