Problem 72 - Brief Applied Calculus

Question

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{3}{x} $$

Step-by-Step Solution

Answer

The simplified difference quotient is \( \frac{-3}{x(x+h)} \).

Step 1: Write Difference Quotient Formula

The problem requires calculating the difference quotient of a function. The difference quotient is given by \( \frac{f(x+h) - f(x)}{h} \). We need to find \( f(x+h) \) first.

Step 2: Substitute into the Function

Start by substituting \( x + h \) into the function \( f(x) = \frac{3}{x} \). This gives us \( f(x+h) = \frac{3}{x+h} \).

Step 3: Set Up the Difference Quotient

Now substitute \( f(x+h) \) and \( f(x) \) into the difference quotient formula:\[ \frac{f(x+h) - f(x)}{h} = \frac{\frac{3}{x+h} - \frac{3}{x}}{h}. \]

Step 4: Simplify the Numerator

To simplify the expression, find a common denominator for the fractions within the numerator:\[ \frac{3}{x+h} - \frac{3}{x} = \frac{3x - 3(x+h)}{x(x+h)} = \frac{3x - 3x - 3h}{x(x+h)} = \frac{-3h}{x(x+h)}. \]

Step 5: Simplify the Whole Expression

Insert the simplified numerator into the difference quotient and cancel \( h \) in the numerator and denominator:\[ \frac{-3h}{h \cdot x(x+h)} = \frac{-3}{x(x+h)}. \] Thus, the simplified form of the difference quotient is \( \frac{-3}{x(x+h)} \).

Key Concepts

Understanding Rational Functions

Rational functions are a fascinating type of mathematical function. Simply put, they are the ratio of two polynomial functions. A typical rational function can be written in the form \( \frac{p(x)}{q(x)} \), where \( p(x) \) and \( q(x) \) are polynomials, and \( q(x) \) is not zero.
For example, the function \( f(x) = \frac{3}{x} \) used in the problem is a classic illustration of a rational function. Here, the numerator is the constant polynomial 3, and the denominator is the linear polynomial \( x \).

• Why Are Rational Functions Important? They are everywhere in real life! From physics to economics, whenever we model relations that involve rates or fractions, rational functions can be a vital tool.
• What Do We Learn from Them? They teach us about behavior near points where the denominator approaches zero – these are known as vertical asymptotes where the function value can shoot towards positive or negative infinity.

Understanding their behavior and how to manipulate them is essential for tackling more complex mathematical problems.

The Art of Simplifying Expressions

Simplifying expressions is a key skill in mathematics because it helps to reveal the foundational beauty and simplicity hidden within complex formulas. In our problem, when simplifying the difference quotient formula, we must perform several algebraic manipulations:
By substituting \( x+h \) and then simplifying the fractions involved in the expression \( \frac{\frac{3}{x+h} - \frac{3}{x}}{h} \), we reach the form \( \frac{-3h}{h \cdot x (x+h)} \). The goal is to make the expression as clear and manageable as possible.

• Cancel Common Terms: One of the most powerful methods is cancelling out common terms like \( h \) in the numerator and denominator.
• Find Common Denominators: This helps rearrange complex fractions into a single fraction, making it easier to simplify.

Through simplification, you can uncover the typically hidden logic and structure within a mathematical expression.

Rate of Change and Differentiation

The rate of change is central to understanding how functions behave, especially when variables are interconnected as they are in calculus. The difference quotient \( \frac{f(x+h) - f(x)}{h} \) is a conceptual tool used to approximate the rate of change or the slope of the function \( f(x) \) at any given point.
By analyzing the expression, we can determine how small changes in \( x \) (specifically \( h \)) affect \( f(x) \). This becomes the backbone of differentiation in calculus, leading to finding the derivative which tells us the exact rate of change at any particular point.

• Understanding Through Visualization: Imagine zooming in on a curve; at infinitesimally small intervals, the curve appears straight, and the slope of this line is the derivative.
• Applications: The rate of change tells us how quickly or slowly something is changing, which is crucial in fields like physics (speed), biology (population growth), and economics (trends over time).

Mastering the difference quotient not only aids in solving mathematical exercises but also enhances the understanding of how the world changes and adapts.