Problem 81 - 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. $$f(x)=1-x^{2}$$

Step-by-Step Solution

Answer

The simplified difference quotient is \(-2x - h\).

Step 1: Identify the Function and Expression

You are given the function \( f(x) = 1 - x^2 \). Your task is to find the difference quotient \( \frac{f(x+h) - f(x)}{h} \).

Step 2: Evaluate \(f(x+h)\)

Substitute \( x+h \) into the function \( f(x) \) to find \( f(x+h) \):\[ f(x+h) = 1 - (x+h)^2 = 1 - (x^2 + 2xh + h^2) \]Simplify further to get: \[ f(x+h) = 1 - x^2 - 2xh - h^2 \]

Step 3: Compute \(f(x+h) - f(x)\)

Subtract \( f(x) = 1 - x^2 \) from \( f(x+h) = 1 - x^2 - 2xh - h^2 \):\[ f(x+h) - f(x) = (1 - x^2 - 2xh - h^2) - (1 - x^2) \]This simplifies to:\[ f(x+h) - f(x) = -2xh - h^2 \]

Step 4: Formulate the Difference Quotient

Place \( f(x+h) - f(x) \) over \( h \) to form the difference quotient:\[ \frac{f(x+h) - f(x)}{h} = \frac{-2xh - h^2}{h} \]

Step 5: Simplify the Difference Quotient

Factor \( h \) out from the numerator:\[ \frac{-2xh - h^2}{h} = \frac{h(-2x - h)}{h} \]Cancel \( h \) from numerator and denominator (since \( h eq 0 \)):\[ -2x - h \]

Step 6: Provide the Simplified Form

The simplified form of the difference quotient is:\[ -2x - h \]

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions that involve sums of powers of variables. A typical polynomial function can be written as:

• The function has coefficients and variables like \( a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \),
• where \( a_n, a_{n-1}, ..., a_1, a_0 \) are constants.

In our exercise, the polynomial is \( f(x) = 1 - x^2 \).This is a quadratic polynomial because the highest power of \( x \) is \( 2 \).

Working with polynomial functions often involves tasks such as finding roots or simplifying expressions like the difference quotient. Understanding polynomial functions is critical as they appear in many areas of mathematics and science.

Simplifying Expressions

Simplifying expressions involves reducing them to their most concise and manageable form. This often means:

• Identifying common factors,
• Using proper algebraic techniques, and
• Eliminating terms where possible.

The key to successfully simplifying expressions is practice and familiarity with the rules and operations involved. In the exercise, simplifying the expression occurred in several places, such as when expanding \((x+h)^2\) into \(x^2 + 2xh + h^2\), then subtracting and factoring to simplify the difference quotient further. Reducing such complex expressions to simpler forms helps us understand and work with mathematical concepts more effectively.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations or expressions using algebraic principles. Techniques include:

• Expanding expressions using distributive property (e.g., \((x+h)^2\) to \(x^2 + 2xh + h^2\)),
• Combining like terms,
• Factoring,
• Canceling common terms,
• Rewriting in an equivalent convenient form.

In the given problem, effective algebraic manipulation allowed for the simplification of the difference quotient \(\frac{-2xh - h^2}{h}\) to \(-2x - h\).Understanding algebraic manipulation is a foundational skill in algebra that facilitates solving equations, interpreting graphs, and working through algebraic procedures.