Deriving Forward Euler from Tangent-Line Approximation

From a Local Slope to a Numerical Step

A differential equation specifies how a solution changes, but it does not always give the solution’s values in a convenient explicit form. A numerical method can estimate a value at a nearby time by using information available at the current time. For forward Euler, that information is the solution’s current value and its slope there.

The geometric idea is simple: near a point on a differentiable curve, the curve is approximated by its tangent line. The numerical step follows that tangent line for a chosen time increment. The approximation is local: it uses the slope at the start of the step, not the slope at the end.

We will derive the update and make precise what is discarded when the curved solution is replaced by its tangent. In particular, differentiability alone gives a first-order approximation; a bound on the solution’s second derivative gives a quantitative bound on the error in that one step.

The Tangent Line at a Known Point

Consider a scalar initial-value problem

Here \(t_n\) is the current time, \(y_n\) is the value of the solution at that time, and \(f(t_n,y_n)\) is the slope specified by the differential equation at the current point. Assume the solution is differentiable near \(t_n\). The tangent line to its graph at \((t_n,y_n)\), written as a function of time \(t\), is

The differential equation gives \(y'(t_n)=f(t_n,y_n)\). If the next time is \(t_{n+1}=t_n+h\), then the tangent line’s value at that time is

Using this tangent-line value as the approximation at the next time gives the forward Euler update.

The value \(y_{n+1}\) produced by the rule is generally an approximation, not an assertion that the exact solution has that value. To keep the distinction clear, write \(y(t_{n+1})\) for the exact solution value and \(y_{n+1}\) for the Euler approximation. Their difference measures what the tangent-line step leaves out at that step.

Why the Update Is a Tangent-Line Approximation

The derivative describes the limiting rate of change as the time increment shrinks. In precise terms, differentiability at \(t_n\) means

Thus, for small \(h\), the difference between the actual solution value and its tangent-line value is smaller than a fixed multiple of \(|h|\) in the limiting sense. The tangent-line expression is therefore the natural first-order estimate of the solution at the nearby time. Substituting \(y(t_n)=y_n\) and \(y'(t_n)=f(t_n,y_n)\) into that expression yields the forward Euler rule.

Quantifying What the Tangent Line Discards

Differentiability explains why the tangent line is a good approximation for sufficiently small steps, but it does not by itself give a numerical error bound. For a bound, suppose the solution has a continuous second derivative on the interval between \(t_n\) and \(t_n+h\), and suppose

throughout that interval, for some constant \(M\ge 0\). The exact difference between the solution and its tangent line at the end of the step is

This difference can be bounded directly. For \(h\ge 0\), the fundamental theorem of calculus gives

For each \(s\) between \(0\) and \(h\), the mean value theorem applied to \(y'\) gives \(|y'(t_n+s)-y'(t_n)|\le Ms\). Therefore

The same bound, with \(|h|\) in place of \(h\), holds for a negative step by integrating over the interval in the reverse direction. Since the Euler value is the tangent-line value, this is also a bound on the difference between the exact next value and the Euler prediction when the starting value is exact.

This bound describes the part discarded by one tangent-line step under the stated assumptions. It is not a statement about the total difference after many Euler steps: that would also depend on how errors from earlier steps affect later values.

Worked Examples

A Common Mistake: Forgetting the Step Length

The slope is a rate of change per unit of time. To convert that rate into a predicted change in the dependent variable, it must be multiplied by the length of the time step. Omitting \(h\) confuses a slope with a change in value.

For example, in the first worked example the starting value is \(2\), the starting slope is \(3\), and the step length is \(0.5\). The incorrect calculation \(y_1=2+3=5\) adds the slope directly to the value. The correct change over this step is \(0.5(3)=1.5\), so

The tangent line check confirms the correction: \(L(1.5)=2+3(1.5-1)=3.5\), not \(5\). Always include the time increment between the starting and next times.

When the Curvature Bound Is Useful

The tangent-line derivation explains the structure of the update: current value plus step length times current slope. The curvature bound adds a way to assess how far a smooth solution might bend away from that line over one step. A larger bound on \(|y''|\) allows a larger possible discrepancy; a shorter step reduces the bound in proportion to \(h^2\).

The estimate has conditions. The solution must have a second derivative on the interval, and \(M\) must bound its absolute value throughout the interval, not merely at the starting point. If those conditions are unavailable, differentiability still justifies the tangent-line approximation in the limiting sense, but the stated numerical bound cannot be claimed. These statements concern a single step begun at an exact solution value; they do not, on their own, bound errors accumulated over a sequence of steps.

Summary

At a known point \((t_n,y_n)\), the differential equation supplies the tangent slope \(f(t_n,y_n)\). Following that tangent line for a time increment \(h\) gives \(y_n+h f(t_n,y_n)\), which is the forward Euler update. If the solution’s second derivative is bounded in magnitude by \(M\) over the step, the difference between the exact next value and the tangent-line prediction is at most \(M h^2/2\). The update is therefore a local linear approximation, with its discarded part controlled by the solution’s curvature when the required smoothness and bound are available.