EM 125 - MATHEMATICAL ANALYSIS I (Análise Matemática I) Differential Calculus in R: The derivative of a function. Geometric interpretation of the derivative as a slope. Basic differentiation rules. Derivatives of inverse functions. The chain rule for differentiating composite functions. Applications of the chain rule. Increments, differentials and linear approximations. The mean-value theorem for derivatives. Polynomial approximations to functions: The Taylor polynomials generated by a function. Taylor’s formula with remainder. Estimates for the error in Taylor’s formula. The Taylor series as a limit of Taylor polynomials. Short study of numerical series. L’Hopital’s rule for the indeterminate form 0/0. Extension of L’Hopital’s rule. Integral Calculus in R: Riemann sums and the integral. Integrability of bounded monotonic functions. The integrability theorem for continues functions. Properties of the integral. Mean-value theorem for integrals. The derivative of an indefinite integral. The first fundamental theorem of calculus. Primitive functions and the second fundamental theorem of calculus. Integration by substitution. Integration by parts. Areas of plane regions. Polar coordinates. Area calculation in polar coordinates. Volume calculations by the method of cross sections. Integration by rational partial fractions. Rational trigonometric integrals. Integrals containing quadratic polynomials. Integral of hyperbolic functions. Improper integral. <<<<<<<<<<  BACK  <<<<<<<<<<