Problem 32 - Calculus Single Variable

Question

A function \(f\) and a point \(P\) are given. Find the slope-intercept form of the equation of the tangent line to the graph of \(f\) at \(P\). $$ f(x)=2 x-2 / x \quad P=(-1 / 2,3) $$

Step-by-Step Solution

Answer

The equation of the tangent line is \( y = 10x + 8 \).

Step 1: Find the derivative of the function

To find the slope of the tangent line, we need the derivative of the function. The function given is \( f(x) = 2x - \frac{2}{x} \). The derivative, \( f'(x) \), is found using the power rule and the quotient rule. The derivative is: \[ f'(x) = 2 + \frac{2}{x^2}. \]

Step 2: Substitute the x-coordinate of P into the derivative

We need the slope of the tangent line at the point \( P(-\frac{1}{2}, 3) \). Substitute \( x = -\frac{1}{2} \) into \( f'(x) \) to find the slope (\[ f'(-\frac{1}{2}) = 2 + \frac{2}{(-\frac{1}{2})^2} = 2 + \frac{2}{\frac{1}{4}} = 2 + 8 = 10. \]

Step 3: Use the point-slope form to find the tangent line equation

The point-slope form of a line is \( y - y_1 = m(x - x_1) \), where \((x_1, y_1)\) is the given point \(P\) and \(m\) is the slope. Substituting the point \((-\frac{1}{2}, 3)\) and the slope \(10\) gives:\[ y - 3 = 10(x + \frac{1}{2}). \]

Step 4: Convert to slope-intercept form

The slope-intercept form is \( y = mx + b \). Simplify the equation from Step 3:\[ y - 3 = 10x + 5, \]\[ y = 10x + 8. \]

Key Concepts

Understanding Derivatives

A derivative is the key to understanding how a function behaves as its input changes. In simpler terms, it tells us how much the output of the function changes for a small change in the input. Derivatives are often denoted by \( f'(x) \) for a function \( f(x) \). It’s the fundamental tool we use to find the slope of the tangent line at any point on the graph of a function. This instantaneous rate of change is what gives us the direction of the function at a specific point. For our exercise, the function is \( f(x) = 2x - \frac{2}{x} \). Finding its derivative involves using the power rule, which applies to terms like \( 2x \), and the quotient rule, which is used when dealing with ratios like \( \frac{2}{x} \). The derivative turns out to be \( f'(x) = 2 + \frac{2}{x^2} \). Remember:

• Use power rule \(\frac{d}{dx}(x^n) = nx^{n-1}\) for terms like \(2x\).
• Apply quotient rule \(\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v{\cdot}u' - u{\cdot}v'}{v^2}\) for terms like \(\frac{2}{x}\).

Understanding derivatives is crucial as it forms the backbone of calculus, helping us explore slopes and tangent lines, like the one at point \( P(-\frac{1}{2},3) \).

Slope-Intercept Form Explained

The slope-intercept form of a line is one of the most familiar ways to express a linear equation. It is written as \( y = mx + b \), where \( m \) is the slope of the line, and \( b \) is the y-intercept - the point where the line crosses the y-axis.When we talk about the slope, \( m \), we are referring to how steep the line is. A larger absolute value of \( m \) indicates a steeper line.
It's the number that multiplies \( x \) and tells us how much \( y \) changes with a tiny change in \( x \).In the given exercise, after finding the slope which is \( 10 \) at point \( P \), we began with the point-slope form:

• \( y - y_1 = m(x - x_1) \)
• Substituting the given point and slope, \( y - 3 = 10 (x + \frac{1}{2}) \)
• Simplifying gives the final slope-intercept form: \( y = 10x + 8 \).

Converting to slope-intercept form is beneficial because it provides a quick reference for the behavior and position of the line relative to the axes.

Exploring Function Analysis

Function analysis involves examining various characteristics of a function, like its behavior, shape, and the relationships between its input and output.When we analyzed the function \( f(x) = 2x - \frac{2}{x} \), we started by looking at the derivative to understand its rate of change.The immediate value of this analysis is seen in finding the tangent line at \( P \). Although the function seems simple, analyzing it lets us break down the roles each term plays.

• The term \( 2x \) shows a straightforward linear increase.
• \( -\frac{2}{x} \) adds complexity by creating a hyperbolic curve.

These combine into a unique function graph.Analyzing a function like this helps with

• Understanding its intercepts and asymptotic behavior.
• Determining intervals of increase or decrease.
• Finding critical points.

Through this deeper analysis, we are able to predict how changes in \( x \) will influence the function, making predictions and calculations about the tangent line incredibly insightful and accurate.