• Explain and use linearization to approximate a function’s value near a specific point
• Calculate and interpret differentials to estimate small changes in function values
• Measure the accuracy of approximations made with differentials by calculating relative and percentage errors

Differentials in Action: From Medical Dosages to Weather Forecasts

In this apply-it task, we’ll explore how differentials and linearization can be used in medical dosage calculations and weather forecasting. These examples will demonstrate the practical applications of these mathematical concepts in real-world scenarios.

Part 1: Medication Dosage Calculation

A doctor is determining the appropriate dosage of a certain medication for patients based on their weight. The recommended dosage $D$ (in milligrams) of the medication can be modeled by a function of the patient’s weight $w$ (in kilograms):

$D(w) = 5w^(\frac{2}{3})$

Part 2: Rainfall Volume Estimation

A meteorologist is predicting the volume of rainfall over a circular region based on the measured radius of the storm. The radius is measured to be $50$ km, with a possible error of $±0.5$ km. The volume $V$ of rainfall is modeled as the volume of a cylinder, where the height $h$ represents the average rainfall depth, which is $2$ cm ($0.02$ km).

The volume $V$ of the rainfall can be expressed as:

$V = πr²h$