Ep 29: Every Smooth Function Is Secretly a Polynomial | Taylor and Maclaurin Series

Linear approximation said: near a point, a smooth function f looks like a line --- f(a) + f'(a) (x - a). Quadratic approximation said: add the curvature term, \tfrac{1}{2} f''(a) (x - a)^2. Today we don't stop. We add the cubic, the quartic, the n-th order term, and keep going forever. The result is the \emph{Taylor series} of f at a --- an infinite polynomial that, for the functions you care about, equals f exactly. This is calculus' grand reveal. The exponential e^x, the sine, the cosine, the logarithm --- every transcendental function you've ever met is, in a precise sense, just a polynomial. Infinite, yes, but with coefficients given by a formula so clean it fits on one line: f^{(n)}(a)/n!. We'll derive the recipe, unroll the famous series, approximate e^{0.1} to five decimal places by hand, state Taylor's theorem with its remainder, and trace the history from Madhava in 14th-century Kerala to Brook Taylor's 1715 treatise. Next episode formalizes \emph{where} these series converge. 📚 Continues in: → 'Power Series and the Radius of Convergence' --- once you know a function has a Taylor series, the next question is: for which x does it actually equal f(x)? Next episode. 🎓 Best for: Calculus 1, 2, and 3 students, AP Calculus AB / BC, anyone reviewing before an exam, and self-learners who want a single coherent course instead of a pile of disconnected YouTube videos. — QED Realized — Clean explanations of the ideas that take a real lecture to land. 🔔 Subscribe so you don't miss the next one:    / @qedrealized   #Calculus #Math #Mathematics #STEM #TaylorSeries #MaclaurinSeries #Calculus2 #IntegralCalculus #PolynomialApproximation — Tags (for search) — taylor series tutorial, maclaurin series tutorial, taylor polynomial of order n, derive taylor series e^x, derive maclaurin series sin x, maclaurin series cos x, geometric series 1 over 1 minus x, ln(1+x) maclaurin series, taylor's theorem with remainder, lagrange remainder formula, taylor series at a equals 0, approximating e to the 0.1 by hand, polynomial approximation of smooth functions, calculus 2 taylor series, AP Calculus BC taylor, MIT taylor series, brook taylor 1715, colin maclaurin 1742, madhava kerala series, james gregory series, linear approximation to all orders, why taylor series works ⏱ Chapters 0:00 Functions as infinite polynomials 1:19 From Episode 10: the line that matches f at a 2:24 Matching all derivatives at a 4:01 Let N \to \infty 5:22 e^x = \sum x^n / n! 6:57 Derivatives cycle: 0, 1, 0, -1, \dots 8:35 Derivative of \sin x --- or directly 10:03 \sum x^n revisited 11:29 Integrate the geometric series 13:09 P_1, P_3, P_5, P_7 near 0 14:31 Five decimal places from four terms 16:17 How good is P_N? --- a formula for the error 18:07 Madhava, Gregory, Newton, Taylor, Maclaurin 19:24 Recap 21:09 Outro 22:03 Thanks for watching