Problem 43 - Calculus Early Transcendentals

Question

Resistors in parallel Two resistors wired in parallel in an electrical circuit give an effective resistance of \(R(x, y)=\frac{x y}{x+y},\) where \(x\) and \(y\) are the positive resistances of the individual resistors (typically measured in ohms).a. Graph the resistance function using the window \([0,10] \times[0,10] \times[0,5].\) b. Estimate the maximum value of \(R\), for \(0

Step-by-Step Solution

Answer

Answer: The symmetry of the resistance function R(x, y) implies that the effective resistance in a parallel circuit does not depend on the order in which the resistors are connected; it only depends on their individual resistances.

Step 1: Graphing the resistance function

To graph the effective resistance function \(R(x, y)=\frac{x y}{x+y}\), we can use a 3D plotting tool like Desmos or GeoGebra. We will plot the function with x ranging from 0 to 10, y ranging from 0 to 10, and R ranging from 0 to 5. This will give us a clear visual representation of how the function behaves in the specified domain.

Step 2: Estimating the maximum value of R

To find the maximum value of \(R(x, y)\) for \(0 < x \leq 10\) and \(0 < y \leq 10\), we can examine the 3D graph we created in Step 1 and make an estimate from there. By observing the graph, we can see that the maximum value of R occurs when both x and y have the same value. Since both x and y can have a maximum value of 10, we can plug in these values into the function to estimate the maximum value of R: \(R(10, 10) = \frac{10 \cdot 10}{10 + 10} = \frac{100}{20} = 5\). So, the maximum value of R is approximately 5 ohms when x and y both equal 10.

Step 3: Explaining the function's symmetry

To say that the resistance function is symmetric in x and y means that if we were to swap the values of x and y in the function, the output (i.e., the effective resistance) would remain the same. In other words, \(R(x, y) = R(y, x)\). This symmetry can be seen from the mathematical expression of the function: \(R(x, y) =\frac{x y}{x+y} =\frac{y x}{y+x} = R(y, x)\). This symmetry implies that the effective resistance in a parallel circuit does not depend on the order in which the resistors are connected; it only depends on their individual resistances.

Key Concepts

Resistor network

Resistor networks involve connecting resistors in a circuit either in series or in parallel. This exercise focuses on resistors in parallel. In a parallel arrangement:

• The total or "effective" resistance is different than if the resistors were in series.
• The formula for calculating the effective resistance R when two resistors with resistances x and y are connected in parallel is given by: \( R(x, y) = \frac{x y}{x + y} \).
• One interesting aspect of parallel resistors is that the effective resistance is always less than the resistance of the smallest resistor involved. This might seem counterintuitive, but it reflects the added paths available for current to flow.

Understanding this behavior is crucial for designing circuits that require specific resistance levels without wasting energy.

Function symmetry

Symmetry in functions is a key concept in mathematics, lending both elegance and practicality to the study of equations and graphs. For the resistance function described:

• The function is given by \( R(x, y) = \frac{x y}{x + y} \). Here, symmetry in the function means \( R(x, y) = R(y, x) \).
• This implies that swapping the resistors in their positions doesn't affect the overall resistance they produce together.
• This property is highly advantageous in electrical engineering, as it simplifies the analysis and design of circuits. Engineers can interchange components without worrying about the impact on the circuit's resistance.

This symmetry relates to the commutative property in mathematics, where the order of operations does not change the outcome.

3D graphing

3D graphing is a powerful tool for visualizing mathematical equations involving two or more variables. For this exercise on resistor networks:

• Graphing the function \( R(x, y) = \frac{x y}{x + y} \) in a 3D space helps us understand how resistance changes with different values of x and y.
• Common tools used for such graphing include software like Desmos and GeoGebra, which allow users to input ranges for variables and visualize the resulting surfaces.
• The window from 0 to 10 for both x and y, and 0 to 5 for R, provides a clear picture of the function's behavior over the given domain.

For those learning calculus and electrical engineering, this 3D perspective provides an intuitive grasp of how variables interrelate.

Maximum value estimation

Estimating the maximum value of a function is a fundamental skill in calculus. For resistor networks:

• The objective is to find the maximum of \( R(x, y) = \frac{x y}{x + y} \) given certain constraints on x and y.
• Visually inspecting the 3D graph assists in pinpointing the highest point within a specified range.
• By inspecting this graph, the maximum value appears when both resistors have equal resistance, i.e., when \( x = y \).

Plugging x and y both equal to 10 into the function gives the maximum effective resistance of 5 ohms, illustrating the usefulness of graphing techniques for approximation.