Problem 52 - Calculus and its applications

Question

Find the simplified difference quotient for each function listed. $$ f(x)=\frac{1}{x^{2}} $$

Step-by-Step Solution

Answer

The simplified difference quotient is \( \frac{-2x - h}{x^2(x+h)^2} \).

Step 1: Write the Difference Quotient Formula

The difference quotient for a function \( f(x) \) is given by the formula \( \frac{f(x+h) - f(x)}{h} \). Our goal is to find and simplify this expression.

Step 2: Substitute the Function into the Formula

Substitute \( f(x) = \frac{1}{x^2} \) into the difference quotient formula. This gives us: \[\frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h}\]We need to simplify this expression.

Step 3: Simplify the Expression

To simplify \( \frac{\frac{1}{(x+h)^2} - \frac{1}{x^2}}{h} \), find a common denominator for the numerators:\[\frac{1}{(x+h)^2} = \frac{x^2}{x^2(x+h)^2}, \quad \text{and} \quad \frac{1}{x^2} = \frac{(x+h)^2}{x^2(x+h)^2}\]Thus, we rewrite the difference:\[\frac{x^2 - (x+h)^2}{x^2(x+h)^2} = \frac{x^2 - (x^2 + 2xh + h^2)}{x^2(x+h)^2} = \frac{-2xh - h^2}{x^2(x+h)^2}\]

Step 4: Divide by h and Further Simplify

Divide the simplified difference by \( h \):\[\frac{-2xh - h^2}{x^2(x+h)^2} \div h = \frac{-2x - h}{x^2(x+h)^2}\]This further simplifies the difference quotient.

Step 5: Write the Final Simplified Form

The simplified difference quotient is:\[\frac{-2x - h}{x^2(x+h)^2}\]This is the final form of the expression.

Key Concepts

Simplifying Rational Expressions

Rational expressions are fractions where the numerator and the denominator are polynomials. When simplifying these expressions, our aim is to transform them into their simplest form. This often includes reducing the fraction by finding a common factor or utilizing algebraic identities.

In our exercise, the difference quotient involves simplifying a rational expression where the numerators are expressed as: \( \frac{1}{(x+h)^2} - \frac{1}{x^2} \). To simplify it, both fractions must have a common denominator. This common denominator is \( x^2(x+h)^2 \).

This approach allows us to accurately combine the fractions and ultimately simplify the expression. When simplifying:

• Ensure both fractions have the same denominator.
• Factor out common terms as seen in \( x^2 - (x+h)^2 \).
• Reduce the expression wherever possible.

This process makes the difference quotient more manageable and closer to obtaining derivatives.

Calculus

Calculus is a branch of mathematics that focuses on rates of change and the study of functions. The difference quotient is a fundamental concept in calculus. It is primarily utilized to approximate the derivative of a function.

The core idea of the difference quotient is to measure how a function changes as its input changes incrementally. This is expressed through the formula: \( \frac{f(x+h) - f(x)}{h} \). The quotient analyzes the rate of change of the function, serving as a stepping stone to understanding derivatives.

• The difference quotient highlights how calculus transitions from studying stationary values to dynamic changes.
• It provides a method to approximate the derivative before limits are introduced.
• Grasping this concept is vital for further exploration when delving into topics like differential calculus and optimization problems.

Understanding calculus and the application of the difference quotient opens doors to comprehending the fundamental processes behind various physical phenomenona and technical applications.

Functions and Their Derivatives

A function is a mathematical entity that relates inputs (independents) to outputs (dependents). The derivative of a function represents its rate of change or how the output variable responds when the input variable is altered.

In our example, \( f(x) = \frac{1}{x^2} \) is the function of interest. To find its derivative, we utilize the difference quotient. By simplifying the expression \( \frac{f(x+h) - f(x)}{h} \), we get ever closer to calculating the exact derivative of the function at a point when \( h \) approaches zero.

Derivatives are essential in:

• Understanding and predicting the behavior of variables.
• Identifying trends or patterns within data.
• Modeling real-life situations where changing conditions are involved, such as velocity and acceleration in physics.

Once the difference quotient is fully simplified, as in this exercise, taking the limit as \( h \rightarrow 0 \) reveals the true derivative. This showcases how functions can behave dynamically, helping support deeper analysis in mathematics and science.