1. What is the Tangent Line Equation?

The tangent line equation \( y - y_0 = f'(x_0)(x - x_0) \) represents the line that touches a curve at exactly one point (x₀, y₀) and has the same slope as the curve at that point.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( f'(x_0) \) — Derivative of the function at point x₀
• \( x_0 \) — x-coordinate of the point of tangency
• \( y_0 \) — y-coordinate of the point of tangency (f(x₀))

Explanation: The equation calculates the line that best approximates the curve near the given point, using the derivative as the slope.

3. Importance of Tangent Line Calculation

Details: Tangent lines are fundamental in calculus for understanding rates of change, optimization problems, and linear approximations of functions.

4. Using the Calculator

Tips: Enter the function f(x), the x-coordinate x₀, and the corresponding y-coordinate y₀. The calculator will determine the tangent line equation at that point.

5. Frequently Asked Questions (FAQ)

Q1: What is the geometric interpretation of a tangent line?
A: A tangent line touches a curve at exactly one point and represents the instantaneous rate of change (slope) of the function at that point.

Q2: How is the derivative f'(x₀) calculated?
A: The derivative is calculated using limit definitions or differentiation rules, representing the slope of the tangent line at point x₀.

Q3: Can a function have multiple tangent lines at a point?
A: For smooth functions, there is exactly one tangent line at each point. Non-differentiable functions may not have a unique tangent line.

Q4: What's the difference between tangent and secant lines?
A: A secant line connects two points on a curve, while a tangent line touches the curve at exactly one point.

Q5: Are tangent lines used in real-world applications?
A: Yes, tangent lines are used in physics for instantaneous velocity, in economics for marginal analysis, and in engineering for approximation techniques.