1. What is the Equation of Tangent?

The equation of the tangent line to a curve at a given point is a straight line that "just touches" the curve at that point and has the same slope as the curve at that point. The general form is:

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( f'(a) \) — Derivative of the function at point a (slope)
• \( a \) — x-coordinate of the point of tangency
• \( f(a) \) — y-coordinate of the point of tangency
• \( x \) — Independent variable
• \( y \) — Dependent variable

Explanation: The equation represents a linear approximation of the function near point a, using the function's derivative (slope) at that point.

3. Importance of Tangent Line Calculation

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

4. Using the Calculator

Tips: Enter the derivative (slope) at point a, the x-coordinate (a), and the y-coordinate (f(a)). The calculator will provide both point-slope and slope-intercept forms.

5. Frequently Asked Questions (FAQ)

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

Q2: How is this useful in Desmos?
A: You can plot both your original function and its tangent line to visualize how well the tangent approximates the function near point a.

Q3: Can I use this for any differentiable function?
A: Yes, as long as you know the derivative at point a, this equation works for any differentiable function.

Q4: What if I don't know the derivative?
A: You'll need to calculate the derivative first using differentiation rules or numerical methods.

Q5: How accurate is the tangent approximation?
A: It's most accurate very close to point a. The approximation error increases as you move farther from a.