1. What is the Tangent Line Equation?

The tangent line to a curve at a given point is the straight line that "just touches" the curve at that point. The equation of the tangent line can be found using the point-slope form of a line.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( m \) — Slope of the tangent line
• \( x_0 \) — x-coordinate of the point of tangency
• \( y_0 \) — y-coordinate of the point of tangency
• \( x \) — Independent variable
• \( y \) — Dependent variable

Explanation: The equation represents a line with slope \( m \) that passes through the point \( (x_0, y_0) \).

3. Importance of Tangent Line Calculation

Details: Tangent lines are fundamental in calculus and are used to approximate functions locally, find instantaneous rates of change (derivatives), and solve optimization problems.

4. Using the Calculator

Tips: Enter the slope (m), the point of tangency (x0, y0), and the x-value where you want to evaluate the tangent line. The calculator will provide the equation in three forms: point-slope, expanded, and slope-intercept.

5. Frequently Asked Questions (FAQ)

Q1: How is the slope (m) of the tangent line determined?
A: The slope is typically found by taking the derivative of the function at the point of tangency.

Q2: What's the difference between tangent line and secant line?
A: A secant line connects two points on a curve, while a tangent line touches the curve at exactly one point.

Q3: Can a tangent line intersect the curve at other points?
A: Yes, a tangent line may intersect the curve elsewhere, but it only "touches" at the point of tangency.

Q4: What if the slope is infinite?
A: This indicates a vertical tangent line, which has an equation of the form x = constant.

Q5: How accurate is the tangent line approximation?
A: The approximation is most accurate very close to the point of tangency and becomes less accurate as you move further away.