1. What is a Tangent Line?

A tangent line to a curve at a given point is a straight line that just "touches" the curve at that point. It represents the instantaneous rate of change of the function at that specific point.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( f(x) \) — The original function
• \( f'(x_0) \) — The derivative of f at point x₀ (slope)
• \( x_0 \) — The point of tangency
• \( f(x_0) \) — The function value at x₀

Explanation: The equation represents a line with slope equal to the derivative at x₀ that passes through the point (x₀, f(x₀)).

3. Importance of Tangent Lines

Details: Tangent lines are fundamental in calculus for understanding derivatives, approximating functions, and solving optimization problems.

4. Using the Calculator

Tips: Enter a valid mathematical function (like "sin(x)", "x^2+3*x-2"), the point of tangency (x₀), and optionally adjust the graph range.

5. Frequently Asked Questions (FAQ)

Q1: What functions can I input?
A: The calculator supports basic operations (+, -, *, /, ^), trigonometric functions (sin, cos, tan), logarithms, and more.

Q2: How accurate is the tangent line?
A: The accuracy depends on precise calculation of the derivative. For complex functions, numerical methods are used.

Q3: Can I see both the function and tangent line?
A: Yes, the calculator graphs both the original function and its tangent line at the specified point.

Q4: What if my function isn't differentiable at x₀?
A: The calculator will notify you if the function isn't differentiable at the given point.

Q5: Can I save or export the graph?
A: Yes, you can download the graph as an image or share it.