Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Determine the difference quotient \(\frac{f(x+h)-f(x)}{h}\) (where \(h \neq 0\) j for each function \(f\). Simplify completely. $$f(x)=x^{3}$$

Step-by-Step Solution

Answer

The simplified difference quotient is \(3x^2 + 3xh + h^2\).

Step 1: Substitute and Expand

Start with the function \(f(x) = x^3\). To find the difference quotient \( \frac{f(x+h) - f(x)}{h} \), first calculate \(f(x+h)\).Substitute in \(x + h\) for \(x\) to get:\[ f(x+h) = (x+h)^3 = x^3 + 3x^2h + 3xh^2 + h^3 \].

Step 2: Compute the Difference

Calculate \( f(x+h) - f(x) \):\[ f(x+h) - f(x) = (x^3 + 3x^2h + 3xh^2 + h^3) - x^3 \]Simplify the expression:\[ = 3x^2h + 3xh^2 + h^3 \].

Step 3: Calculate the Quotient

Now divide by \( h \):\[ \frac{3x^2h + 3xh^2 + h^3}{h} \]Factor \( h \) out of each term and then cancel it:\[ = \frac{h(3x^2 + 3xh + h^2)}{h} = 3x^2 + 3xh + h^2 \].

Step 4: Simplify the Expression

The expression \( 3x^2 + 3xh + h^2 \) is simplified and does not contain any further terms to combine or cancel. This is the difference quotient in its simplest form.

Key Concepts

Calculus

Calculus is a branch of mathematics focused on studying how things change. It's all about understanding motion and change through concepts such as derivatives and integrals. One of the important tools in calculus is the difference quotient. The difference quotient gives us a way to find the slope of the curve of a function at a particular point. It's essentially the foundation of derivatives.

In our exercise, the difference quotient \[ \frac{f(x+h) - f(x)}{h} \] helps us understand how the function’s output changes as the input changes by a small amount \( h \). As \( h \) approaches zero, the difference quotient approximates the derivative, which is the rate of change or slope of the function \( f(x) \).

• The difference quotient covers change over small intervals.
• This concept is the stepping stone to defining derivatives.
• By simplifying the expression, calculus allows us to find precise rates of change.

The exercise shows how calculus takes a cubic function \( f(x) = x^3 \) and dissects its behavior through the difference quotient, setting the stage for deeper analysis.

Functions

Functions are mathematical expressions that describe relationships between variables. When you plug in a value into a function, it outputs a result based on the rule defined by the function. Functions can come in many forms: linear, quadratic, polynomial, and more.

The function in the exercise is a polynomial function, specifically a cubic function, \( f(x) = x^3 \). Its simplicity allows us to practice evaluating the difference quotient effectively.

• A function takes an input, here \( x \), and produces an output \( f(x) \).
• We substitute a small change in input, \( x + h \), to examine behavior through \( f(x + h) \).
• This shift in input helps evaluate how the function evolves with respect to changes.

Understanding functions means grasping how they map inputs to outputs and how changes in inputs influence these outputs. In calculus, knowing these shifts prepares one to explore derivatives and integrals.

Polynomials

Polynomials are expressions consisting of variables, coefficients, and non-negative integer exponents. You can have simple monomials or more complex forms like trinomials or even higher-degree polynomials. In this exercise, the polynomial is \( f(x) = x^3 \), a simple cube function.

Polynomials are essential in many fields of mathematics due to their basic structure and because they can approximate more complex functions under certain conditions.

• Polynomial functions are generally smooth and continuous.
• The function \( x^3 \) rises sharply and becomes steeper as \( x \) increases.
• Operating on polynomials often involves expanding and simplifying, as seen in Step 1 and Step 4 of the solution.

Understanding polynomials can help in various areas like calculus, algebra, and numerical analysis, making them indispensable in both simple and advanced mathematical scenarios. They provide a clear pathway to study function behavior since they respond predictably to operations like differentiation.