Problem 85 - Algebra and Trigonometry Real Mathematics, Real People

Question

Expand the expression in the difference quotient and simplify. \(\frac{f(x+h)-f(x)}{h}, h \neq 0\) \(f(x)=x^{3}\)

Step-by-Step Solution

Answer

After expansion and simplification, the difference quotient for \(f(x) = x^3\) results in \(3x^2 + 3xh + h^2\)

Step 1: Expand \(f(x+h)\) and \(f(x)\)

For \(f(x) = x^3\), compute \(f(x + h) = (x + h)^3\), resulting in \(x^3 + 3x^2h + 3xh^2 + h^3\). Then, subtract \(f(x) = x^3\) to obtain \((x^3 + 3x^2h + 3xh^2 + h^3) - x^3\).

Step 2: Simplify the expression

Simplify the expression to reduce: \(3x^2h + 3xh^2 + h^3\) . Now, this expression must be divided by \(h\).

Step 3: Divide by \(h\)

Divide the above expression by \(h\) to get: \(3x^2 + 3xh + h^2\)

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations (such as addition, subtraction, multiplication, and division). In the context of the difference quotient \(\frac{f(x+h)-f(x)}{h}, h eq 0\), we handle the expression \(f(x) = x^{3}\) as an algebraic expression where \(x\) is the variable.

When working with algebraic expressions, especially polynomial functions like \(x^{3}\), it's crucial to understand how to correctly apply operations like expansion, which uses the distributive property to multiply variables and constants. For example, expanding \(f(x+h)\) requires us to cube the binomial \(x+h\) to get \(x^3 + 3x^2h + 3xh^2 + h^3\), as demonstrated in the original problem.

The key is to perform these operations step by step, ensuring each part of the expression adheres to the fundamental rules of algebra. In the difference quotient, these algebraic portions represents the behavior of the function as \(h\) approaches zero, which is a cornerstone of calculus and its interpretation of rates of change.

Simplifying Expressions

The art of simplifying expressions is to reduce them to their most basic form without changing their value. Simplification often includes combining like terms, which in the given exercise, involves eliminating the \(x^{3}\) term after subtracting \(f(x)\) from \(f(x+h)\) because it appears in both expressions but with opposite signs, hence it cancels out.

Then, you are left with \(3x^2h + 3xh^2 + h^3\). Here, each term shares a common factor of \(h\), which allows further simplification. It's like breaking down a complex machine into its basic components to understand how each part contributes to the whole. By dividing each term by \(h\), you eliminate the common factor, resulting in a simplified form \(3x^2 + 3xh + h^2\). This simplified expression is now ready for further analysis or computation, such as evaluating limits, another core concept in calculus.

Remember, simplification often involves a keen eye for detail and recognizing the appropriate algebraic rules that apply to the terms within the expression.

Polynomial Functions

Polynomial functions are expressions comprised of several terms with variables raised to non-negative integer exponents. The function \(f(x) = x^{3}\) in our exercise is an example of a polynomial function, specifically, a cubic function because the highest exponent is 3.

These functions are fundamental building blocks in algebra and calculus because of their defined structure and continuous nature. When we calculate the difference quotient for a polynomial function, we're investigating the function's instant rate of change or slope at a particular point. In relation to our original problem, by applying the difference quotient to the cubic function, we discern how small changes in \(x\) (\(h\) being the small change) influence the output of \(f(x)\).

Moreover, polynomial functions have various characteristics that define their shape and long-term behavior, such as the leading coefficient and the degree of the polynomial. These features determine the end behavior of the graph, the number of possible real roots, and the function's symmetry. In the realm of calculus and beyond, understanding polynomial functions is essential when interpreting graphs and predicting trends in real-world applications.