Question Video: Finding the Linear Approximation of a Reciprocal Rational Function | Nagwa Question Video: Finding the Linear Approximation of a Reciprocal Rational Function | Nagwa

# Question Video: Finding the Linear Approximation of a Reciprocal Rational Function Mathematics • Higher Education

By finding the linear approximation of the function π(π₯) = 1/π₯ at a suitable value of π₯, estimate the value of 1/4.002.

03:31

### Video Transcript

By finding the linear approximation of the function π of π₯ is equal to one over π₯ at a suitable value of π₯, estimate the value of one divided by 4.002.

The question wants us to estimate the value of one divided by 4.002. And it wants us to do this by finding a linear approximation of the function π of π₯ at a suitable value of π₯. Letβs start by recalling what we mean by a linear approximation of the function π of π₯ at π₯ is equal to π. If our function π of π₯ is differentiable at the point π₯ is equal to π, we can approximate our function π of π₯ with its tangent line. This is our linear approximation of π of π₯ at π₯ is equal to π. We call this πΏ of π₯. Itβs equal to π of π plus π prime of π times π₯ minus π.

The question wants us to approximate one divided by 4.002. And since π of π₯ is one over π₯, this is π evaluated at 4.002. Remember, our linear approximation is more accurate the closer our input is to π₯ is equal to π. And we want to estimate an input of 4.002. So we should pick π close to 4.002. Weβll pick π equal to four. So to find our linear approximation, we need to find π evaluated at four and π prime evaluated at four. Letβs start by finding π evaluated at four.

π of π₯ is one over π₯, so we substitute π₯ is equal to four. We get one-quarter. And weβll write this as 0.25. Next, to find π prime evaluated at π, we need to find an expression for π prime of π₯. So we need to differentiate π of π₯, which is one over π₯. Weβll rewrite this by using our laws of exponents. One over π₯ is equal to π₯ to the power of negative one. And now we can differentiate this by using the power rule for differentiation.

We multiply by our exponent of π₯ and reduce this exponent by one. This gives us π prime of π₯ is equal to negative π₯ to the power of negative two. Weβre now ready to find π prime of π. Remember, π is equal to four, so we substitute π₯ is equal to four into our expression for π prime of π₯. We get negative one times four to the power of negative two. Weβll then evaluate this by using our laws of exponents. π to the power of negative π is equal to one divided by π to the πth power.

Using this, we get π prime of four is equal to negative one divided by four squared, which is equal to negative one over 16. Weβll then write this in its decimal expansion, negative 0.0625. Weβre now ready to find our linear approximation of π of π₯ is equal to one over π₯ at π₯ is equal to four. We get πΏ of π₯ is equal to π of four plus π prime of four times π₯ minus four. And substituting in our values for π of four and π prime of four, we get πΏ of π₯ is equal to 0.25 minus 0.0625 times π₯ minus four.

Weβre now ready to find an estimate for one divided by 4.002. Itβs approximately equal to our linear approximation evaluated at π₯ is equal to 4.002. Substituting π₯ is equal to 4.002 into our linear approximation, we get 0.25 minus 0.0625 times 4.002 minus four. And then we can just calculate this expression. We get 0.249875. Therefore, by finding a linear approximation of the function π of π₯ is equal to one over π₯ at π₯ is equal to four, weβve shown one divided by 4.002 is approximately equal to 0.249875.

## Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

• Interactive Sessions
• Chat & Messaging
• Realistic Exam Questions