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 and has the same slope as the curve at that point. The equation is derived from the point-slope form of a line.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( f'(x_0) \) — Derivative of the function at point x₀ (slope)
• \( x_0 \) — X-coordinate of the point of tangency
• \( f(x_0) \) — Y-coordinate of the point of tangency
• \( x \) — X-value where you want to evaluate the tangent line

Explanation: The equation gives the y-value of the tangent line at any given x, using the slope of the curve at the point of tangency.

3. Importance of Tangent Lines

Details: Tangent lines are fundamental in calculus for approximating functions, finding instantaneous rates of change, and solving optimization problems.

4. Using the Calculator

Tips: Enter the derivative at the point, the coordinates of the point of tangency, and optionally an x-value to evaluate the tangent line.

5. Frequently Asked Questions (FAQ)

Q1: What if I don't know the derivative?
A: You'll need to calculate the derivative of your function first. This calculator requires the derivative value at the point of tangency.

Q2: Can this calculator find the derivative for me?
A: No, this calculator only computes the tangent line equation once you provide the derivative value.

Q3: What's the difference between secant and tangent lines?
A: A secant line connects two points on a curve, while a tangent line touches the curve at exactly one point with matching slope.

Q4: Can I use this for implicit functions?
A: Yes, as long as you can determine the derivative (slope) at the point of tangency.

Q5: How accurate is the tangent line approximation?
A: It's most accurate near the point of tangency. Accuracy decreases as you move further from x₀.