Taylor Mimics a Curve with a Polynomial
A curve is complicated; a polynomial is easy to handle. Taylor's idea is simple. Pick a point, then build a polynomial that matches the curve there: same height, same slope, same bending (second derivative), and so on, matching more and more derivatives. The higher the degree, the more faithfully it mimics the curve near that point. Even transcendental functions like sin, cos and ex can be written as an infinite sum of polynomial terms (a power series), and that is exactly how a calculator finds their values. But move far from the center and the approximation falls apart. The size of that breakdown is the remainder (the truncation error), and every numerical method in this bundle is, at heart, a story about fighting that error.
Raise the degree slider. Degree 1 is just the tangent line, so it leaves the curve quickly. Push it to 3, 5, 7 and the polynomial embraces the sine curve over a wider and wider stretch. The key phrase is "near the center." For any degree, the closer you are to the center (here 0) the better the fit, and toward the ends it eventually peels away, because a polynomial must blow up far out. Yet add enough terms and across the window you see, it becomes almost indistinguishable from sine.
Now stack ex one term at a time. The first term is 1 (a constant), then x, then x²/2, then x³/6 — each term is xᵏ/k!. The bars below are the size of each term at x=2: they grow at first, then shrink fast once the factorial (k!) in the denominator takes over. That is why the partial sum settles on one value. Watch T(2) creep toward the true e²≈7.39 as you add terms. A function that settles like this for every x is called analytic, and ex, sin and cos are the headline examples. The next lesson asks how far out the gathering reaches (the radius of convergence).
The red bar is the gap between sine and the polynomial — the error. Park the probe near the center 0 and the bar is almost invisible. But drag the probe toward either end and, past a certain point, the bar suddenly grows long. Raising the degree widens the well-fit zone, yet beyond it the error still explodes. This is the key instinct for using a Taylor approximation: what sets the error is not "how accurate is it" in the abstract but "how far from the center are you." That is why, in practice, you place the center close to the point you actually care about.
Take just one step of a degree-1 Taylor — the tangent line. Measure the slope at a point and ride that straight line forward by h to predict the next value. The dashed line is that prediction; the curve is the truth. When h is small the two nearly agree, but as h grows the red gap opens up. That gap is the remainder of the first-order Taylor, the truncation error. And this is exactly what Euler's method (lesson 5 of this bundle) does at every step: take the slope from a differential equation and march forward one straight step at a time. That is why approximation and numerical methods share one root — the remainder of a polynomial approximation becomes the error of a numerical solver.
Pick a function. Choose sin and set the terms to 1 and you get sin x ≈ x. That is the small-angle approximation physics always reaches for with pendulums and small oscillations, and it fits better the smaller the angle. Cosine starts as 1 − x²/2, and ex as 1 + x + x²/2. Calculators and computers evaluate sin, cos, ex and logarithms with polynomials like these (plus a little correction). That is the trick by which a machine that can only multiply and add handles transcendental functions. The principle never changes: decide the accuracy you need, then spend that many terms.