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 𝑓(π‘₯) = 1/π‘₯ at a suitable value of π‘₯, estimate the value of 1/4.002.

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Video Transcript

By finding the linear approximation of the function 𝑓 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 𝑓 of π‘₯ at a suitable value of π‘₯. Let’s start by recalling what we mean by a linear approximation of the function 𝑓 of π‘₯ at π‘₯ is equal to π‘Ž. If our function 𝑓 of π‘₯ is differentiable at the point π‘₯ is equal to π‘Ž, we can approximate our function 𝑓 of π‘₯ with its tangent line. This is our linear approximation of 𝑓 of π‘₯ at π‘₯ is equal to π‘Ž. We call this 𝐿 of π‘₯. It’s equal to 𝑓 of π‘Ž plus 𝑓 prime of π‘Ž times π‘₯ minus π‘Ž.

The question wants us to approximate one divided by 4.002. And since 𝑓 of π‘₯ is one over π‘₯, this is 𝑓 evaluated at 4.002. Remember, our linear approximation is more accurate the closer our input is to π‘₯ is equal to π‘Ž. And we want to estimate an input of 4.002. So we should pick π‘Ž close to 4.002. We’ll pick π‘Ž equal to four. So to find our linear approximation, we need to find 𝑓 evaluated at four and 𝑓 prime evaluated at four. Let’s start by finding 𝑓 evaluated at four.

𝑓 of π‘₯ is one over π‘₯, so we substitute π‘₯ is equal to four. We get one-quarter. And we’ll write this as 0.25. Next, to find 𝑓 prime evaluated at π‘Ž, we need to find an expression for 𝑓 prime of π‘₯. So we need to differentiate 𝑓 of π‘₯, which is one over π‘₯. We’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 𝑓 prime of π‘₯ is equal to negative π‘₯ to the power of negative two. We’re now ready to find 𝑓 prime of π‘Ž. Remember, π‘Ž is equal to four, so we substitute π‘₯ is equal to four into our expression for 𝑓 prime of π‘₯. We get negative one times four to the power of negative two. We’ll then evaluate this by using our laws of exponents. π‘Ž to the power of negative 𝑛 is equal to one divided by π‘Ž to the 𝑛th power.

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

We’re now ready to find an estimate for one divided by 4.002. It’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 𝑓 of π‘₯ is equal to one over π‘₯ at π‘₯ is equal to four, we’ve shown one divided by 4.002 is approximately equal to 0.249875.

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