Linear approximation a. Find the linear approximation for...

Linear approximation a. Find the linear approximation for the following functions at the given point. b. Use part (a) to estimate the given function value. $$f(x, y)=\sqrt{x^{2}+y^{2}} ;(3,-4) ; \text { estimate } f(3.06,-3.92)$$

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Key Concepts

Linear Approximation

Linear approximation is a technique used to approximate the value of a function near a given point by using the tangent line (or tangent plane in multiple dimensions). It relies on the idea that if a function is differentiable at a point, then it can be well approximated by its linearization in a small region around that point.

Partial Derivatives

Partial derivatives measure the rate at which a function changes as one variable changes while keeping the other variables constant. They are central to finding the linear approximation of functions of several variables, as they help form the gradient vector which determines the slope in each direction.

Gradient Vector

The gradient vector is a vector of all partial derivatives of a function and points in the direction of the steepest ascent. Its components are used in constructing the tangent plane, which serves as the linear model for the function near a point.

Tangent Plane

The tangent plane to a multivariable function at a point is the plane that best approximates the function near that point. It is derived using the function’s value and its partial derivatives at the point, and it forms the basis of the linear approximation.

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