Question

The following table shows values of a function \(f(x, y)\) for values of \(x\) from 2 to 2.5 and values of \(y\) from 3 to \(3.5 .\) Use this table to estimate the values of the following partial derivatives. $$\begin{array}{|l|l|l|l|l|l|l|}\hline y\ x & 2 & 2.1 & 2.2 & 2.3 & 2.4 & 2.5 \\\\\hline 3 & 4.243 & 4.347 & 4.450 & 4.550 & 4.648 & 4.743 \\\\\hline 3.1 & 4.384 & 4.492 & 4.598 & 4.701 & 4.802 & 4.902 \\\\\hline 3.2 & 4.525 & 4.637 & 4.746 & 4.853 & 4.957 & 5.060 \\\\\hline 3.3 & 4.667 & 4.782 & 4.895 & 5.005 & 5.112 & 5.218 \\\\\hline 3.4 & 4.808 & 4.930 & 5.043 & 5.156 & 5.267 & 5.376 \\\\\hline 3.5 & 4.950 & 5.072 & 5.191 & 5.308 & 5.422 & 5.534 \\\\\hline\end{array}$$ $$f_{x}(2,3)$$

Short answer

Question: Using the given table values, estimate the partial derivative \(f_x(2,3)\) using the forward difference method. Answer: The estimated value of the partial derivative \(f_x(2,3)\) is approximately 1.04.

Step-by-step solution

Identify the required function values

We need to find the approximate value of \(f_x(2,3)\), which can be calculated using the forward difference method. We will use the values \(f(2,3)\) and \(f(2.1,3)\) from the given table.

Calculate the forward difference quotient

The forward difference quotient for the partial derivative \(f_x(2,3)\) is given by: $$ f_x(2,3) \approx \frac{f(2.1,3) - f(2,3)}{2.1 - 2} $$

Substitute the given values and find the approximate partial derivative

Using the values given in the table, we can calculate the approximate value of \(f_x(2,3)\): $$ f_x(2,3) \approx \frac{4.347 - 4.243}{0.1} = \frac{0.104}{0.1} = 1.04 $$ Hence, the estimated value of the partial derivative \(f_x(2,3)\) is approximately 1.04.

Key concepts

Forward Difference Method

The forward difference method is a numerical technique to estimate the slope of a function at a particular point. It's particularly useful when we have discrete data points or when calculating the exact derivative is impractical.

Conceptually, it's like drawing a straight line between two points on a graph and calculating the line's slope, which gives us an approximation for the derivative at the starting point. You can think of this line as a 'tangent' that travels forward from the point.

The formula for the forward difference method is: \[f'(x) \approx \frac{f(x+h) - f(x)}{h}\]
where \(f'(x)\) is the approximate derivative of the function \(f\) at point \(x\), \(h\) is a small increment in \(x\), and \(f(x+h)\) is the function value at point \(x+h\).

Partial Derivative Approximation

When dealing with functions of multiple variables, we use partial derivatives to study how a function changes with respect to one variable while keeping the others constant. The approximation of a partial derivative follows the same logic as the forward difference method, but it is applied with respect to one variable at a time in multivariable functions.

For a function \(f(x, y)\), the partial derivative with respect to \(x\) at a point \((a, b)\) can be estimated as follows: \[f_x(a,b) \approx \frac{f(a+h,b) - f(a,b)}{h}\]
Here, only \(x\) is incremented by a small value \(h\), and \(y\) remains unchanged. This gives us an approximate rate of change of \(f\) in the direction of the \(x\)-axis at point \((a, b)\).

Multivariable Calculus

In multivariable calculus, we extend the concepts of single-variable calculus to functions that depend on two or more variables. This involves different types of derivatives, such as partial and directional derivatives, as well as multiple integrals.

Understanding how a function behaves in multiple dimensions is crucial for fields like physics, economics, and engineering. With multivariable calculus, instead of looking at just a curve on a graph, we explore surfaces and higher-dimensional shapes to analyze various rates of change and accumulations.

Partial derivatives, integral theorems, and vector calculus are some of the core concepts explored within multivariable calculus. These concepts allow us to calculate, among other things, the gradient of a function, which indicates the direction of the greatest rate of increase of the function and the magnitude of that increase.

Function Values Estimation

Estimating function values is a fundamental aspect of numerical analysis, particularly when exact values are difficult or impossible to obtain. We use various estimation techniques, like interpolation, extrapolation, and numerical differentiation, to predict these values based on known data points.

In cases where the function is complex or only partially known, we might estimate the function's values by fitting a simpler function to the known data points and using it to estimate the values at other points. The accuracy of these estimations depends on the method used and the quality and density of the known data points.

The forward difference method is an example of such an estimation technique for derivatives. By using neighboring function values, we can estimate the derivative even if we don't have an explicit formula for the entire function.