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. $$\begin{array}{|l|l|l|l|l|l|l|}\hline y\ x & 2 & 2.1 & 2.2 & 2.3 & 2.4 & 2.5 \\\\\hline 3 & 4.243 & 4.347 & 4.450 & 4.550 & 4.648 & 4.743 \\\\\hline 3.1 & 4.384 & 4.492 & 4.598 & 4.701 & 4.802 & 4.902 \\\\\hline 3.2 & 4.525 & 4.637 & 4.746 & 4.853 & 4.957 & 5.060 \\\\\hline 3.3 & 4.667 & 4.782 & 4.895 & 5.005 & 5.112 & 5.218 \\\\\hline 3.4 & 4.808 & 4.930 & 5.043 & 5.156 & 5.267 & 5.376 \\\\\hline 3.5 & 4.950 & 5.072 & 5.191 & 5.308 & 5.422 & 5.534 \\\\\hline\end{array}$$ $$f_{y}(2.4,3.3)$$

Short answer

Question: Estimate the value of the partial derivative \(f_y(2.4, 3.3)\) using the given table of values for a function \(f(x, y)\). Answer: The estimated value of the partial derivative \(f_y(2.4, 3.3)\) is \(1.55\).

Step-by-step solution

Understanding the problem

In this exercise, we need to estimate the value of the partial derivative \(f_y(2.4, 3.3)\) using the given table. The partial derivative \(f_y\) represents the rate at which the function changes with respect to the variable \(y\). So, we need to find the rate of change of \(f(x, y)\) in the vertical (y) direction at the point \((2.4, 3.3)\).

Finding the required points

To estimate the partial derivative at the given point, we need the values of the function at a point just above and just below the point. We can find these points in the table: - \(f(2.4, 3.2) = 4.957\) - \(f(2.4, 3.4) = 5.267\)

Estimating the partial derivative using the difference quotient

We will now estimate the partial derivative \(f_y(2.4, 3.3)\) using the difference quotient formula: \(f_y(2.4, 3.3) \approx \dfrac{f(2.4,3.4)-f(2.4,3.2)}{3.4-3.2}\)

Calculate the difference quotient

Plug in the values of \(f(2.4, 3.4)\) and \(f(2.4, 3.2)\) into the formula: \(f_y(2.4, 3.3) \approx \dfrac{5.267 - 4.957}{3.4 - 3.2} = \dfrac{0.310}{0.2}\)

Simplify and find the partial derivative

Simplify the expression for \(f_y(2.4, 3.3)\) to get the final answer: \(f_y(2.4, 3.3) \approx \dfrac{0.310}{0.2} = 1.55\) Therefore, the estimated value of the partial derivative \(f_y(2.4, 3.3)\) is \(1.55\).

Key concepts

Multivariable Calculus

Multivariable calculus extends the concepts of single variable calculus to functions that depend on more than one variable. This branch of mathematics is crucial for analyzing situations where multiple factors influence an outcome.

For instance, if we consider the temperature in a room to be a function of both the location within the room and time, we are dealing with a multivariable function. Calculating derivatives in multivariable calculus enables us to understand how small changes in these variables affect the temperature. While in single-variable calculus we deal with the slope of the curve, in multivariable calculus we concern ourselves with slopes in different directions, known as partial derivatives. These derivatives represent the rate of change with respect to each independent variable, holding others constant.

Difference Quotient

The difference quotient is a method used to approximate the derivative of a function.

It's a fundamental concept in calculus, expressing the ratio of the difference of function values to the difference of the points at which they are evaluated. For a function of a single variable, the difference quotient is given by \( \frac{f(x + h) - f(x)}{h} \), where \( h \) is a small increment in the variable \( x \).

Application to Partial Derivatives

In the context of partial derivatives for multivariable functions, the difference quotient is used along a specific direction of a given variable while keeping the others fixed. It serves as the fundamental building block for estimating derivatives when exact values are difficult to obtain analytically.

Function Rate of Change

In calculus, the rate of change of a function gives us the speed at which the function's value is changing with respect to changes in its variables.

Single Variable Functions

For single variable functions, it is simply the derivative. It tells us, for example, how fast a car is moving if the function represents its distance over time.

Multivariable Functions

For functions of several variables, like \( f(x, y) \), we have multiple rates of change - one for each variable. These are known as partial derivatives and symbolized as \( f_x \) and \( f_y \) for the variables \( x \) and \( y \) respectively. They highlight how the function's value alters in response to slight variations in each direction.

Estimating Derivatives

Estimating derivatives is crucial when dealing with complex functions or when data points are all that we have. In the absence of an exact algebraic function, we use available data points and the concept of the difference quotient to estimate the derivative.

Finding an Approximated Rate of Change

With partial derivatives, this involves approximating how much the function changes as one variable changes and the others remain constant. Calculus provides us with techniques for estimation, and as seen in the exercise, we can use the difference quotient even in multivariable contexts to estimate the partial derivative at a specific point. Such estimations are invaluable for real-world applications where exact calculations aren't possible or practical.