# Linear approximation and percentage error

• ver_mathstats
In summary: For the linearization of ##f(x) = 1/x## at ##x = 100##, you should have a linear function that when you put in ##x = 100##, you get ##f(100)##, not some other value.In summary, the conversation discussed the finding of a linearization, L(x) = -0.0001x+0.2 and the calculation of L(1/99) = 0.0199989899. However, there was confusion around the value of L(1/99) as it seemed off by quite a bit. The correct linearization should give back the correct value at the point being linearized around. Thus, it is important to
ver_mathstats
Homework Statement
Approximate 1/99 to four decimal places using the linearization L(x) of f(x)=1/x at a=100 and use a calculator to compute the percentage error.
Relevant Equations
L(x) of f(x)=1/x
I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899.

Then I tried to put that value into my percentage error formula along with 1/99 and got:

the absolute value of (1/99)-L(1/99) and then we divide that by our actual value which is 1/99, then I multiply everything by 100.

I got the answer 97.99% and I got this wrong, but I am unsure of where I went wrong and how to fix my mistake.

Thank you.

ver_mathstats said:
I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899.
Can you show your work in getting this? Your value for L(1/99) seems off by quite a bit. ##1/99 \approx .01## and your answer is almost twice this.
ver_mathstats said:
the absolute value of (1/99)-L(1/99) and then we divide that by our actual value which is 1/99, then I multiply everything by 100.
That's the idea, but since you didn't show your figures, it's possible that some of them are incorrect.

Last edited:
FactChecker
ver_mathstats said:
Homework Statement: Approximate 1/99 to four decimal places using the linearization L(x) of f(x)=1/x at a=100 and use a calculator to compute the percentage error.
Homework Equations: L(x) of f(x)=1/x

I found the linearization, L(x) = -0.0001x+0.2 and I found L(1/99) = 0.0199989899.
A good check to make for a linearization is that it gives you back the correct value at the point you are linearlizing around.

FactChecker
Where did you get the 0.2 in L(x)= -0.0001x+0.2?

## What is linear approximation?

Linear approximation is a method used to estimate the value of a function at a specific point by using a linear function that closely approximates the true function. It is based on the tangent line of the function at the given point.

## How is linear approximation used in real-world applications?

Linear approximation is commonly used in engineering and science to estimate values of physical quantities, such as temperature or velocity, at a specific point. It is also used in economics and finance to make predictions and analyze trends.

## What is percentage error and how is it calculated?

Percentage error is a measure of how accurate a calculation or measurement is compared to its true or expected value. It is calculated by taking the absolute value of the difference between the measured value and the true value, divided by the true value, and multiplied by 100%.

## How is linear approximation related to percentage error?

Linear approximation can be used to estimate the percentage error in a calculation or measurement. By comparing the linear approximation to the true value, the percentage error can be determined and used to evaluate the accuracy of the estimation.

## What are some limitations of linear approximation?

Linear approximation is only accurate for small intervals around the given point and may not accurately estimate the function outside of this range. It also assumes that the function is continuous and differentiable, which may not always be the case in real-world applications.

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