Taylor Series
Build a better guess, one layer of motion at a time.
Single player · 2DLoading…
Function + approximation
Blue: function · Orange dashed: approximation · Click to inspect
Build your approximation
At the inspection point
- Terms / degree
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- Function f(x)
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- Approximation T(x)
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- Absolute error
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The layers in your guess
| Term | Car analogy | Derivative at a | Contribution at x |
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Why the car analogy works
Each derivative describes how the previous layer changes at one instant. Each term f⁽ⁿ⁾(a)(x − a)ⁿ/n! turns that layer into a position contribution. One term means position only; four terms reach the cubic layer. Zero coefficients still count as terms.
Speed here means signed velocity. The gas pedal is an analogy: in a simplified car whose acceleration is proportional to pedal depression, pedal speed corresponds to jerk, foot acceleration to the fourth derivative, and each next layer is one more rate of change. A real engine is more complicated.
Taylor polynomials are local guesses. More terms need not improve the approximation far from a, especially outside the convergence interval. Trigonometric inputs are radians. Shaded regions are outside the open convergence interval; curves leaving the vertical plot range are clipped.