# Differentials find maximum percentage error

• naspek
In summary, the maximum percentage error in the calculated value of T for a simple pendulum with small oscillations can be approximated using differentials. The error is affected by the errors in the measurements of L and g, which should be at most 0.5% and 0.1% respectively. The maximum percentage error is 1.2%.
naspek

## Homework Statement

The period T of a simple pendulum with small oscillations is calculated from the
formula T = 2pie (L / g)^1/2 . Suppose that measurements of L and g have errors of at
most 0.5% and 0.1% respectively. Use differentials to approximate the maximum
percentage error in the calculated value of T.

## The Attempt at a Solution

dT = (dT/dL)dL + (dT/dg)dg
= 2pie (1/2g(L/g)^1/2) dL + 2pie (1/2g(L/g^2)^1/2) dg
i don't know how am i going to proceed..

naspek said:

## Homework Statement

The period T of a simple pendulum with small oscillations is calculated from the
formula T = 2pie (L / g)^1/2 . Suppose that measurements of L and g have errors of at
most 0.5% and 0.1% respectively. Use differentials to approximate the maximum
percentage error in the calculated value of T.

## The Attempt at a Solution

dT = (dT/dL)dL + (dT/dg)dg
= 2pie (1/2g(L/g)^1/2) dL + 2pie (1/2g(L/g^2)^1/2) dg
i don't know how am i going to proceed..
You're not using the chain rule correctly. Also, the name of the Greek letter is pi, not pie.
dT/dt = d/dt(2 pi sqrt(L/g)) = 2 pi d/dt( L^(1/2)/g^(1/2))
Now use the quotient rule to complete the differentiation on the right. After you get that, you can multiply both sides of your equation by dt to get an equation that involves dT, dL, and dg.

ok.. got it already.. maximum percentage is 1.2% ^_^

## What is a differential?

A differential is a mathematical concept that represents the instantaneous rate of change of a function. It is typically denoted by the symbol "dx" and is used in calculus to calculate the slope of a curve at a specific point.

## What is maximum percentage error?

Maximum percentage error is a measure of the largest possible error or deviation from the true value when performing a calculation or measurement. It is expressed as a percentage of the true value.

## How is the maximum percentage error calculated?

The maximum percentage error is calculated by taking the absolute value of the difference between the approximate value and the true value, dividing it by the true value, and then multiplying by 100 to convert it to a percentage.

## Why is it important to calculate maximum percentage error?

Calculating maximum percentage error allows us to understand the potential for error in our calculations or measurements. It helps to determine the accuracy and reliability of our results and allows us to make adjustments or improvements in our methods if necessary.

## How is the concept of differentials related to maximum percentage error?

The concept of differentials is closely related to maximum percentage error because it allows us to approximate the maximum possible error in a calculation or measurement. By using differentials, we can estimate the error in our results and determine the maximum percentage error.

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