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Linear Approximations

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...

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Related Experiment Video

Updated: Jun 15, 2026

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Composition and Distribution Analysis of Bioaerosols Under Different Environmental Conditions

Published on: January 7, 2019

Low-Cost Indoor Sensor Deployment for Predicting PM_2.5 Exposure.

Shahar Tsameret¹, Daniel Furuta¹, Provat Saha^2,3

¹Department of Mechanical & Aerospace Engineering, University of Miami, Coral Gables, Florida 33146, United States.

|August 15, 2024

Summary

This summary is machine-generated.

Indoor air quality significantly impacts health. This study used low-cost sensors to predict daily fine particulate matter (PM2.5) exposure for different groups, emphasizing sensor placement and correction factors.

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Area of Science:

Environmental Health Sciences
Air Quality Monitoring
Exposure Science

Background:

Indoor air quality (IAQ) is crucial, with people spending ~90% of their time indoors.
Limited deployment of indoor particulate matter (PM) sensor networks compared to outdoor networks.
Particulate matter (PM2.5) poses significant health risks, necessitating accurate exposure assessment.

Purpose of the Study:

Investigate indoor PM2.5 exposure using low-cost sensor networks in Pittsburgh.
Predict daily PM2.5 exposure for diverse demographics.
Evaluate the impact of different correction factors on sensor data.

Main Methods:

Deployment of two low-cost indoor PM2.5 sensor networks.
Utilized Monte Carlo simulation for exposure prediction.
Compared empirical and physics-based correction factors for PurpleAir sensor data.

Main Results:

Mean PM2.5 exposure varied by 1.5 μg/m³ or less when indoor and outdoor concentrations were similar.
Increased time outdoors raised exposure by up to 3 μg/m³ when indoor PM was lower than outdoor.
Correction factors influenced reported PM2.5 concentrations.

Conclusions:

Sensor site selection is critical for accurate indoor air quality assessment.
Robust indoor and outdoor low-cost sensor networks are valuable for understanding exposure.
Accurate PM2.5 exposure data is essential for public health interventions.