Linear approximation a. Find the linear approximation to the...

Linear approximation
a. Find the linear approximation to the function $f$ at the given point.
b. Use part (a) to estimate the given function value.
$$\begin{aligned}
&f(x, y, z)=\ln (1+x+y+2 z) ;(0,0,0) ; \text { estimate }\\
&f(0.1,-0.2,0.2)
\end{aligned}$$

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Key Concepts

Tangent Plane

The tangent plane is the plane that touches a surface at a given point and approximates the surface near that point. In the context of linear approximation, the tangent plane provides a linear function whose value closely matches the original function when the inputs are near the point of tangency.

Gradient Vector

The gradient vector is a collection of all the partial derivatives of a multivariable function. It points in the direction of the greatest rate of increase of the function and its components are used in constructing the equation of the tangent plane for linear approximation.

Partial Derivatives

Partial derivatives are the derivatives of a multivariable function with respect to one variable while holding the other variables constant. They are crucial in computing the coefficients for the linear approximation and help in understanding the function’s behavior along each axis independently.

Linear Approximation

Linear approximation is a method used in calculus to estimate the value of a function near a point using the tangent plane. It relies on the idea that for small changes around a specific point, the function behaves almost linearly, and so a first order Taylor expansion (or the equation of the tangent plane) can approximate the function’s value.

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