Problem 15 - Precalculus: Mathematics for Calculus

Question

Find an equation of the tangent line to the curve at the given point. Graph the curve and the tangent line. $$y=\frac{x}{x-1}, \quad \text { at }(2,2)$$

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = -x + 4 \).

Step 1: Differentiate the Function

First, find the derivative of the function to get the slope of the tangent line at any point. The given function is \( y = \frac{x}{x-1} \). Apply the quotient rule: \[ y' = \frac{(x-1)(1) - x(1)}{(x-1)^2} = \frac{x-1-x}{(x-1)^2} = \frac{-1}{(x-1)^2} \].

Step 2: Find the Slope at the Given Point

Plug the x-coordinate of the given point into the derivative to find the slope of the tangent line at that point. The x-coordinate is 2.\[ y'(2) = \frac{-1}{(2-1)^2} = \frac{-1}{1} = -1 \].

Step 3: Use the Point-Slope Form

Now that we have the slope and a point on the tangent line (2,2), we can use the point-slope form of a line to find the equation. The point-slope form is given by: \[ y - y_1 = m(x - x_1) \], where \(m\) is the slope, and \((x_1, y_1)\) is the point. Substituting the given values, we have:\[ y - 2 = -1(x - 2) \].

Step 4: Simplify the Equation

Simplify the equation to get the equation of the tangent line in slope-intercept form \(y = mx + b\).\[ y - 2 = -x + 2 \].Add 2 to both sides:\[ y = -x + 4 \].

Step 5: Graph the Function and the Tangent Line

Plot the original function \( y = \frac{x}{x-1} \) and the tangent line \( y = -x + 4 \) on the same graph. At the point (2, 2), you will see that the tangent line just touches the curve, illustrating the concept of tangency.

Key Concepts

Quotient Rule

The quotient rule is a valuable tool used in calculus to find the derivative of a function that is the quotient of two differentiable functions. Think of a function like \( \frac{u(x)}{v(x)} \) where both \(u(x)\) and \(v(x)\) are differentiable functions. The rule itself is:\[(y') = \frac{v(x) \cdot u'(x) - u(x) \cdot v'(x)}{[v(x)]^2}\]Here's a breakdown of what each part means:

• \(u(x)\): the numerator function
• \(v(x)\): the denominator function
• \(u'(x)\): the derivative of the numerator function
• \(v'(x)\): the derivative of the denominator function

By applying this formula, you're essentially stating the derivative of a quotient is the bottom function times the derivative of the top minus the top function times the derivative of the bottom, all over the bottom squared. This rule is particularly useful when dealing with fractions in calculus, simplifying the process considerably.

Derivative

A derivative represents the rate at which a function is changing at any given point, and it is a fundamental concept in calculus. It tells us how steep the slope of the function is at a particular point or how fast something is growing or shrinking.To calculate a derivative, you use the rules of differentiation, such as the power rule, product rule, chain rule, and the quotient rule. The derivative of a function,\(f(x)\), is denoted as \(f'(x)\) or \(\frac{dy}{dx}\).Finding the derivative involves computing this rate of change, and it is essential when you're interested in understanding how something evolves over time—or, in our context, the slope of a tangent line to a curve.

Point-Slope Form

The point-slope form is a way to express the equation of a line when you know the slope and a point on the line. It's particularly convenient for finding equations of lines tangent to curves because the information at our disposal is usually a slope (from a derivative) and a specific point. The general format is given by:\[ y - y_1 = m(x - x_1) \]where:

• \(m\): slope of the line
• \((x_1, y_1)\): coordinates of the given point on the line

The point-slope form helps us quickly establish an equation without having to rearrange everything initially into the slope-intercept form. It's a straightforward method to formulate a line that passes through a given point with the slope helping guide its direction.

Slope-Intercept Form

Slope-Intercept Form is often the go-to format for expressing the equation of a straight line and is incredibly useful for graphing. The slope-intercept form is:\[ y = mx + b \]where:

• \(m\): slope of the line
• \(b\): y-intercept, the point where the line crosses the y-axis

This form makes it easy to see how a line behaves as \(x\) varies—the \(m\) value gives the line its steepness and direction, while \(b\) shows where the line starts on the graph. Transforming a point-slope equation to the slope-intercept form is simply a matter of rearranging it to solve for \(y\). Doing this allows for straightforward graphing, illustrating the relationship between \(x\) and \(y\) clearly and simply.