91Ó°ÊÓ

Partial derivatives are essential tools in calculus for functions of several variables. When we talk about partial derivatives, we refer to the rate at which a function changes as one of its variables changes, while keeping the other variables constant. This is slightly different from a regular derivative which deals with functions of a single variable.
For the function \(f(x, y) = x^3 y + x y^3\), we find the partial derivative with respect to \(x\), denoted as \(f_x(x,y)\), and with respect to \(y\), denoted as \(f_y(x,y)\). Partial derivatives help us understand how sensitive the function is to a slight change in one direction.

• \(f_x(x, y) = 3x^2y + y^3\): This shows how the function changes with \(x\) when \(y\) is constant.
• \(f_y(x, y) = x^3 + 3xy^2\): This shows how the function changes with \(y\) when \(x\) is constant.

Once we have these, we can use them to build Taylor polynomials which approximate the function around a given point.

The first-order Taylor polynomial of a function provides a linear approximation to the function near a specific point. This approximation is particularly useful when dealing with functions that are too complex to evaluate directly or when you only need a rough estimate.
For a function \(f\) at a point \((a, b)\), the first-order Taylor polynomial, \(T_1(x, y)\), is given by: \[T_1(x, y) = f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b)\]
In our specific example of \(f(x, y) = x^3 y + x y^3\), and considering the point \((1, 1)\):

• Compute \(f(1, 1)\) which is straightforwardly \(1^3 \cdot 1 + 1 \cdot 1^3 = 2\).
• Calculate partial derivatives at \((1, 1)\): \(f_x(1, 1) = 4\) and \(f_y(1, 1) = 4\).
• The first-order Taylor polynomial becomes:\[T_1(x, y) = 2 + 4(x-1) + 4(y-1)\]
• This eventually simplifies to: \( T_1(x, y) = 4x + 4y - 6\).

It creates a plane in the \(xy\)-space, providing local linear approximation rather than an exact value. Simple yet powerful!

The second-order Taylor polynomial extends the linear approximation to include curvature, offering a more accurate representation of the function around a point. This is crucial for functions with significant non-linear behavior.
The formula for the second-order Taylor polynomial, \(T_2(x, y)\), expands on the first-order by incorporating second partial derivatives:\[ T_2(x, y) = T_1(x, y) + \frac{1}{2}f_{xx}(a, b)(x-a)^2 + \frac{1}{2}f_{yy}(a, b)(y-b)^2 + f_{xy}(a, b)(x-a)(y-b)\]
For the function given in our exercise:

• Calculate second derivatives:
  • \(f_{xx}(1, 1) = 6\)
  • \(f_{yy}(1, 1) = 6\)
  • \(f_{xy}(1, 1) = 6\)
• Combine these into the formula, using:\[T_2(x, y) = 4x + 4y - 6 + 3(x-1)^2 + 3(y-1)^2 + 6(x-1)(y-1)\]
• This simplifies to:\( T_2(x, y) = 3x^2 + 3y^2 + 6xy - 2x - 2y + 6 \).

This polynomial gives a better approximation for values close to \((1,1)\) by taking the curvature of the surface into account.

Function approximation using Taylor polynomials is a powerful mathematical tool, letting us estimate complex functions at select points. This process involves creating a simpler model that closely matches the behavior of our target function near a specified point of interest.
By evaluating the first-order and second-order Taylor polynomials at \( (1.2, 1.2) \), we can predict the behavior of \(f(x, y) \) without directly calculating its complex form:

• By using \(T_1(1.2, 1.2)\), we get approximately \(3.6\).
• For \(T_2(1.2, 1.2)\), the estimate shifts dramatically to \(18.48\).

Interestingly, the actual function value at \( (1.2, 1.2) \) is closely \(3.456\), showing that the first-order polynomial's simpler, linear approach provided a better approximation in this scenario. This demonstrates how even more complex polynomial approximations can become overly fit for problems too close to studied points, illustrating the need for careful selection of approximation orders based on the function's stability and behavior.