1. What is the Tangent Line Equation?

The tangent line equation represents a straight line that touches a curve at a single point without crossing it. The equation is derived using the point-slope form of a line and the slope between two points on the curve.

2. How Does the Calculator Work?

The calculator uses the following equations:

Where:

• \( (x_1, y_1) \) — First point on the curve
• \( (x_2, y_2) \) — Second point on the curve
• \( m \) — Slope of the tangent line

Explanation: The calculator first computes the slope between the two points, then uses the point-slope form to generate the equation of the tangent line.

3. Importance of Tangent Line Calculation

Details: Tangent lines are fundamental in calculus and physics, representing instantaneous rates of change. They're used in optimization problems, motion analysis, and curve sketching.

4. Using the Calculator

Tips: Enter the coordinates of two points on the curve. The points must be distinct (x1 ≠ x2) to avoid division by zero. The calculator will provide the slope and both point-slope and slope-intercept forms of the equation.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between tangent and secant lines?
A: A tangent touches the curve at exactly one point, while a secant intersects the curve at two points.

Q2: Can I use this for vertical tangent lines?
A: No, this calculator can't handle vertical lines (infinite slope) which occur when x1 = x2.

Q3: How accurate is the tangent line approximation?
A: The approximation is best when the two points are very close together, becoming exact as the distance approaches zero.

Q4: Can I use this for 3D curves?
A: No, this calculator only handles 2D curves. Tangent lines in 3D require vector equations.

Q5: What if my points are on a straight line?
A: The tangent line will coincide with the line itself, as the slope between any two points on a straight line is constant.