Shrink an interval without dividing by zero
When does an average rate become a unique local rate?
Entering midway? An interval slope is output change divided by input change. A derivative is a limit of these slopes.
Start with a concrete question
Compare x² with |x| using left and right secants. The second function has a corner at zero.
Try the initial inputs, then write a prediction before changing one setting. All calculations run locally on your CPU.
Go directly to the experiment · Review prerequisite mathematics and return here
What do you expect?
Will a continuous graph always have a derivative?
Change one thing. Inspect what happens.
- For x² at x=1, shrink h and compare the two interval slopes with 2.
- Set h=0 and explain why the quotient is undefined rather than zero.
- Choose |x| at x=0; compare left and right slopes.
- Explain why averaging those opposite slopes would not prove a derivative exists.
Your settingsMechanism
Calculation belowEvidence
Compare outcomes
Curve, interval slopes and tangent where defined
Horizontal: Input x. Vertical: Function output.
- 1. Function (solid line)
- 2. Left secant (dashed line)
- 3. Right secant (dashed line)
- 4. Local tangent (dashed line)
Axes use the displayed ranges.
Read plotted values as a table
| Series | Input x | Function output |
|---|---|---|
| Function | -3 | 9 |
| Function | -2.95 | 8.703 |
| Function | -2.9 | 8.41 |
| Function | -2.85 | 8.123 |
| Function | -2.8 | 7.84 |
| Function | -2.75 | 7.563 |
| Function | -2.7 | 7.29 |
| Function | -2.65 | 7.022 |
| Function | -2.6 | 6.76 |
| Function | -2.55 | 6.502 |
| Function | -2.5 | 6.25 |
| Function | -2.45 | 6.003 |
| Function | -2.4 | 5.76 |
| Function | -2.35 | 5.523 |
| Function | -2.3 | 5.29 |
| Function | -2.25 | 5.063 |
| Function | -2.2 | 4.84 |
| Function | -2.15 | 4.622 |
| Function | -2.1 | 4.41 |
| Function | -2.05 | 4.202 |
| Function | -2 | 4 |
| Function | -1.95 | 3.802 |
| Function | -1.9 | 3.61 |
| Function | -1.85 | 3.423 |
| Function | -1.8 | 3.24 |
| Function | -1.75 | 3.063 |
| Function | -1.7 | 2.89 |
| Function | -1.65 | 2.722 |
| Function | -1.6 | 2.56 |
| Function | -1.55 | 2.403 |
| Function | -1.5 | 2.25 |
| Function | -1.45 | 2.103 |
| Function | -1.4 | 1.96 |
| Function | -1.35 | 1.823 |
| Function | -1.3 | 1.69 |
| Function | -1.25 | 1.563 |
| Function | -1.2 | 1.44 |
| Function | -1.15 | 1.322 |
| Function | -1.1 | 1.21 |
| Function | -1.05 | 1.103 |
| Function | -1 | 1 |
| Function | -0.95 | 0.9025 |
| Function | -0.9 | 0.81 |
| Function | -0.85 | 0.7225 |
| Function | -0.8 | 0.64 |
| Function | -0.75 | 0.5625 |
| Function | -0.7 | 0.49 |
| Function | -0.65 | 0.4225 |
| Function | -0.6 | 0.36 |
| Function | -0.55 | 0.3025 |
| Function | -0.5 | 0.25 |
| Function | -0.45 | 0.2025 |
| Function | -0.4 | 0.16 |
| Function | -0.35 | 0.1225 |
| Function | -0.3 | 0.09 |
| Function | -0.25 | 0.0625 |
| Function | -0.2 | 0.04 |
| Function | -0.15 | 0.0225 |
| Function | -0.1 | 0.01 |
| Function | -0.05 | 0.0025 |
| Function | 0 | 0 |
| Function | 0.05 | 0.0025 |
| Function | 0.1 | 0.01 |
| Function | 0.15 | 0.0225 |
| Function | 0.2 | 0.04 |
| Function | 0.25 | 0.0625 |
| Function | 0.3 | 0.09 |
| Function | 0.35 | 0.1225 |
| Function | 0.4 | 0.16 |
| Function | 0.45 | 0.2025 |
| Function | 0.5 | 0.25 |
| Function | 0.55 | 0.3025 |
| Function | 0.6 | 0.36 |
| Function | 0.65 | 0.4225 |
| Function | 0.7 | 0.49 |
| Function | 0.75 | 0.5625 |
| Function | 0.8 | 0.64 |
| Function | 0.85 | 0.7225 |
| Function | 0.9 | 0.81 |
| Function | 0.95 | 0.9025 |
| Function | 1 | 1 |
| Function | 1.05 | 1.102 |
| Function | 1.1 | 1.21 |
| Function | 1.15 | 1.323 |
| Function | 1.2 | 1.44 |
| Function | 1.25 | 1.563 |
| Function | 1.3 | 1.69 |
| Function | 1.35 | 1.822 |
| Function | 1.4 | 1.96 |
| Function | 1.45 | 2.103 |
| Function | 1.5 | 2.25 |
| Function | 1.55 | 2.402 |
| Function | 1.6 | 2.56 |
| Function | 1.65 | 2.723 |
| Function | 1.7 | 2.89 |
| Function | 1.75 | 3.063 |
| Function | 1.8 | 3.24 |
| Function | 1.85 | 3.422 |
| Function | 1.9 | 3.61 |
| Function | 1.95 | 3.803 |
| Function | 2 | 4 |
| Function | 2.05 | 4.202 |
| Function | 2.1 | 4.41 |
| Function | 2.15 | 4.623 |
| Function | 2.2 | 4.84 |
| Function | 2.25 | 5.063 |
| Function | 2.3 | 5.29 |
| Function | 2.35 | 5.522 |
| Function | 2.4 | 5.76 |
| Function | 2.45 | 6.003 |
| Function | 2.5 | 6.25 |
| Function | 2.55 | 6.502 |
| Function | 2.6 | 6.76 |
| Function | 2.65 | 7.023 |
| Function | 2.7 | 7.29 |
| Function | 2.75 | 7.563 |
| Function | 2.8 | 7.84 |
| Function | 2.85 | 8.122 |
| Function | 2.9 | 8.41 |
| Function | 2.95 | 8.703 |
| Function | 3 | 9 |
| Left secant | 0.5 | 0.25 |
| Left secant | 1 | 1 |
| Right secant | 1 | 1 |
| Right secant | 1.5 | 2.25 |
| Local tangent | 0.5 | 0 |
| Local tangent | 1.5 | 2 |
Follow the calculation
- Function f(x)=x² has derivative 2x.
- Current x=1, h=0.5.
- Left slope = (1.00000−0.25000)/0.5=1.50000.
- Right slope = (2.25000−1.00000)/0.5=2.50000.
A derivative is the limiting local rate. A nonzero-width secant is an interval average and can differ from it.
Captures keep your inputs, prediction and observation. Results are recalculated from those inputs. Resetting inputs keeps your captures and challenge history.
Explain the mechanism
Start with the meaning
An interval slope is output change divided by input change. A derivative is a limit of these slopes.
Compare x² with |x| using left and right secants. The second function has a corner at zero.
A worked explanation
For f(x)=x², the right difference quotient is ((x+h)²−x²)/h=2x+h when h≠0. Its limit is 2x.
At x=1 and h=0.5, left and right slopes are 1.5 and 2.5. Both approach 2 as h tends to zero, giving a unique tangent.
For |x| at zero, the left slope is −1 and the right +1. The curve is continuous but has no unique tangent slope there. A central difference returns zero and can hide this failure.
Symbols and relationship
f′(x)=lim h→0 [f(x+h)−f(x)]/h when the two-sided limit exists. Finite h gives an approximation, not the definition at h=0.
Use the graph and trace together
The graph and calculations use your current inputs. Read axis labels and units before comparing positions or slopes. Change one input, predict the effect, and use the numerical trace to verify the plotted behavior. A displayed curve is an illustration; the formula and its assumptions determine which conclusions are justified.
Scope and edge cases
- Only two elementary functions; analytic derivatives establish correctness here.
- Zero h is explicitly undefined for the quotient; floating-point cancellation for extremely tiny h is avoided by bounded controls.
Try a fresh case
For f(x)=x² at x=2, what is the right interval slope with h=0.5?
This is a public practice check. Retries support learning; matching a number does not certify independent proficiency.
Take the investigation further
Choose any extension. These are project prompts for you to carry out; they are not automatically assessed.
School investigation
Sketch a tangent and two secants, labeling their different meanings.
College investigation
Compare forward and central differences on smooth functions and nondifferentiable counterexamples.
Workplace investigation
Specify differentiability and step-size assumptions before using a numerical sensitivity estimate.
Keep the evidence, choose your next step
Completion records your own decision that you have worked through this investigation. Practice checks and independent project evidence are separate.