Numerical differentiation and integration

Measured motion often arrives as a table of positions and times rather than a formula. Numerical differentiation estimates velocity from nearby position samples. Numerical integration estimates displacement from velocity samples. Both operations approximate calculus with finite data, and their errors behave very differently.

From average slope to a derivative

For a continuous position function, instantaneous velocity is

\[v(t)=\frac{\mathrm{d}x}{\mathrm{d}t}=\lim_{h\to0}\frac{x(t+h)-x(t)}{h}\]

Sampled data cannot take the limit because the interval is finite. A forward difference uses the current and next sample:

\[v(t)\approx\frac{x(t+h)-x(t)}{h}\]

A backward difference uses the current and previous sample, while a centered difference uses one on each side:

\[v(t)\approx\frac{x(t+h)-x(t-h)}{2h}\]

For a smooth function and exact arithmetic, forward and backward differences have leading truncation error proportional to \(h\). The centered difference cancels that first-order term and has leading error proportional to \(h^2\). This advantage does not make it usable at an endpoint where a sample on one side is missing.

A derivative example

Let position be

\[x(t)=t^2\,\mathrm{m}\]

where the numerical time is expressed in seconds and the coefficient implicitly carries the units needed to make the result a length. The exact velocity at \(t=2.0\,\mathrm{s}\) is \(4.0\,\mathrm{m/s}\). With \(h=0.5\,\mathrm{s}\), the forward difference gives \(4.5\,\mathrm{m/s}\), the backward difference gives \(3.5\,\mathrm{m/s}\), and the centered difference gives the exact \(4.0\,\mathrm{m/s}\).

The centered result is exact here because the symmetric formula differentiates this quadratic without error. It is not exact for every function.

import numpy as np

times_s = np.linspace(0, 4, 41)
positions_m = 2 * times_s**2
index = 20
assert len(times_s) == len(positions_m)
assert np.all(np.diff(times_s) > 0)
velocity_m_per_s = (positions_m[index + 1] - positions_m[index - 1]) / (
    times_s[index + 1] - times_s[index - 1]
)
print(f"At {times_s[index]:.1f} s: {velocity_m_per_s:.3f} m/s")
assert np.isclose(velocity_m_per_s, 4 * times_s[index])

The denominator should come from the timestamps. Replacing it with a configured nominal interval gives a wrong slope when samples are irregular or one sample is missing.

Differentiation amplifies noise

A difference subtracts nearby position measurements and divides by a small time. If each position includes noise of typical size \(\sigma_x\), the noise contribution to a simple derivative estimate grows roughly like \(\sigma_x/h\). Reducing the interval lowers truncation error but magnifies measurement noise and floating-point cancellation.

There is therefore no universal instruction to use the smallest possible interval. Plot results across several intervals, compare with instrument resolution, and state any smoothing or fitted model used before differentiation. Smoothing changes the data and can erase real fast motion, so its window and assumptions belong in the report.

At a data boundary, choose an explicit policy: use a one-sided difference, return no estimate, or fit a local model. Copying the nearest interior derivative silently invents information.

Integration as signed area

Velocity is the rate of change of position, so displacement over an interval is

\[\Delta x=\int_{t_0}^{t_1}v(t)\,\mathrm{d}t\]

On a velocity-versus-time graph, this integral is signed area. Regions with negative velocity subtract from displacement. Integrating speed instead gives distance travelled, which is a different quantity.

For samples separated by a constant interval, the left-rectangle rule is

\[\Delta x\approx\sum_{n=0}^{N-1}v_n\Delta t\]

The trapezoid rule averages the velocities at both ends of each interval:

\[\Delta x\approx\sum_{n=0}^{N-1}\frac{v_n+v_{n+1}}{2}(t_{n+1}-t_n)\]

Writing the actual time difference supports irregular samples and makes the units visible.

from dataclasses import dataclass


@dataclass(frozen=True)
class MotionSample:
    time_seconds: float
    velocity_metres_per_second: float


def trapezoid_displacement(
    samples: list[MotionSample],
) -> float:
    displacement_metres = 0.0

    for current, next_sample in zip(samples, samples[1:]):
        interval_seconds = next_sample.time_seconds - current.time_seconds

        if interval_seconds <= 0:
            raise ValueError("sample times must increase")

        average_velocity = (
            current.velocity_metres_per_second + next_sample.velocity_metres_per_second
        ) / 2

        displacement_metres += average_velocity * interval_seconds

    return displacement_metres

An integration example

Suppose

\(v(t)=3t^2\,\mathrm{m/s}\)

from zero to \(2.0\,\mathrm{s}\). The exact displacement is

\[\int_0^2 3t^2\,\mathrm{d}t=8.0\,\mathrm{m}\]

Using trapezoids with \(h=0.5\,\mathrm{s}\) gives \(8.25\,\mathrm{m}\). The positive difference is consistent with trapezoids lying above this convex velocity curve. Refining the spacing should reduce the error quadratically while the function remains smooth and other error sources are negligible.

Bias accumulates under integration

Integration is less sensitive to high-frequency random noise than differentiation, but it accumulates bias. If a velocity sensor has a constant offset \(b\), the displacement error after duration \(T\) contains

\[\Delta x_{\mathrm{bias}}=bT\]

A small offset can therefore create a large long-duration drift. More samples do not remove that systematic error. Calibration, independent position observations, or a state-estimation model may be needed.

Missing samples create gaps whose area is unknown. Interpolating across a gap is an assumption about the unobserved velocity. Record the gap and interpolation method rather than treating the reconstructed values as measurements.

Differentiate and integrate as cross-checks

For a smooth analytic test trajectory, differentiate sampled positions and compare with the known velocity. Integrate sampled velocity and compare endpoint displacement. Then repeat with smaller intervals and observe the expected error trend.

Differentiating and then integrating sampled data does not necessarily recover the original samples exactly. Boundary choices, noise, constants of integration, and both numerical approximations matter. Integration recovers position only up to an initial position:

\[x(t)=x(t_0)+\int_{t_0}^{t}v(\tau)\,\mathrm{d}\tau\]

The starting value is not optional state.

Separate truncation, roundoff, and data error

The difference between a computed result and the desired physical quantity can have several sources. Truncation error comes from replacing a derivative or integral with a finite formula. Roundoff comes from finite arithmetic. Data error includes measurement noise, timestamp uncertainty, bias, and quantization. Modeling error appears when the quantity being processed does not represent the intended physical system.

A refinement study primarily diagnoses truncation error. Starting from one data set and interpolating it onto an increasingly fine grid does not create new measurement information. The numerical quadrature over the interpolant may converge, while the interpolant remains a poor account of what happened between observations.

For differentiation, decreasing the interval initially reduces truncation error, but subtraction of nearby noisy or rounded values eventually dominates. Plotting estimated velocity against interval size can reveal a region where results are relatively stable. That stable region is evidence for a useful scale, not a universal optimum.

For integration, compare multiple quadrature rules and refinements on a known smooth function before applying them to measurements. If results disagree more than the reported measurement uncertainty, investigate the sampling and method. If they agree far below a known sensor bias, do not claim the many matching digits as physical accuracy.

Record each component that was estimated and each one that was omitted. A total error cannot be inferred solely from the numerical method's formal order.

What to report

A numerical derivative or integral should include input units, timestamp policy, method, spacing, boundary treatment, missing-data policy, smoothing or interpolation, and an error comparison. Preserve unrounded samples and report uncertainty when the source data is measured.

These operations translate geometric ideas - slope and signed area - into array algorithms. Their apparent simplicity should not hide the decisions made between the samples.

A worked Python notebook

Differentiate a quadratic position curve and integrate its slope to recover displacement. The question for this calculation is precise: how does position change when we change time, while keeping the other choices fixed? The earlier sections explain the physical relationship. Here we turn one part of that relationship into a calculation that you can run, inspect, and change in Google Colab.

This smooth reference is deliberately free of measurement noise. Keep that sentence beside the results. A formula can return a number outside its useful range, and a plotting library will usually draw that number without asking whether the physical assumptions still hold. The selected interval is from 0 to 4 s. It gives us a specific domain to discuss instead of leaving the graph's range to a software default.

Use the notebook link above the lesson, run the setup cell, and then run the cells in order. The code below is also included there. NumPy handles the arrays of numerical values, SymPy keeps the equation symbolic, and Matplotlib draws the results. If these libraries or notebook cells are unfamiliar, start with Your first physics notebook.

State the relationship before choosing a graph

In this small example, q stands for time, expressed in s. The output is position, expressed in m. The equation below uses the numerical values in those units. Any fixed coefficients therefore carry the units needed to make the relation consistent. A value supplied in another unit must be converted before it enters this expression.

\[f(q) = 2 q^{2}\]

This is a function of one changing input. It is not a statement that the physical system has only one variable. We deliberately hold the other inputs fixed so that one cause of variation is visible. Changing several inputs at once would make it harder to tell which change produced the output difference. Write down the held-fixed choices before trying a different experiment.

At 2 s, the expression gives 8 m. This is our reference calculation. It is a prediction of the stated example, not a new measurement. Its displayed digits help us compare implementations. They should not be read as a claim that physical inputs are known to six significant figures.

import numpy as np
import sympy as sp
import matplotlib.pyplot as plt

q = sp.Symbol("q", real=True)
expression = sp.sympify("2*q**2", locals={"q": q})
reference_input = 2.0
reference_output = float(expression.subs(q, reference_input))
print("Input:", reference_input, "s")
print("Predicted position:", reference_output, "m")

sp.Symbol declares a mathematical symbol. It does not assign a numerical measurement to that symbol. subs then replaces the symbol with our chosen input, and float converts the resulting symbolic number into a standard Python numerical value. Keeping these steps separate lets us inspect the equation before numerical substitution. It also prevents a later plot from becoming the only surviving record of the model.

Compare a few cases you can explain

Before drawing hundreds of points, calculate three. The lower endpoint gives 0 m. The midpoint, at 2 s, gives 8 m. The upper endpoint gives 32 m. Read these together with the assumptions: they describe different choices of time within the same example, with the other inputs held fixed.

These three values are a useful first check, but they do not establish the shape between them. Two endpoints can be equal even when the curve bends or oscillates in between. Likewise, a large change between endpoints does not tell us whether the relationship is linear. We need the equation and intermediate samples to answer those questions.

evaluate = sp.lambdify(q, expression, modules="numpy")
inputs = np.array([0.0, 2.0, 4.0])
outputs = np.asarray(evaluate(inputs), dtype=float)
outputs = np.broadcast_to(outputs, inputs.shape)
for input_value, output_value in zip(inputs, outputs):
    print(f"{input_value:.6g} s -> {output_value:.6g} m")
assert np.all(np.isfinite(outputs))

lambdify turns the symbolic expression into a numerical function. We explicitly select NumPy as its numerical backend so it can accept an array. The same expression now has two useful forms: one for algebra and one for calculating many cases. The assertion checks that the chosen inputs produce finite outputs. It would catch an infinity or an undefined numerical result, but a finite value can still be physically inappropriate.

Make a graph that answers the question

The horizontal axis shows time; the vertical axis shows position. Every plotted point is a prediction of the same expression. We mark the reference case separately so that a reader can connect the earlier printed value to its place on the curve. The axis labels include units because a shape without a scale cannot tell us the size of a physical effect.

input_grid = np.linspace(0.0, 4.0, 301)
output_grid = np.asarray(evaluate(input_grid), dtype=float)
output_grid = np.broadcast_to(output_grid, input_grid.shape)
fig, ax = plt.subplots(figsize=(7, 4))
ax.plot(input_grid, output_grid, label="Stated model")
ax.scatter([reference_input], [reference_output], label="Reference case", zorder=3)
ax.set_xlabel("Time (s)")
ax.set_ylabel("Position (m)")
ax.legend()
ax.grid(alpha=0.25)
fig.tight_layout()
plt.show()

Look for a sign change, a turning point, or a region where the curve becomes steep. Connect each feature to the expression rather than to the colour of the line. If the curve has a minimum or maximum, changing the input in opposite directions may have similar effects on the output near that point. If it is nearly straight over a short interval, a local linear approximation may be useful even when the complete expression is nonlinear.

Increasing the number of samples makes the line smoother and may reveal features that a coarse grid missed. It does not improve the physical inputs or enlarge the model's validity range. For quantities that can take only discrete values, use the line as a guide to the relationship and interpret only the allowed input values. A plotting function does not enforce such restrictions for you.

Measure the local sensitivity

A derivative asks a more specific question than the whole graph: how quickly does position change near the reference input? Its units are the output unit divided by the input unit. For this calculation, the derivative at 2 s is 8 in those ratio units. Locally, the output increases as the input increases. This statement is local; the sign or size of the slope can be different elsewhere on the curve.

\[f'(q) = 4 q\]

For a small input change, multiply the derivative by that change to estimate the output change. Then compare that estimate with a fresh evaluation of the full expression. The distinction between these two calculations matters: the derivative gives a tangent approximation, while substitution evaluates the chosen expression at the new point.

slope_expression = sp.diff(expression, q)
slope_at_reference = float(slope_expression.subs(q, reference_input))
changed_input = 2.02
linear_prediction = reference_output + slope_at_reference * (
    changed_input - reference_input
)
full_prediction = float(evaluate(changed_input))
print("Symbolic derivative:", slope_expression)
print("Tangent prediction:", linear_prediction, "m")
print("Full expression:", full_prediction, "m")
print("Approximation difference:", full_prediction - linear_prediction, "m")

Here the changed input is 2.02 s. The tangent estimate is 8.16 m, while the full expression gives 8.1608 m. If the two values are close, it means the chosen input change is small enough for this local approximation at this point. It does not mean the entire model is linear. If the slope is zero, a quadratic or higher-order term may supply the leading change instead.

This is also the beginning of uncertainty propagation. If the input has a small stated standard uncertainty, the magnitude of the slope times that uncertainty estimates one contribution to output uncertainty. Do not automatically use the same rule for a wide interval, a discontinuity, a turning point with zero first derivative, or several correlated uncertain inputs. Those situations need more of the model than one slope.

Check the derivative with numerical differences

We can estimate the same slope from values on either side of the reference point. Subtract the lower output from the upper output and divide by twice the input step. Parentheses are essential: the entire output difference must be divided by the entire input interval. Without them, Python can execute a different mathematical expression while still producing a plausible number.

The calculation below uses three decreasing step sizes. Each estimate is compared with the symbolic derivative, so the reference is independent of the numerical difference formula. Near an endpoint, it is acceptable here to evaluate a small extension only because the displayed mathematical expression is smooth there; such a check does not assert that extrapolated values describe additional physical cases.

input_span = 4.0
for fraction in [1e-2, 1e-3, 1e-4]:
    step = input_span * fraction
    upper = float(evaluate(reference_input + step))
    lower = float(evaluate(reference_input - step))
    numerical_slope = (upper - lower) / (2 * step)
    difference = numerical_slope - slope_at_reference
    print(f"step={step:.6g}: slope={numerical_slope:.6g}, difference={difference:.6g}")

For a smooth function, reducing the step often reduces the approximation error at first. Very small steps can eventually make subtraction of nearly equal floating-point values less reliable. Therefore “smaller” is not an unlimited guarantee of “better.” Compare the observed behaviour over a useful range of steps instead of choosing the smallest number the computer can represent.

An absolute slope difference is useful when the expected slope is near zero. A relative difference would divide by a very small reference and could look enormous even when the absolute discrepancy is tiny. Away from zero, a relative difference can help compare quantities with very different scales. The tolerance should reflect the question and the numerical method, rather than a universal decimal threshold copied into every test.

Change the example and preserve the reasoning

Make one copy of the reference calculation before editing it. Choose a second value of time inside the stated interval, predict whether position will rise or fall, and then evaluate the expression. Place the prediction and the result in a text cell. If they disagree, identify whether the mistake was in the sign, the unit conversion, the algebra, or the assumption about how the curve behaves.

Next choose a shorter plotting interval around the reference point. Compare the local curve with its tangent approximation. Then widen the interval again. The difference shows why a useful local estimate need not describe the whole range. Keep the axis labels and numerical scale visible when comparing the plots; automatic axis limits can make two different effects appear equally large.

Finally, return to the physical limit stated at the start: This smooth reference is deliberately free of measurement noise. Describe one change to the setup that would violate that limit. State what new input, equation, or observation would be needed before trusting an answer in the changed situation. This last step connects the Python experiment to the physics instead of treating successful execution as the only criterion for a useful result.

Before saving, restart the runtime and run all cells in order. A result that depends on an earlier, now-deleted input cell is not reproducible from the saved notebook. Keep unrounded values for later calculations, round only the reported output, and label any new data as measured, assumed, or generated. The graph and printed values should remain traceable to the same expression and the same choices of inputs.

Recover the output change by integrating the slope

There is another way to check that the expression and its derivative describe the same relationship. Add up the small output changes predicted by the slope across the chosen input interval. In the continuous limit, this integral equals the output at the upper endpoint minus the output at the lower endpoint. The result has units of m: the input unit in the integration cancels the input unit in the denominator of the slope.

This statement is about the mathematics of this particular relationship. It does not assign a new physical meaning to an arbitrary area under a graph. For example, an area under force versus distance represents work, while an area under a different pair of quantities may represent something else or have no standard name. Always carry the axis units through the calculation before naming an integral.

slope_function = sp.lambdify(q, slope_expression, modules="numpy")
endpoint_change = float(evaluate(4.0) - evaluate(0.0))
for points in [101, 201, 401]:
    grid = np.linspace(0.0, 4.0, points)
    slopes = np.asarray(slope_function(grid), dtype=float)
    slopes = np.broadcast_to(slopes, grid.shape)
    integrated_change = np.trapezoid(slopes, grid)
    print(
        points,
        "integrated change:",
        integrated_change,
        "endpoint change:",
        endpoint_change,
    )

The trapezoidal rule joins neighbouring slope samples with straight segments and adds their signed areas. A negative contribution remains negative; replacing every slope by its magnitude would calculate a different quantity. Increasing the sample count provides a practical refinement check. If the values do not settle toward the endpoint difference, first inspect the interval, singularities, and array ordering.

Do not conclude that two agreeing calculations prove the model is physically complete. Both use the same selected relationship, so an omitted effect would be omitted from both. Their agreement is evidence that the numerical tools implement that relationship consistently. Comparing with an independent experiment or a more appropriate physical model is a separate test.

Keep this comparison in the notebook when you change the expression. It is particularly useful after adding a coefficient, changing a sign convention, or converting an input unit. A code cell that produces a smooth graph can conceal these mistakes; a second calculation of the same output change gives you another place to notice them.

Sources

• OpenStax, University Physics Volume 1, Sections 3.1-3.3, position, velocity, and acceleration.
• MIT OpenCourseWare, 8.01SC Classical Mechanics, Chapter 4, "One Dimensional Kinematics".
• Benjamin Crowell, Light and Matter, Sections 3.4-3.7, varying acceleration, graphical integration, and calculus.
• The Physics Hypertextbook, "Calculus".