Name: tangent plane estimates for two variables
Uploaded: 2026-06-08T09:38:27.256Z
Duration: 1 min 23 s
Channel: JoVE

Suppose pollution concentration across a city is modeled as a differentiable function of x and y and represented by a 3D surface above the xy-plane. Because the concentration changes smoothly, a nearby value can be estimated from a known measurement.This process is called linear approximation. It uses a local flat model to estimate the pollution level near a specific point, such as a monitoring station where the concentration is known.While the pollution surface may curve across the city, it acts like a flat sheet very close to that station. This sheet is called the tangent plane, defined by the known concentration and the partial derivatives in the x and y directions.So, the concentration at a nearby location is estimated using the tangent plane rather than the curved surface. When the location is very close to the monitoring station, the estimate is usually accurate.But as the location moves farther away, the approximation becomes less accurate because the pollution surface no longer stays close to that flat model. In this way, linear approximation gives a quick local estimate of a complex surface.

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 fy(x, y) = 6y. At (2, 1), fx(2, 1) = 4 and fy(2, 1) = 6. Therefore, the linear approximation is\begin{equation*}L(x,y)=7+4(x-2)+6(y-1)\end{equation*}To estimate f(2.1, 0.9), substitute x = 2.1 and y = 0.9:\begin{equation*}L(2.1,0.9)=7+4(2.1-2)+6(0.9-1) = 6.8\end{equation*}The linear approximation estimates that f(2.1, 0.9) is approximately 6.8. The exact value is\begin{equation*}f(2.1,0.9)=(2.1)^2+3(0.9)^2=4.41+2.43=6.84\end{equation*}Thus, the tangent plane estimate is close to the actual value because (2.1, 0.9) is near (2, 1).

Linear Approximations

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Education

JoVE Core

Calculus

حساب التفاضل والتكامل

Unit

وحدة

Partial Derivatives and Gradients

المشتقات الجزئية والتدرجات

KEY TERMS AND CONCEPTS

المصطلحات والمفاهيم الأساسية

التقريب الخطي

tangent plane estimates for two variables

Tangent plane estimates help approximate a differentiable function of two variables near a known point. Instead of using the curved surface directly, the function is replaced by its tangent plane. This gives a fast estimate for values close to the point being studied.

For the function f(x,y)=x^2+3y^2, the point (2, 1) is used as the reference. The exact value there is f(2,1)=2^2+3(1)^2=4+3=7. The linear approximation near (a,b) is L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b).

To build the estimate, the partial derivatives are found first. For this function, f_x(x,y)=2x and f_y(x,y)=6y. At (2,1), the partial derivatives are f_x(2,1)=4 and f_y(2,1)=6, so the tangent plane formula becomes L(x,y)=7+4(x-2)+6(y-1).

This plane can be used to estimate nearby values. For example, substituting x=2.1 and y=0.9 gives L(2.1,0.9)=7+4(2.1-2)+6(0.9-1)=6.8. The exact value is f(2.1,0.9)=(2.1)^2+3(0.9)^2=4.41+2.43=6.84, so the estimate is very close because the point is near (2,1).

Tangent Plane Estimates for Two Variables

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Tangent plane estimates help approximate a differentiable function of two variables near a known point. Instead of using the curved surface directly, the function is replaced by its tangent plane. This gives a fast estimate for values close to the point being studied. For the function f(x,y)=x^2+3y^2, the point (2, 1) is used as the reference. The exact value there is f(2,1)=2^2+3(1)^2=4+3=7. The linear approximation near (a,b) is L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b). To build the...

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Use tangent plane estimates to approximate a two-variable function near a known point, find partial derivatives, and compare the estimate with the exact value.

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The JoVE Core Calculus collection provides a comprehensive visual guide to the entire calculus sequence, seamlessly connecting mathematical theory to real-world applications in physics and economics. Through dynamic animated lessons, the course transitions smoothly from foundational single-variable concepts to advanced multivariable topics. Students will explore everything from standard differentiation and integration to spatial vectors, multi-dimensional derivatives, and complex vector fields. By breaking down abstract coordinate systems, infinite series, and spatial motion, this structured approach builds the robust analytical skills and advanced mastery essential for STEM success.

calculus

Tangent Plane Estimates for Two Variables

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