Problem 72 - Precalculus with Limits

Question

In Exercises 67 - 72, expand the expression in the difference quotient and simplify. \( \dfrac{f\left(x + h\right) - f\left(x\right)}{h} \quad \quad \) Difference quotient \( f(x) = \dfrac{1}{x} \)

Step-by-Step Solution

Answer

The simplified form of the difference quotient \(\dfrac{f\left(x + h\right) - f\left(x\right)}{h}\) for the function \(f(x) = \dfrac{1}{x}\) is \(\dfrac{-1}{x*(x + h)}\)

Step 1: Substituting into the function

We begin by substituting \(x + h\) and \(x\) into \(f\), which gives us \(f(x + h) = \dfrac{1}{x + h}\) and \(f(x) = \dfrac{1}{x}\).

Step 2: Applying the difference quotient formula

Next, we substitute these values of \(f(x + h)\) and \(f(x)\) into the difference quotient formula to get: \(\dfrac{f(x + h) - f(x)}{h} = \dfrac{\dfrac{1}{x + h} - \dfrac{1}{x}}{h}\).

Step 3: Simplifying the expression

We simplify the expression by multiplying the numerator and denominator by \(x(x + h)\), which is the least common multiple of \(x\) and \(x + h\). This gives us: \(\dfrac{x - (x + h)}{h * x*(x + h)} = \dfrac{-h}{h * x*(x + h)}\). Canceling out \(h\) from the numerator and denominator, we find the simplified expression to be: \(\dfrac{-1}{x*(x + h)}\)

Key Concepts

Calculus

Calculus is a branch of mathematics that studies how things change. It's like a mathematical telescope and microscope; it helps us zoom in and out on the behavior of functions to understand motion, growth, and areas under curves. In calculus, one of the foundational tools is the difference quotient, which measures the rate at which a function's value changes as its input changes.

This concept is crucial when you're dealing with more complicated scenarios, such as finding the instantaneous rate of change of a function at a certain point, which leads us to derivatives. In our current exercise, we take the first steps towards this concept by finding out how the function changes over a small 'h' interval. It sets the stage for understanding the core principles of calculus and how functions behave on a deeper level.

Limits

The idea of limits is central to calculus and involves approaching a certain value closely but not necessarily reaching it. Limits can be thought of as the behavior of a function as the input gets closer to a particular value. In many cases, this behavior can reveal the function's value at a point where it might otherwise be undefined.

For instance, with the difference quotient, when we want to find out the derivative of a function—which is what the difference quotient represents as 'h' approaches zero—we're effectively taking a limit. In layman's terms, we're asking, 'What happens to our expression as 'h' gets smaller and smaller?' Understanding limits is crucial for figuring out many key aspects of functions in calculus, such as continuity, derivatives, and integrals.

Simplifying Expressions

Simplifying expressions is part of the bread and butter of any calculus work. It involves rewriting an expression into a more manageable or aesthetically pleasing form without changing its value. When you're dealing with rational functions, such as the one in our exercise, simplifying often requires finding a common denominator and then reducing the resulting expression.

As seen in the solution, to simplify the difference quotient, we multiplied the numerator and denominator by the least common multiple of the denominators in our expression. By doing so, we eliminate the complex fraction and make the expression much less daunting to work with. Simplification is incredibly important, not just for making an expression look 'nice,' but also for making further mathematical processes, like taking limits, more straightforward.

Rational Functions

Rational functions are ratios of two polynomials—just like fractions are ratios of numbers. In the exercise, we're given a simple rational function, which models various real-world phenomena, such as rates and proportional relationships.

In the world of calculus, rational functions can be intriguing to work with because they may have points where they are not defined (such as division by zero) or where their behavior changes dramatically. This is why simplifying expressions involving rational functions can be so valuable—it helps us get to the heart of what the function is all about and understand its behavior across its entire domain, especially as 'x' approaches significant values, or 'h' approaches zero in the case of the difference quotient.