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Mathematics
Calculus grapher
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Shape the graph of f by dragging it directly, then watch the derivative f′, the second derivative f″ and the integral ∫f change with it; use the tangent, the signed area and the reference line to read the relations at a point, or sketch a prediction before looking at f.
Enter or Space: grab or release the curve. Arrow keys: move by 0.1 (Shift: 0.01). While grabbed, the curve follows the handle.
Hill: Drag up or down to make a smooth hill at the pointer. Press or drag anywhere on the graph of f.
Display options
On: numbers on the axes, readouts and meters; off: read by sign only (positive, negative, zero).
Variable
Derivative notation
Discontinuities
How the graphs relate
- At x = 2.50 the tangent slopes up, f′ = 1.46 > 0, so f is increasing.
- Where f′ > 0 (orange) f increases, where f′ < 0 (light blue) f decreases; at a smooth peak or valley of f, f′ = 0 and changes sign.
Readings and scale comparison
Readings at the probe
Tangent at x = 2.50
- f
- 0.73
- f′
- 1.46
Slope of tangent
210−1−2
f′(x) = 1.46Same scale as the vertical axis of f′(x).
Model notes
- f is represented by 1,251 equally spaced points on 0 ≤ x ≤ 10 (Δx = 0.008); each point is marked smooth, corner or discontinuous by the shape that made it.
- f′ is the mean of the two secant slopes next to each point (one side at the ends or next to a corner), f″ is a central difference and ∫f is a cumulative sum of trapezoids from x = 0: numerical approximations on the 0.008 grid, not symbolic expressions.
- At a corner or a jump the derivative is not defined; the simulation says 'not defined' and gives the slopes on the left and right instead of a number.
- The graph of f shows −2 ≤ y ≤ 2 but the model does not limit y: repeated tilts or shifts can move the curve off the chart. The vertical axes of f′, f″ and ∫f zoom through six levels from ±20 to ±0.5.
- Smooth is a Gaussian-weighted average (σ = 0.05); Freeform joins and smooths between pointer positions, so its result depends on how you drag.