Problem 69 - Brief Applied Calculus

Question

$$ \begin{array}{l} \text { For each function, find and simplify }\\\ \frac{f(x+h)-f(x)}{h} . \quad(\text { Assume } h \neq 0 .) \end{array} $$ $$ \begin{array}{l} f(x)=x^{3} \\ {\left[\text { Hint: Use } \quad(x+h)^{3}=x^{3}+3 x^{2} h+3 x h^{2}+h^{3} .\right]} \end{array} $$

Step-by-Step Solution

Answer

\(3x^2 + 3xh + h^2\)

Step 1: Apply the Difference Quotient Formula

Begin by applying the difference quotient formula to the function \( f(x) = x^3 \). This requires finding \( f(x+h) \) and substituting into the formula: \[ \frac{f(x+h) - f(x)}{h} = \frac{(x+h)^3 - x^3}{h} \]

Step 2: Expand \((x+h)^3\) Using the Given Hint

Using the hint provided, expand \((x+h)^3\):\[ (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \]

Step 3: Substitute the Expansion into the Difference Quotient

Substitute the expanded form \(x^3 + 3x^2h + 3xh^2 + h^3\) in place of \((x+h)^3\) in the difference quotient:\[ \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h} \]

Step 4: Simplify the Expression

Subtract \(x^3\) from the expression:\[ \frac{x^3 + 3x^2h + 3xh^2 + h^3 - x^3}{h} = \frac{3x^2h + 3xh^2 + h^3}{h} \]Cancel out common factor \(h\):\[ = 3x^2 + 3xh + h^2 \]

Step 5: Write the Final Simplified Form

The simplified version of the difference quotient is:\[ 3x^2 + 3xh + h^2 \]

Key Concepts

Difference Quotient

The difference quotient is a crucial concept in differential calculus, providing a formula to compute the slope of the tangent line at any given point on a curve. It's represented by the expression: \[ \frac{f(x+h) - f(x)}{h} \]Here:

• \(f(x+h)\) is the function evaluated at\( x+h \).
• \(f(x)\) is the function evaluated at \(x\).
• \(h\) is a small increment, approaching zero.

When we apply the difference quotient to a specific function, like \(f(x) = x^3\), we are looking for the rate at which the function's output changes with respect to its input, \(x\).

This formula effectively helps us to find the derivative of the function, capturing its rate of change and slope of the curve at any specific \(x\) point.

Function Simplification

Function simplification is essential in calculus to make complex expressions more manageable. To simplify a function within the context of the difference quotient, you first perform algebraic manipulations to break down the expression into simpler components.

Consider the expression:\[ \frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h} \]

• Start by canceling out common terms—in this case,\(x^3\), as it appears in both parts and thus simplifies out.
• Next, factor out units of \(h\) to simplify the division by \(h\). This leads to \(\frac{3x^2h + 3xh^2 + h^3}{h} = 3x^2 + 3xh + h^2\).

Simplifying functions in this way is pivotal for uncovering the behavior and characteristics of the curve represented by \(f(x)\).

It removes unnecessary complexity and highlights the core terms affecting the difference quotient's value.

Polynomial Expansion

Polynomial expansion plays a significant role in the process of computing and simplifying the difference quotient, particularly when dealing with powers of binomials.In this context, expanding expressions such as \((x+h)^3\) is vital. The given hint for expanding \((x+h)^3\) provides:

• \(x^3\)
• \(3x^2h\)
• \(3xh^2\)
• \(h^3\)

These terms are derived using the Binomial Theorem, which helps calculate powers of sums by expanding them into a series of terms. Each part of the expansion represents various ways \(x\) and \(h\) interact.

Understanding polynomial expansion is essential as it allows us to break down what seems complicated, revealing the nature of the polynomial's behavior when summed over a particular increment, \(h\).

This is crucial for recognizing how small changes in \(x\) become identifiable patterns, which lead to function simplification and derivative computation.