1. What is 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 tangent line equals the derivative of the function at that point.

2. How Does the Calculator Work?

The calculator finds points where the derivative of a function equals a given slope:

Where:

• \( f'(x) \) — Derivative of the function
• \( m \) — Given slope value
• \( x_0 \) — x-coordinate of tangency point

Explanation: The calculator solves the equation f'(x) = m to find all x-values where the tangent line has the specified slope.

3. Mathematical Foundation

Details: The derivative f'(x) represents the instantaneous rate of change of the function. Setting it equal to a specific slope value identifies points where the tangent line has that exact slope.

4. Using the Calculator

Tips: Enter the mathematical function using standard notation (e.g., x^2, sin(x), exp(x)) and the desired slope value. The calculator will attempt to solve f'(x) = m.

5. Frequently Asked Questions (FAQ)

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

Q2: Can a function have multiple points with the same slope?
A: Yes, many functions can have multiple points where the derivative equals the same value, especially periodic functions.

Q3: What if no solution exists?
A: The calculator will indicate that no real solutions were found for the given function and slope combination.

Q4: How accurate are the results?
A: The accuracy depends on the complexity of the function and the numerical methods used to solve the equation.

Q5: Can I use this for implicit functions?
A: Currently, the calculator is designed for explicit functions of the form y = f(x). Implicit functions require different approaches.