1. What is the Tangent Line Formula?

The tangent line formula calculates the equation of a line that touches a curve at a specific point without crossing it. This line represents the instantaneous rate of change of the function 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

Explanation: The equation uses the point-slope form with the derivative as the slope to find the tangent line at the specified point.

3. Importance of Tangent Line Calculation

Details: Tangent lines are fundamental in calculus for understanding rates of change, optimization problems, and approximating functions near specific points.

4. Using the Calculator

Tips: Enter the derivative value at the point, and the coordinates (x₀, y₀) of the point of tangency. All values must be valid numbers.

5. Frequently Asked Questions (FAQ)

Q1: What is a tangent line?
A: A tangent line is a straight line that touches a curve at exactly one point without crossing it, representing the instantaneous slope at that point.

Q2: How is the derivative related to the tangent line?
A: The derivative at a point gives the slope of the tangent line to the curve at that specific point.

Q3: Can this calculator handle any function?
A: This calculator requires you to provide the derivative value. For complex functions, you may need to calculate the derivative separately first.

Q4: What if the derivative is zero?
A: If the derivative is zero, the tangent line will be horizontal (slope = 0) at that point.

Q5: How accurate is the tangent line approximation?
A: The tangent line provides a linear approximation that is most accurate very close to the point of tangency.