1. What is the Point of Tangency?

The point of tangency is the point where a line touches a curve without crossing it. At this point, the slope of the curve (given by its derivative) equals the slope of the line.

2. How Does the Calculator Work?

The calculator solves the equation:

Where:

• \( f'(x) \) — Derivative of the function f(x)
• \( m \) — Slope of the tangent line

Explanation: The calculator finds the x-coordinate where the derivative of the function equals the slope of the given line, then calculates the corresponding y-coordinate using the original function.

3. Importance of Finding Points of Tangency

Details: Points of tangency are crucial in optimization problems, curve sketching, physics (instantaneous rates of change), and engineering applications where precise contact points are required.

4. Using the Calculator

Tips: Enter the function f(x) using standard mathematical notation (e.g., x^2, sin(x), exp(x)). Enter the line equation in the form y=mx+b. The calculator will find where the function's derivative equals the line's slope.

5. Frequently Asked Questions (FAQ)

Q1: What types of functions can I input?
A: The calculator supports polynomial, trigonometric, exponential, and logarithmic functions using standard mathematical notation.

Q2: What if there are multiple points of tangency?
A: The calculator will return all solutions found. Some functions may have multiple points where the derivative equals a given slope.

Q3: How accurate are the results?
A: Results are calculated with high precision using symbolic differentiation and equation solving algorithms.

Q4: Can I use this for implicit functions?
A: Currently, the calculator is designed for explicit functions y=f(x). Implicit functions require different approaches.

Q5: What if no point of tangency exists?
A: The calculator will indicate that no solution was found if the line's slope never equals the function's derivative.