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 and has the same slope as the curve at that point. In Desmos, this equation helps visualize the tangent line to functions.

2. How Does the Calculator Work?

The calculator uses the tangent line equation:

Where:

• \( m \) — Slope of the tangent line (derivative at point x₀)
• \( x \) — x-value where you want to evaluate the tangent line
• \( x_0 \) — x-coordinate of the point of tangency
• \( y_0 \) — y-coordinate of the point of tangency (f(x₀))

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

3. Importance of Tangent Lines

Details: Tangent lines are fundamental in calculus for understanding derivatives, approximating functions, and analyzing instantaneous rates of change.

4. Using the Calculator

Tips:

• Enter the slope (m) which you can find using the derivative of your function
• Provide the point of tangency (x₀, y₀)
• Enter the x-value where you want to evaluate the tangent line

5. Frequently Asked Questions (FAQ)

Q1: How do I find the slope (m) for my function?
A: The slope is the derivative of your function evaluated at x₀. In Desmos, you can use the derivative notation (e.g., f'(x₀)).

Q2: Can I use this for any function?
A: Yes, as long as the function is differentiable at the point of tangency.

Q3: Why does the calculator show two forms of the equation?
A: The point-slope form shows the relationship to the point of tangency, while the simplified form is easier to graph.

Q4: What if my tangent line is vertical?
A: Vertical lines have undefined slope and require a different approach (x = x₀).

Q5: How accurate is this calculator?
A: It's mathematically precise for the given inputs, but remember the tangent is only a good approximation near the point of tangency.