Calculator Form
Enter the curve, choose a point on the x-axis, adjust graph settings, and calculate the slope of the normal line.
Example Data Table
| Curve | x Value | Point on Curve | Tangent Slope | Normal Slope | Normal Equation |
|---|---|---|---|---|---|
| y = x² | 2 | (2, 4) | 4 | -0.25 | y - 4 = -0.25(x - 2) |
| y = sin(x) | 1 | (1, 0.841471) | 0.540302 | -1.850816 | y - 0.841471 = -1.850816(x - 1) |
| y = ln(x) | 1 | (1, 0) | 1 | -1 | y = -x + 1 |
| y = x³ | 0 | (0, 0) | 0 | Vertical | x = 0 |
Formula Used
1) Point on the Curve
For a selected x-value x₀, first compute the point y₀ = f(x₀).
2) Tangent Slope
The tangent slope is the derivative at the point: mt = f′(x₀).
3) Numerical Derivative
This tool estimates the derivative with the central difference formula: f′(x₀) ≈ [f(x₀ + h) − f(x₀ − h)] / (2h).
4) Normal Slope
The normal slope is the negative reciprocal of the tangent slope: mn = −1 / mt, provided mt ≠ 0.
5) Special Case
If the tangent is horizontal, then mt = 0 and the normal line is vertical, so its equation is x = x₀.
6) Normal Line Equation
When the normal slope is finite, the equation is y − y₀ = mn(x − x₀).
How to Use This Calculator
- Enter the curve function using x as the variable, such as x^2 + 3*x - 1.
- Type the x-value where you want the normal line.
- Set a small derivative step size h for numerical differentiation.
- Choose graph minimum and maximum x-values for the chart window.
- Pick the number of graph samples and decimal places.
- Press Calculate Normal Slope to show the result above the form.
- Review the slope, equations, angles, and graph.
- Use the CSV and PDF buttons to export the result.
Frequently Asked Questions
1) What is the slope of the normal line?
It is the slope of the line perpendicular to the tangent at a chosen point on the curve. You compute it from the tangent slope by taking the negative reciprocal, when that tangent slope is not zero.
2) How is the normal slope related to the tangent slope?
If the tangent slope is m, the normal slope is −1/m. This relationship comes from perpendicular lines in analytic geometry. The product of their slopes equals −1 whenever both slopes are finite.
3) What happens when the tangent slope is zero?
A zero tangent slope means the tangent line is horizontal. The normal line then becomes vertical. Vertical lines do not have a finite slope, so the calculator shows the normal equation as x = x₀.
4) Why does this calculator use a step size h?
The derivative is estimated numerically with the central difference method. The step size h controls how closely the tool samples nearby function values. Smaller values often improve accuracy, but extremely tiny values may increase rounding noise.
5) Can I use trigonometric or logarithmic functions?
Yes. You can enter functions such as sin(x), cos(x), tan(x), ln(x), log(x), sqrt(x), and exp(x). Trigonometric expressions are evaluated in radians, and multiplication should be written explicitly, such as 2*x.
6) Why might my result differ slightly from a textbook answer?
This tool estimates the derivative numerically, so tiny differences can appear from step size, rounding, or domain restrictions. Increasing decimal places and choosing a sensible h usually brings the answer closer to the expected analytical value.
7) What does the graph show?
The chart displays the original curve, the tangent line at the chosen point, the normal line, and the plotted point itself. This visual comparison helps you confirm whether the normal line is correctly perpendicular to the tangent.
8) What do the CSV and PDF exports include?
They export the main calculation summary, including the function, point coordinates, tangent slope, normal slope, tangent equation, normal equation, and angle values. This makes it easier to save, print, or share your work.