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Linear approximation: We learned how to use linear approximation to approximate the value of a multivariable function near a given point. This involved finding the tangent plane to the surface at the given point and using it to estimate the function value. To do this, we used the differentials dx and dy to do the approximation, and ignored any nonlinear terms like dxdy. We also discussed relative error or measuer $dx/x$ and $dy/y$ and relative error $dz/z$, which allowed us to quantify the accuracy of our approximation. A key ingredient in the estimation is trigonometric inequality.
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Chain rule: We introduced the chain rule for multivariable functions, which allowed us to compute the derivative of a composition of functions. We used a diagram to understand the formula for the chain rule and applied it to various examples. We also discussed how the chain rule affected the relative error of our approximation. Specifically, we found that the relative error of the composite function was proportional to the product of the relative errors of the individual functions in the composition.
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Gradient: We explored the concept of the gradient, which was a vector that pointed in the direction of maximum increase of a function. The length of the gradient represented the rate of change of the function in that direction. We learned how to compute the gradient of a function of two variables and interpret it geometrically. The gradient was also used to find the direction of steepest ascent or descent for a function.s