Problem 81 - College Algebra with Modeling and Visualization

Question

Complete the following for the given \(f(x)\) (a) Find \(f(x+h)\) (b) Find the difference quotient of \(f\) and simplify. $$ f(x)=x^{3} $$

Step-by-Step Solution

Answer

(a) \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\); (b) Difference quotient: \(3x^2 + 3xh + h^2\).

Step 1: Substitute for f(x+h)

To find \(f(x+h)\), substitute \(x+h\) into the function \(f(x)\). Since \(f(x) = x^3\), replace \(x\) with \(x+h\): \[f(x+h) = (x+h)^3\]

Step 2: Expand (x+h)^3

We need to expand \((x+h)^3\). Use the binomial theorem to expand the expression:\[(x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3\]

Step 3: Find the Difference Quotient

The difference quotient is given by:\[\frac{f(x+h) - f(x)}{h}\]Substitute \(f(x+h) = x^3 + 3x^2h + 3xh^2 + h^3\) and \(f(x) = x^3\):\[\frac{(x^3 + 3x^2h + 3xh^2 + h^3) - x^3}{h}\]

Step 4: Simplify the Expression

Subtract \(x^3\) from \(x^3 + 3x^2h + 3xh^2 + h^3\): \[\frac{3x^2h + 3xh^2 + h^3}{h}\]Factor out \(h\) from the numerator:\[\frac{h(3x^2 + 3xh + h^2)}{h}\]Cancel \(h\) from the numerator and the denominator:\[3x^2 + 3xh + h^2\]

Step 5: Finalize the Difference Quotient

The simplified difference quotient, which is derived from the previous step after cancelling \(h\), is:\[3x^2 + 3xh + h^2\]

Key Concepts

Function Substitution

Function substitution is a straightforward yet essential technique in mathematics, especially when working with functions. When given a function like \( f(x) = x^3 \), and asked to find \( f(x+h) \), you are essentially replacing the variable \( x \) with the expression \( x+h \). This substitution allows you to explore how the function behaves when its input is altered by a small amount \( h \).
To perform this substitution:

• Identify the expression for function substitution, such as \( f(x+h) \).
• Replace every instance of \( x \) in the function with \( x+h \). For instance, if you have \( f(x) = x^3 \), substitute to get \( f(x+h) = (x+h)^3 \).

This step sets the stage for further calculations, such as finding the difference quotient. Always make sure you execute the substitutions correctly, as any mistake here can throw off your entire calculation.

Binomial Theorem

The Binomial Theorem provides a fast way to expand expressions that are raised to a power, such as \((x+h)^3\). It expresses this expansion as a sum of terms involving coefficients, powers of \( x \), and powers of \( h \).
For an expression like \((x+h)^3\), you expand it using:

• First, recognize that \( (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \).
• Here, each term combines different powers of \( x \) and \( h \) multiplied by a binomial coefficient. These coefficients can be understood using Pascal's triangle or calculated as "n choose k" from combinatorial mathematics.

The Binomial Theorem simplifies the process of expansion and is crucial for evaluating polynomials and working with expressions in algebra and calculus.
Understanding this theorem reduces computational burden and helps clarify why each term appears in the result.

Simplification of Expressions

Simplifying expressions is the process of rewriting them in the most reduced form. This is especially useful when dealing with difference quotients. Once you calculate \( f(x+h) - f(x) \) and set up the difference quotient \( \frac{f(x+h) - f(x)}{h} \), it becomes necessary to simplify.
Here’s how simplification unfolds:

• Start by simplifying the numerator. In this example, combine like terms after substitution: \( 3x^2h + 3xh^2 + h^3 \).
• Factor out \( h \) from the numerator: \( h(3x^2 + 3xh + h^2) \).
• Cancel \( h \) in the numerator and the denominator, if \( h eq 0 \), since you've factored it out: resulting in \( 3x^2 + 3xh + h^2 \).

By simplifying, you reveal the structure of the expression and ensure it is in the most usable form, which is helpful for subsequent calculations and interpretations in calculus.