Match derivatives with a polynomial

Try

The tangent approximation to e^x at zero is 1+x. It matches the function’s value and first derivative there. Add a quadratic term ax² and choose a so the second derivative also matches. Since the new polynomial’s second derivative is 2a and e^0=1, the required coefficient is 1/2.

Predict

Predict the coefficient of x³ by differentiating ax³ three times at zero. The factor is 3!, not 3. Continue this reasoning for a general nth power before using the Taylor formula. Then predict whether a polynomial centered at 1 should use powers of x or powers of x−1.

Experiment

Live experiment: approximation with a justified remainder

Approximate eˣ using Pₙ(x)=1+x+x²/2!+…+xⁿ/n!. Predict whether the same degree gives a smaller error near zero or near an endpoint, then compare x=0.5 with x=2.

Evaluation point x Polynomial degree n 012346810Reset experiment

Polynomial: 2.5. eˣ: 2.718282. Polynomial minus eˣ: -0.218282.

Every derivative of eˣ is eˣ. Between0 and x, its magnitude is at most e^|x|. Taylor's theorem therefore gives |error| ≤ e^|x| · |x|^(n+1)/(n+1)!, bounded above by 0.453047 after upward decimal rounding.

Inspect terms and the factorial

Terms, from degree0 through degree2: 1 + 1 + 0.5. Their signs are included. The remainder uses (2+1)! = 6, not n!.

Displayed values are rounded to six decimal places, so a displayed zero error need not mean exact equality. The derivative bound guarantees the mathematical truncation error for the exact polynomial. Browser calculations also have floating-point roundoff, which this bound does not include; at very small errors roundoff can dominate. A close-looking graph alone supplies neither guarantee. This experiment concerns eˣ, whose Taylor series converges everywhere; other functions can have finite convergence intervals. Settings are temporary.

The live investigation uses e^x. Change degree while holding the target input fixed, then move the target farther from zero. Inspect each term as well as the total approximation. The following derivative table is a separate worked construction for sin x, with radians assumed.

Derivative order at zero	Value	Polynomial contribution
0	0	0
1	1	x
3	−1	−x³/6
5	1	x⁵/120

The omitted even-order values are zero. Write the derivative cycle to justify those omissions; the absence of an even power is not an assumption that only odd powers are allowed in Taylor polynomials.

Understand

The degree-n Taylor polynomial centered at a is Pₙ(x)=Σₖ₌₀ⁿ f^(k)(a)(x−a)^k/k!, provided these derivatives exist. A Maclaurin polynomial is the special case a=0. Successive differentiation at the center isolates each coefficient, because lower-degree terms vanish after enough derivatives and higher-degree terms still contain a factor x−a. This establishes the matching coefficients.

For e^x at zero all derivative values are 1, giving 1+x+x²/2+x³/6 and so on. For e^(2x), the kth derivative value at zero is 2^k, so the cubic coefficient is 8/6=4/3. For ln x centered at 1, derivative values through order three are 0,1,−1,2. The resulting polynomial is (x−1)−(x−1)²/2+(x−1)³/3. The logarithm’s domain and the chosen center both matter.

The polynomial matches local derivatives, but its finite-step error requires separate analysis. A degree-2 approximation to e^x at x=1/2 is 13/8, not exactly e^(1/2). The live component displays the actual error for its known teaching function; for a different function an exact value might not be available, making a theorem-based bound especially useful.

Even infinitely many matching derivatives do not automatically establish equality with an infinite series. The smooth function e^(−1/x²) away from zero, extended by zero at zero, has all derivatives zero at the center but positive values elsewhere. The fact about all derivatives can be proved by bounding an exponential against reciprocal powers; it is presented here as a counterexample whose full proof belongs to analysis. Its Taylor series is identically zero and does not represent the function off the center.

Apply

Construct the practice polynomials from derivative values and factorial denominators. Label the center and degree. A polynomial with a highest nonzero fifth-degree term can also serve as a sixth-degree Taylor polynomial when that coefficient vanishes; the error theorem still needs the degree being invoked to be stated clearly.

Continue

Next use Taylor’s remainder theorem to justify finite accuracy and, for suitable functions, prove equality with the infinite series. Local matching and global error claims require different evidence.