91Ó°ÊÓ

To begin understanding tangent lines, imagine a point touching a curve only at one specific place. This touching line is your tangent line. It represents the slope or steepness of the curve at that point.
When we talk about tangent lines in functions, we refer to drawing them on the graph at given x-coordinates. In our example, for the function \( f(x) = \frac{1}{x} \), we aim to draw tangent lines at \(-2, 0\), and \(1\).
Suppose you plot the graph and choose a point like \(x = -2\). Approximating the slope there will give you the right tangent. Likewise, do it for other points.
It's crucial to understand tangent lines visually to help with grasping how derivatives behave and change across different parts of a function.

The derivative measures how a function changes as its input changes. The limit definition of the derivative tells us how to find it exactly.
For a function \(f(x)\), the derivative is found using the formula:
\[ f^{\text{prime}} (x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
This formula is crucial because it calculates the precise rate of change or slope of the function at any point. Let's break it down: The value \(h\) represents a tiny change in \(x\).

• When \(x\) shifts by \(h\), the corresponding shift in \(f(x)\) gets measured.
• The smaller \(h\) becomes, the closer \(f(x+h) - f(x)\) approximates the exact change rate at \(x\).
  Keep this in mind for understanding the behavior of functions closely.

Calculating derivatives follows from the limit definition, but let's simplify processes used in practice. For the function \(f(x) = \frac{1}{x}\), let's derive its derivative step by step:
Start with the limit definition:
\[ f^{\text{prime}}(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \]
Substitute \(f(x) = \frac{1}{x}\):
\[ f^{\text{prime}}(x) = \lim_{h \to 0} \frac{\frac{1}{x+h} - \frac{1}{x}}{h} \]
Simplify the fraction within the limit:
\[ f^{\text{prime}}(x) = \lim_{h \to 0} \frac{\frac{x - (x+h)}{x(x+h)}}{h} = \lim_{h \to 0} \frac{-h}{h \cdot x \(x+h\)} = \lim_{h \to 0} \frac{-1}{x \(x+h\)} \]
Finally, solve the limit:
\[ f^{\text{prime}}(x) = \frac{-1}{x^2} \]
This result tells us how \(f(x) = \frac{1}{x}\) changes at any given \(x\) point. After that, calculate specific values like \(f^{\text{prime}}(-2), f^{\text{prime}}(0)\), and \(f^{\text{prime}}(1)\).
Note: \(f^{\text{prime}}(0)\) is undefined since \(-1\) divided by \(0\) is not possible.

Graphing functions helps visualize their behavior and properties. Let's consider \(f(x) = \frac{1}{x}\).
This particular function is a hyperbola, a unique type of curve with two branches. To accurately graph it, remember some key points:
- \(x = 0\) is a vertical asymptote, meaning the function approaches this line but never intersects it.
- Similarly, \(y = 0\) is a horizontal asymptote.

• Sketch the curve by drawing two parts approaching these asymptotes.
• This visualization aids in understanding how values rapidly change near \(x = 0\).

Practicing graphing various functions can immensely deepen your understanding of their dynamics.
For our example, adding tangent lines at \(-2, 0, 1\) on your graph will enhance your insight into how derivatives represent slopes at specific points.