Problem 73 - Brief Applied Calculus

Question

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

\(\frac{-2x - h}{(x + h)^2 x^2}\)

Step 1: Find the Expression for f(x + h)

Start by replacing every instance of \(x\) in the function \(f(x) = \frac{1}{x^2}\) with \(x + h\). This gives us \(f(x + h) = \frac{1}{(x + h)^2}\).

Step 2: Write the Difference Quotient

Using the formula \(\frac{f(x+h)-f(x)}{h}\), substitute the expressions we have for \(f(x + h)\) and \(f(x)\): \[\frac{f(x+h)-f(x)}{h} = \frac{\frac{1}{(x + h)^2} - \frac{1}{x^2}}{h}\] Now we have the difference of two fractions in the numerator.

Step 3: Simplify the Numerator

Find a common denominator for the two fractions in the numerator. The common denominator is \((x + h)^2 x^2\). Thus, rewrite the expression as: \[\frac{x^2 - (x + h)^2}{(x + h)^2 x^2}\] Keep in mind we're focusing on simplifying the expression within the numerator. Start expanding \((x + h)^2\): \[(x + h)^2 = x^2 + 2xh + h^2\] Substitute back into the expression: \[x^2 - (x^2 + 2xh + h^2) = x^2 - x^2 - 2xh - h^2 = -2xh - h^2\]

Step 4: Express the Difference Quotient as a Simplified Fraction

Substitute the simplified numerator into the whole difference quotient: \[\frac{-2xh - h^2}{h(x + h)^2 x^2}\] Factor out \(h\) from the numerator: \[\frac{h(-2x - h)}{h(x + h)^2 x^2}\] Cancel the \(h\) in the numerator and the \(h\) in the denominator, simplifying to: \[\frac{-2x - h}{(x + h)^2 x^2}\]

Step 5: Final Simplified Expression

The simplified form of the expression \(\frac{f(x+h)-f(x)}{h}\) results in: \[\frac{-2x - h}{(x + h)^2 x^2}\] This is the final result of the exercise in its simplest form.

Key Concepts

Calculus Problem-Solving

When tackling calculus problems, especially finding the difference quotient, you're essentially exploring how a function behaves as it changes. This involves analyzing small changes in the input to see how they affect the output. The difference quotient is a crucial first step in understanding derivatives and represents the average rate of change of the function over an interval.To solve these problems, begin by correctly identifying the function in question. Then, replace all occurrences of the variable with the expression \(x + h\). This shows the function's value at a point slightly away from \(x\). Next, plug this new function into the difference quotient formula, which is:

• \(f(x + h)\)
• Subtract \(f(x)\)

Place the result over \(h\), ensuring to simplify correctly. Understanding each step helps you predict how the function behaves as \(x\) changes by an infinitesimal amount.

Rational Functions

Rational functions feature prominently in calculus and involve expressions that are ratios of polynomials. In our exercise, the function \(f(x) = \frac{1}{x^2}\) is a simple rational function. These functions are defined everywhere except where the denominator equals zero, which creates vertical asymptotes, leading to undefined values.Understanding rational functions is key in calculus problem solving because:

• They often appear in real-world applications such as physics and engineering.
• Their algebraic properties help in simplifying complex expressions.
• They require manipulation like finding a common denominator for adding or subtracting fractions, as seen in our solution.

By mastering operations on rational functions, especially simplification techniques, you become adept at handling various calculus challenges.

Algebraic Simplification

Algebraic simplification is vital when working with difference quotients. This process involves making expressions simpler without changing their value, allowing for easier interpretation and problem solving. In our solution, we started with two fractions in the numerator of the difference quotient. The key step was finding a common denominator to combine these fractions into a single expression. Simplifying involved:

• Expanding powers, such as \((x + h)^2 = x^2 + 2xh + h^2\).
• Canceling common terms to simplify \(x^2 - (x^2 + 2xh + h^2)\) to \(-2xh - h^2\).
• Factoring out similar terms, like factoring out \(h\) from \(-2xh - h^2\).

After simplification, we can clearly see how the expression changes, producing a more manageable form of the result \(\frac{-2x - h}{(x + h)^2 x^2}\). This prepares us for further calculus explorations, such as finding derivatives.