Calculator Inputs
Formula Used
The secant slope between two nearby points on a function is
msecant = [f(a + h) − f(a)] / h
To estimate the limit numerically, choose a point a, then use smaller and smaller values of h. If the computed secant slopes settle toward one number, that value is the numerical limit and usually matches the derivative at x = a.
For left-hand estimates, h is negative. For right-hand estimates, h is positive. A two-sided numerical limit is more believable when both sides approach the same value.
How to Use This Calculator
- Enter a function using
xas the variable. - Choose the target point
awhere you want the limit. - Enter a positive starting
h. - Choose a reduction factor between 0 and 1.
- Set how many steps the calculator should generate.
- Select left-hand, right-hand, or both sides.
- Press the calculate button.
- Review the result card, secant table, and graphs.
- Export the table to CSV or PDF if needed.
Example Data Table
Example for f(x) = x² at a = 2 using right-hand intervals.
| Step | h | x₂ | f(2) | f(2 + h) | Secant slope |
|---|---|---|---|---|---|
| 1 | 1 | 3 | 4 | 9 | 5 |
| 2 | 0.5 | 2.5 | 4 | 6.25 | 4.5 |
| 3 | 0.25 | 2.25 | 4 | 5.0625 | 4.25 |
| 4 | 0.125 | 2.125 | 4 | 4.515625 | 4.125 |
These slopes approach 4, which is the derivative of x² at x = 2.
Frequently Asked Questions
1) What does this calculator estimate?
It estimates the numerical limit of secant slopes as the interval width h approaches zero. That helps you approximate the derivative at a chosen point.
2) Why are smaller h values important?
Smaller h values move the second point closer to the target point. That usually makes the secant slope behave more like the local instantaneous rate of change.
3) Why compare left-hand and right-hand values?
A reliable two-sided numerical limit usually needs both sides to approach the same number. If they disagree, the derivative may fail to exist or the step settings may be too coarse.
4) What reduction factor should I use?
A factor like 0.5 is a practical starting choice because it shrinks h smoothly. Smaller factors can converge faster, but they may also increase rounding sensitivity.
5) Can I enter trigonometric or logarithmic functions?
Yes. The calculator supports common expressions such as sin, cos, tan, sqrt, abs, exp, log, and ln. Use x as the variable and standard parentheses.
6) Why might I get an evaluation error?
Errors can happen when the function is undefined near the target point, such as square roots of negative values or logarithms of nonpositive values. Syntax issues can also cause failures.
7) Does this always equal the exact derivative?
Not always. It is a numerical estimate, so the result depends on the function, the selected steps, and floating-point rounding. Good convergence usually gives a strong approximation.
8) What do the exports include?
The CSV export saves the numerical table for spreadsheets. The PDF export creates a printable report with the summary values and the secant table shown on the page.