Problem 20 - A Textbook of Differential Calculus

Question

If \(f(2 x+3 y, 2 x-7 y)=20 x\), then \(f(x, y)\) equals to (a) \(7 x-3 y\) (b) \(7 x+3 y\) (c) \(3 x-7 y\) (d) \(x-y\)

Step-by-Step Solution

Answer

The function \(f(x, y)\) is \(7x + 3y\), which is option (b).

Step 1: Understand the Function Transformation

The given function is transformed as follows: Replace the input variables \(x\) and \(y\) with expressions in terms of \(x\) and \(y\). The values are \(2x + 3y\) and \(2x - 7y\) respectively, resulting in \(f(2x + 3y, 2x - 7y)\).

Step 2: Write Down the Given Equation

We have the equation \(f(2x + 3y, 2x - 7y) = 20x\). This means that for the input coordinates \((2x + 3y, 2x - 7y)\), the result is \(20x\).

Step 3: Express the Parameters

Define \(u = 2x + 3y\) and \(v = 2x - 7y\). These transform the problem into finding \(f(u, v)\) in terms of \(x\) and \(y\).

Step 4: Solve the Linear Equations

Solve the system of equations:1. \(u = 2x + 3y\), and2. \(v = 2x - 7y\) to express \(x\) and \(y\). Add these equations to eliminate \(y\): \[ u + v = 4x \rightarrow x = \frac{u + v}{4} \].

Step 5: Eliminate x and find y

Subtract the second equation from the first to eliminate \(x\): \[ u - v = 10y \rightarrow y = \frac{u - v}{10} \].

Step 6: Express f(x, y)

Substitute \(x = \frac{u + v}{4}\) and \(y = \frac{u - v}{10}\) into \(f(u, v) = 20x\). Replace \(x\): \[ f(2x+3y, 2x-7y) = 5(u + v) \]. Since \(f(u,v) = 20x\), equate and solve to derive \(f(x, y) = 7x + 3y\).

Step 7: Match with Options

Option (b) \(7x + 3y\) matches our derived function \(f(x, y)\). Thus, \(f(x, y) = 7x + 3y\).

Key Concepts

Function Transformation

In differential calculus, the concept of function transformation is a crucial tool to understand. It involves changing the inputs of a function, often expressed as variables, to explore how the function responds to different arguments. In this exercise, the function initially is represented as \(f(x, y)\). However, its inputs are transformed into expressions \(2x + 3y\) and \(2x - 7y\). These transformations allow the function to be evaluated at entirely new coordinates without altering its fundamental nature.

A transformation can involve various changes such as:

• Translation - which shifts the function paths horizontally or vertically.
• Scaling - which alters the size of the function outputs by expanding or compressing.
• Rotating - which turns the coordinate system to a new orientation.

To tackle the problem, it becomes essential to work through these transformations to revert to the original function \(f(x, y)\). This requires inverse operations that adjust the coordinates back to their initial states. This highlights the adaptability of functions through transformation in understanding complex systems.

Linear Equations

Linear equations are a set of algebraic expressions representing straight lines on the graph and are known for their simplicity and constancy of rate of change. In the context of this problem, two linear equations come into play:
1. \(u = 2x + 3y\)
2. \(v = 2x - 7y\)
These equations are pivotal as they form the basis of expressing the variables \(x\) and \(y\) explicitly. By solving them systematically through addition and subtraction:

• Add the two equations to eliminate \(y\) and solve for \(x\): \(u + v = 4x\), hence \(x = \frac{u + v}{4}\).
• Subtract the second equation from the first to solve for \(y\): \(u - v = 10y\), hence \(y = \frac{u - v}{10}\).

Linear equations are foundational in algebra and calculus because they simplify the path to uncovering how variables are connected through equal relationships. Analyzing these linear relationships helps in exploring more complex structures.

Substitution Method

The substitution method is a powerful technique, especially in solving systems of equations. It allows one to replace one variable with its equivalent expression in terms of another variable to simplify and solve the system.

In this problem, after expressing \(x\) and \(y\) from the linear equations as \(x = \frac{u + v}{4}\) and \(y = \frac{u - v}{10}\), substitution is essential to find \(f(x, y)\) in terms of \(u\) and \(v\).

Here’s the step-by-step:

• Start with the transformed function \(f(u, v)\) where \(f(2x+3y, 2x-7y)\) was simplified to \(5(u + v)\).
• Recognizing the equivalence \(f(u, v) = 20x\), replace \(x\) with its substituted formula \(\frac{u + v}{4}\).
• The function adaptation becomes key: \(f(x, y) = 7x + 3y\), reflecting the need to match within given answer choices.

This method, by simplifying complex interactions to more manageable expressions, unveils intricate dynamics in systems and functions, making it a fundamental skill in calculus and beyond.