Fundamental ideas of analysis
The ideas and methods of mathematics, long central to the physical sciences, now play an increasingly important role in a wide variety of disciplines. Analysis provides theorems that prove that results are true and provides techniques to estimate the errors in approximate calculations. The ideas and methods of analysis play a fundamental role in ordinary differential equations, probability theory, differential geometry, numerical analysis, complex analysis, partial differential equations, as well as in most areas of applied mathematics.
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approximate Bolzano-Weierstrass theorem calculate called Cauchy sequence choose code words complete complex numbers compute continuous function continuously differentiable continuously differentiable function converges to zero converges uniformly countable decimal expansion define definition denote density derivative difference quotient differential equation Dom(f equal estimate Euler's method Example exists Figure finite interval fn(x formula Fourier series func Fundamental Theorem given graph Hint hypothesis implies improper Riemann integral iterates lim sup limit point linear Mean Value Theorem metric space Newton's method one-to-one partial sums partition piecewise continuous pointwise Poisson random variable polynomials power series probability proof of Theorem properties Prove radius of convergence real numbers Riemann integral Riemann sum satisfies Section sequence of functions sequence of real series converges solution subinterval subset sup norm Suppose Taylor's theorem Theorem of Calculus tion triangle inequality upper bound