Question

Simplify the difference quotients\(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) for the following functions. $$f(x)=x^{3}-2 x$$

Short answer

Question: Simplify the two difference quotients for the function \(f(x)=x^3-2x\): \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\). Answer: After simplification, we have: 1. \(\frac{f(x+h)-f(x)}{h} = 3x^2+3xh+h^2-2\) 2. \(\frac{f(x)-f(a)}{x-a} = x^2+ax+a^2-2\)

Step-by-step solution

Evaluate f(x+h) and f(a)

In order to simplify the expressions, we first need to find \(f(x+h)\) and \(f(a)\). For \(f(x)=x^3-2x\), we have: 1. \(f(x+h)=(x+h)^3-2(x+h)=(x^3+3x^2h+3xh^2+h^3)-(2x+2h)\). 2. \(f(a)=a^3-2a\).

Compute f(x+h)-f(x) and f(x)-f(a)

Next, we have to find the difference between the function values \(f(x+h) - f(x)\) and \(f(x) - f(a)\). 1. \(f(x+h)-f(x) = (x^3+3x^2h+3xh^2+h^3)-(2x+2h) - (x^3-2x) = 3x^2h+3xh^2+h^3-2h\). 2. \(f(x)-f(a) = (x^3-2x) - (a^3-2a) = x^3 - a^3 - 2(x-a)\).

Simplify both difference quotients

Finally, we can simplfy the two difference quotients by plugging in the expressions for \(f(x+h)-f(x)\) and \(f(x)-f(a)\): 1. \(\frac{f(x+h)-f(x)}{h} = \frac{3x^2h+3xh^2+h^3-2h}{h} = \frac{h(3x^2+3xh+h^2-2)}{h}\). Cancelling out the common factor \(h\) in the numerator and denominator, we get: \( \frac{f(x+h)-f(x)}{h} = 3x^2+3xh+h^2-2\). 2. \(\frac{f(x)-f(a)}{x-a} = \frac{x^3 - a^3 - 2(x-a)}{x-a} = \frac{(x-a)(x^2+ax+a^2-2)}{x-a}\). Cancelling out the common factor \((x-a)\) in the numerator and denominator, we get: \( \frac{f(x)-f(a)}{x-a} = x^2+ax+a^2-2\).

Key concepts

Polynomial Functions

Polynomial functions are expressions that involve powers of the variable. In the given exercise, the function is given by \( f(x) = x^3 - 2x \).
This is a cubic polynomial because the highest power of the variable \(x\) is 3. Polynomial functions can be understood as the sum of monomials, which are individual terms that consist of a constant multiplied by a variable raised to an integer power.
Breaking down this specific polynomial:

• The term \( x^3 \) is the cubic term, contributing to the cubic nature of the function.
• \( -2x \) is a linear term because the power of \( x \) is 1.

Polynomial functions show up frequently in algebra, not only in theoretical exercises but also in real-world applications, like modeling curves or optimizing areas. To master polynomial functions, it is crucial to understand the properties of exponents and how different terms interact as you manipulate the function.

Simplification

Simplification is the process of reducing an expression to its simplest form. This process is indispensable in mathematics, making expressions easier to work with or analyze.
In this exercise, we simplify difference quotients to understand the behavior of the polynomial function near a point. Difference quotients, like \( \frac{f(x+h) - f(x)}{h} \) and \( \frac{f(x) - f(a)}{x-a} \), are often encountered when studying changes in functions or in calculus as definitions of derivatives.
Simplifying these expressions involves:

• Combining like terms to lower the expression's complexity.
• Canceling common factors from the numerator and denominator, which often helps to reveal the function's gradient or rate of change.

For example, remove the common factor \( h \) in the expression \( \frac{3x^2h+3xh^2+h^3-2h}{h} \) to simplify to \( 3x^2+3xh+h^2-2 \). This step reduces computational complexity and aids in conceptual understanding.

Algebraic Manipulation

Algebraic manipulation is the art of rewriting expressions in different, yet equivalent forms. In the context of this exercise, we perform algebraic manipulation by finding the differences \( f(x+h) - f(x) \) and \( f(x) - f(a) \).
Effective manipulation involves being adept with:

• Expansion: Using binomials like \( (x+h)^3 \) and expanding them to \( x^3 + 3x^2h + 3xh^2 + h^3 \).
• Factoring: Such as recognizing \( (x-a)(x^2+ax+a^2-2) \) can simplify factoring and canceling.
• Canceling: Simplifying expressions by removing terms common across fractions.

Algebraic manipulation enables you to transform complicated expressions into simpler ones, facilitating problem-solving. This skill is fundamental in calculus and other advanced math topics, where expressing functions in their simplest form aids in easier calculation and interpretation.