Question

Simplify the difference quotients\(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\) for the following functions. $$f(x)=-\frac{4}{x^{2}}$$

Short answer

Question: Simplify the following difference quotients for the function \(f(x)=-\frac{4}{x^{2}}\): \(\frac{f(x+h)-f(x)}{h}\) and \(\frac{f(x)-f(a)}{x-a}\). Answer: The simplified forms of the difference quotients are: 1. \(\frac{f(x+h)-f(x)}{h} = \frac{-8x - 4h}{x^2(x + h)^2}\) 2. \(\frac{f(x)-f(a)}{x-a} = \frac{4(x+a)}{a^2x^2}\)

Step-by-step solution

Substitute the function into the first difference quotient

We need to find \(f(x+h)\) and \(f(x)\) and substitute them into the first difference quotient: 1. Substitute \((x+h)\) into the function \(f(x) = -\frac{4}{x^2}\): $$f(x + h) = -\frac{4}{(x + h)^2}$$ 2. Then, substitute these values into the first difference quotient and simplify: $$\frac{-\frac{4}{(x+h)^2} - (-\frac{4}{x^2})}{h}$$ #Step 2: Simplifying the first difference quotient#

Simplify the first difference quotient

To simplify this expression, we find a common denominator and simplify the numerator: 1. Common denominator: \((x^2(x + h)^2)\) 2. Rewrite the fractions with the common denominator: $$\frac{\frac{4x^2 - 4(x + h)^2}{x^2(x + h)^2}}{h}$$ 3. Simplify the numerator further: $$\frac{\frac{4x^2 - 4x^2 - 8xh - 4h^2}{x^2(x + h)^2}}{h}$$ 4. Cancel out the \(4x^2\) terms: $$\frac{\frac{-8xh - 4h^2}{x^2(x + h)^2}}{h}$$ 5. Divide both numerator and denominator by \(h\): $$\frac{-8x - 4h}{x^2(x + h)^2}$$ This is the simplified form of the first difference quotient. #Step 3: Substituting the function into the second difference quotient#

Substitute the function into the second difference quotient

We need to find \(f(x)\) and \(f(a)\) and substitute them into the second difference quotient: 1. Substitute \(a\) into the function \(f(x) = -\frac{4}{x^2}\): $$f(a) = -\frac{4}{a^2}$$ 2. Then, substitute these values into the second difference quotient and simplify: $$\frac{-\frac{4}{x^2} - (-\frac{4}{a^2})}{x - a}$$ #Step 4: Simplifying the second difference quotient#

Simplify the second difference quotient

To simplify this expression, we find a common denominator and simplify the numerator: 1. Common denominator: \(a^2x^2\) 2. Rewrite the fractions with the common denominator: $$\frac{\frac{4a^2 - 4x^2}{a^2x^2}}{x - a}$$ 3. Simplify the numerator further:$$\frac{\frac{4(a^2-x^2)}{a^2x^2}}{x-a}$$ 4. Factor out \(4\) from the numerator:$$\frac{\frac{4(x^2-a^2)}{a^2x^2}}{x-a}$$ 5. Use difference of squares in the numerator:$$\frac{\frac{4(x-a)(x+a)}{a^2x^2}}{x-a}$$ 6. Divide both numerator and denominator by \((x - a)\): $$\frac{4(x+a)}{a^2x^2}$$ This is the simplified form of the second difference quotient.

Key concepts

Function Simplification

Function simplification refers to the process of rewriting a function in a simpler or more efficient form without changing its value. This often involves operations like combining like terms, canceling out terms, or using algebraic identities. Simplifying functions helps in solving equations more easily and aids in clarity, making it much easier to understand the underlying relationships in the function.
For example, when working with difference quotients involving complex rational expressions, it's often necessary to simplify the function to make further calculations manageable. This is particularly helpful in calculus, where evaluating limits and understanding rate of change concepts benefit from simplified expressions.
When simplifying a complex expression, it's essential to look for possible factorization or terms that can cancel out. Keeping an eye on common denominators or algebraic identities like the difference of squares can also greatly facilitate this process.

Rational Functions

Rational functions are fractions that involve polynomials in both the numerator and the denominator. They are defined for all values of the variable that do not make the denominator zero. These functions can exhibit interesting behavior, such as vertical asymptotes, horizontal asymptotes, and holes (removable discontinuities).
To simplify or work with rational functions, it often involves finding a common denominator, which lets you combine terms under a single fraction. This can make it easy to perform operations such as addition, subtraction, multiplication, or division between rational expressions.
In the context of difference quotients, simplifying rational functions is crucial because it generally condenses the problem, allowing for straightforward limit evaluation. For example, in the first difference quotient given, we combined the expressions to get a single rational expression, making it possible to further factor and simplify.

Difference of Squares

The difference of squares is a valuable algebraic identity, used when you have an expression in the form of \(a^2 - b^2\). This identity can be factored into two binomials: \((a - b)(a + b)\). This property is commonly used in algebra to simplify expressions and solve equations.
In the original exercise, the difference of squares technique was applied to simplify the numerator of the second difference quotient. Recognizing that the difference of squares formula can be applied helped transform this part of the expression into a factorable form: \((x^2 - a^2) = (x - a)(x + a)\). This then allows for easy cancellation with similar terms in the denominator, facilitating the simplification process.
Understanding and utilizing the difference of squares can significantly simplify algebraic manipulation and can often reveal underlying factors that are crucial for solving an equation or simplifying an expression.