Problem 25 - Precalculus

Question

Find the limit of the difference quotient of the given function to obtain a function that represents the slope of a line drawn tangent to the curve at \(x.\) $$f(x)=\frac{2}{x-1}$$

Step-by-Step Solution

Answer

The derivative is \( f'(x) = \frac{-2}{(x-1)^2} \).

Step 1: Define the Difference Quotient

First, identify the difference quotient formula for a function \( f(x) \). The difference quotient is given by: \[ \frac{f(x+h) - f(x)}{h} \] where \( h \) is a small increment. For our function \( f(x) = \frac{2}{x-1} \), we substitute as follows: \[ \frac{\frac{2}{x+h-1} - \frac{2}{x-1}}{h} \].

Step 2: Simplify the Numerator of the Difference Quotient

To simplify the numerator \( \frac{2}{x+h-1} - \frac{2}{x-1} \), find a common denominator: \((x+h-1)(x-1)\). Thus, the expression becomes: \[ \frac{2(x-1) - 2(x+h-1)}{(x+h-1)(x-1)} \]. Simplifying the numerator, we have: \[ 2x - 2 - 2x - 2h + 2 = -2h \].

Step 3: Simplify the Full Difference Quotient

Now substitute the simplified numerator back into the difference quotient: \[ \frac{-2h}{h((x+h-1)(x-1))} \]. The \( h \) in the numerator and denominator cancels out, giving: \[ \frac{-2}{(x+h-1)(x-1)} \].

Step 4: Take the Limit as h Approaches Zero

Find the limit of the simplified difference quotient as \( h \) approaches zero: \[ \lim_{h \to 0} \frac{-2}{(x+h-1)(x-1)} = \frac{-2}{(x-1)^2} \].

Step 5: Conclude with the Derivative

The limit obtained represents the derivative of the function \( f(x) = \frac{2}{x-1} \), which describes the slope of the tangent line to the curve at any point \( x \). Thus, the derivative is \( f'(x) = \frac{-2}{(x-1)^2} \).

Key Concepts

Limit of a Function

The limit of a function is a fundamental concept in calculus that helps us understand the behavior of functions as close as possible to a particular point. Imagine you've got a road that narrows down to a single lane. The limit is like watching how cars merge onto that single lane from both sides as they get closer to the bottleneck. In mathematical terms, limits help us analyze what value a function gets closer to, as the input approaches some point. For example, if given the function \( f(x) = \frac{2}{x-1} \), to find the slope of the tangent line, we compute the limit of the difference quotient as \( h \) approaches zero. This essentially allows us to find how a function behaves at an infinitely small difference from the point \( x \).

• Limits provide the basis for defining derivatives.
• They assist in understanding function continuity.
• Set the stage for more advanced concepts like integrals in calculus.

Tangent Line

In mathematics, a tangent line is a straight line that "just touches" a curve at a given point. If you think of a roller coaster track, the tangent line at any peak or dip would be like a short, straight piece of track just touching the curve. This line suggests the slope or direction the track is going at that specific point. To find the equation of a tangent line at a point \( x \) on a curve, such as for our function \( f(x) = \frac{2}{x-1} \), you need to determine the derivative at that point, which gives the slope of this line.

• The slope of the tangent line indicates the rate of change at the point of contact.
• Tangent lines are essential in calculus for understanding instant changes in functions.
• Finding the tangent line aids in approximating function values close to a known point.

Derivative

The derivative of a function tells us the function's rate of change with respect to an independent variable. Picture driving a car and glancing at your speedometer – the reading gives you the car's speed, which is the rate at which your position changes over time. In calculus, the derivative functions similarly, providing us the "speed" or slope of the function at a given point. For the function \( f(x) = \frac{2}{x-1} \), the derivative is found by using the limit of the function's difference quotient, resulting in \( f'(x) = \frac{-2}{(x-1)^2} \). This derivative tells us how sharply the curve is rising or falling at any specific point \( x \).

• Derivatives are used to find tangent lines to curves.
• Applications include optimization, motion analysis, and determining function behavior.
• Derivatives help us understand and model real-world phenomena mathematically.

Rational Function

Rational functions are expressions created by dividing two polynomials. It's like a fraction, but with polynomials in the numerator and denominator. A simple example is \( f(x) = \frac{2}{x-1} \), which is the function we're examining here. These functions can display a variety of behaviors like asymptotes, where the function approaches a line but never actually touches it. They are often used in mathematical modeling to represent scenarios where division-based ratios are involved in describing phenomena.

• Rational functions are versatile in representing complex relationships.
• They can have discontinuities, which are points where the function is not defined.
• Understanding their properties is crucial for graphing and solving real-life problems.