• 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 [latex]D[/latex] (in milligrams) of the medication can be modeled by a function of the patient’s weight [latex]w[/latex] (in kilograms):

[latex]D(w) = 5w^(\frac{2}{3})[/latex]

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 [latex]50[/latex] km, with a possible error of [latex]±0.5[/latex] km. The volume [latex]V[/latex] of rainfall is modeled as the volume of a cylinder, where the height [latex]h[/latex] represents the average rainfall depth, which is [latex]2[/latex] cm ([latex]0.02[/latex] km).

The volume [latex]V[/latex] of the rainfall can be expressed as:

[latex]V = πr²h[/latex]