Taylor polynomial example for f(x,y), Real Analysis II

In this video, we build the second-order Taylor polynomial for a scalar-valued function of two variables: f(x, y) = 1 / (x² + y² + 1), centered at the origin. First, we compute the function value, the gradient, and the Hessian at the origin. Then we put everything together to write out the Taylor polynomial! We use that approximation to estimate the value of the function at (0.2, 0.3), compare it with the exact value, and talk about why the approximation works well near the center. After that, we revisit the polynomial using a clever shortcut: recognizing that this function fits the form of a geometric series. By rewriting the denominator, we see that the second-order Taylor polynomial we just computed also matches the beginning of that series expansion. #mathematics #math #taylorpolynomial #multivariablecalculus #gradient #calculus3 #realanalysis #advancedcalculus