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In this video, we explore an example of linearization in multivariable calculus, using U(x, t) = 1 / sqrt(t) * e^(x^2 / 4t). This function arises in the study of the heat equation, though we set that context aside to concentrate on the linear approximation at (0,1). We derive the first-order linear approximation by computing the gradient, discussing the relative ease of differentiating with respect to x versus t, and leveraging the heat equation to simplify our work. This approach allows us to approximate U(x, t) near (0,1) with a more manageable function. #multivariablecalculus #Linearization #HeatEquation #Mathematics #partialderivatives #Gradient #MathAnalysis #Calculus #calculus3 #maths #vectorcalculus
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