Problem 72 - Applied Calculus

Question

For each function, find and simplify \(\frac{f(x+h)-f(x)}{h} .\) (Assume \(\left.h \neq 0 .\right)\) (See instructions on previous page.) $$ f(x)=\frac{3}{x} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding the Difference Quotient

The difference quotient of a function \( f(x) \) is given by the formula: \[\frac{f(x+h) - f(x)}{h},\] where \( h eq 0 \). We need to compute this for the function \( f(x) = \frac{3}{x} \).

Step 2: Substitute \( f(x+h) \) and \( f(x) \)

Substitute \( x+h \) into the function: \( f(x+h) = \frac{3}{x+h} \).Now, substitute \( f(x) \): \( f(x) = \frac{3}{x} \).

Step 3: Apply the Difference Quotient Formula

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

Step 4: Find a Common Denominator

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

Step 5: Simplify the Numerator

Simplify the expression by subtracting the numerators: \[\frac{3x - 3(x+h)}{x(x+h)} = \frac{3x - 3x - 3h}{x(x+h)} = \frac{-3h}{x(x+h)}\]

Step 6: Cancel \( h \)

Substitute the simplified expression back into the difference quotient:\[\frac{-3h}{x(x+h)h} = \frac{-3}{x(x+h)}\]Canceling \( h \) in the numerator and denominator provides the final simplified form.

Key Concepts

Function Simplification

Function simplification is all about rewriting mathematical expressions in a more concise or useful way, which often makes solving calculus problems easier. In this exercise, we are focusing on simplifying the difference quotient of the function \( f(x) = \frac{3}{x} \). This involves tactical algebraic manipulations.
Initially, we put \( x + h \) into our function, getting \( f(x + h) = \frac{3}{x + h} \). Then we apply the difference quotient formula: \( \frac{f(x+h) - f(x)}{h} \). The simplified form requires carefully working through algebraic steps, including handling complex fractions.

Common Denominator

When dealing with fractions, especially in calculus, finding a common denominator is a crucial step. It allows for smooth subtraction or addition of fractions, which is often necessary for solving difference quotients.
For the expression \( \frac{\frac{3}{x + h} - \frac{3}{x}}{h} \), the common denominator is \( x(x+h) \). This means rewriting each fraction to have this same denominator:

• \( \frac{3}{x+h} \) becomes \( \frac{3x}{x(x+h)} \)
• \( \frac{3}{x} \) becomes \( \frac{3(x+h)}{x(x+h)} \)

This step helps unify the fractions, making it easier to perform the next step, which is subtraction.

Fraction Subtraction

Subtracting fractions is another essential part of simplifying the difference quotient. Once the fractions are rewritten with a common denominator, you can subtract them easily.In our problem, after establishing the common denominator \( x(x+h) \), we subtract the numerators:

• \( 3x - 3(x+h) \)

This simplifies to \( 3x - 3x - 3h = -3h \). The process of simplifying fractions and combining terms thoroughly is fundamental in calculus as it frequently enables further reduction or cancellation of terms.

Calculus Problems

The difference quotient is central to understanding calculus problems, specifically derivatives. In this problem, the simplified difference quotient function represents a preliminary step toward finding the derivative of \( f(x) = \frac{3}{x} \).Cancelling terms is a common goal: after subtracting, we reach \( \frac{-3h}{x(x+h)} \). From this, the \( h \) cancels each other out when divided by the denominator's \( h \):

• This results in \( \frac{-3}{x(x+h)} \)

This final expression shows the slope of the function at any point and is crucial for understanding rates of change in calculus. It's an example of breaking down complex calculus problems into smaller, manageable algebraic steps.