91Ó°ÊÓ

In multivariable calculus, partial derivatives are crucial for understanding how a function changes as each variable changes independently. Consider the function \(f(x, y) = e^{x} \cos y\). To linearize it, we need to first calculate its partial derivatives. This requires differentiating \(f\) while treating one variable as constant at a time. - **For \(x\):** The partial derivative \(\frac{\partial f}{\partial x}\) involves differentiating \(e^x\), while keeping \(\cos y\) constant, resulting in \(e^x \cos y\). - **For \(y\):** The partial derivative \(\frac{\partial f}{\partial y}\) results from differentiating \(\cos y\), with \(e^x\) as a constant factor, yielding \(-e^x \sin y\). These derivatives tell us the rate of change of \(f(x, y)\) with respect to each variable around the point \(P_0(0,0)\). Evaluating these at \(P_0\), we find \(\frac{\partial f}{\partial x} = 1\) and \(\frac{\partial f}{\partial y} = 0\). Partial derivatives are essential in analyzing surface behavior in a multidimensional space.

Linearization helps simplify complex functions near a certain point, transforming them into linear approximations. For a multivariable function \(f(x, y)\), the linearization at point \(P_0(0,0)\) can be represented as \(L(x, y) = f(P_0) + \frac{\partial f}{\partial x}(P_0)x + \frac{\partial f}{\partial y}(P_0)y\). This uses the function's value and partial derivatives calculated at a specific point. For our function \(f(x, y) = e^{x} \cos y\) evaluated at \(P_0(0,0)\), we use: - \(f(0,0) = 1\) since \(e^0 \cos 0 = 1\). - From the partial derivatives, \(\frac{\partial f}{\partial x} = 1\) and \(\frac{\partial f}{\partial y} = 0\). Thus, the linear approximation \(L(x, y) = 1 + x\) gives us a simple expression to approximate \(f(x, y)\) near \(P_0\). Linearization is ideal for estimating small increments in functions and reduces computational complexity in various applications.

Error estimation quantifies the difference between the exact value of a function and its linear approximation over a defined range. To assess this for \(f(x, y)\), one can apply bounds on the derivatives around the point of linearization within a rectangle \(R\) defined by \(|x| \leq 0.1\) and \(|y| \leq 0.1\). Taylor's theorem helps set this bound by considering second-order derivatives, which describe curvature, potentially introducing more error. Here, those derivatives are: - \(\frac{\partial^2 f}{\partial x^2} = e^x \cos y\), - \(\frac{\partial^2 f}{\partial y^2} = -e^x \cos y\), and - \(\frac{\partial^2 f}{\partial x \partial y} = -e^x \sin y\). If \(|e^x \cos y| \leq 1.11\) and \(|e^x \sin y| \leq 1.11\), the maximum error \(|E|\) from linearization is controlled by \(|E| \leq \frac{1}{2} \times 1.11 \times 0.1^2 \), giving \(|E| \leq 0.00555\). Estimate errors correctly ensures the reliability of linear approximations.

Taylor's theorem extends functions into a series of polynomials, modeling changes around a point. For multivariable calculus, it provides a powerful way to approximate functions with fewer terms but captures critical behaviors, including curvature and error. For \(f(x, y)\), Taylor's theorem allows us to extend beyond linearization by integrating second-order derivatives, which contribute to the error term \(|E|\). These second derivatives reflect how the function bends, accounting for potential inaccuracies in the linear approximation. By evaluating these derivatives and setting bounds based on known constraints \(|x|, |y| \leq 0.1\), Taylor's theorem offers an upper-error margin using simple calculations. By truncating these higher-order terms, we achieve a balance between accuracy and computational simplicity. Taylor’s theorem is thus a fundamental framework in calculus, guiding both linear and nonlinear approximations.