Linear approximation and differentials Limits and one-sided limits1.1 Limit laws and algebraic methods1.2 Squeeze theorem and trigonometric limits1.3 Infinite limits, limits at infinity and asymptotes1.4 Continuity and discontinuities1.5 Intermediate value theorem1.6 Problems1.7 Definition and geometry of the derivative2.1 Differentiation rules2.2 Chain rule2.3 Derivatives as functions and higher derivatives2.4 Problems2.5 Trigonometric derivatives3.1 Inverse functions and inverse trigonometric derivatives3.2 Exponential, logarithmic and hyperbolic functions and their derivatives3.3 Implicit differentiation3.4 Logarithmic differentiation3.5 Problems3.6 Linear approximation and differentials4.1 Related rates4.2 Extreme values and critical numbers4.3 Rolle’s theorem and mean value theorem4.4 Increasing, decreasing and concavity4.5 Indeterminate forms and l’Hôpital’s rule4.6 Curve sketching4.7 Applied optimization4.8 Problems4.9 Antiderivatives and indefinite integrals5.1 Area and Riemann sums5.2 Definite integrals and their properties5.3 Fundamental theorem of calculus: differentiating accumulation5.4 Fundamental theorem of calculus: evaluating definite integrals5.5 Basic integration formulas5.6 Problems5.7 Substitution6.1 Integration by parts6.2 Trigonometric substitution6.3 Problems6.4 Net change and average value7.1 Areas between curves7.2 Improper integrals over unbounded intervals7.3 Improper integrals with unbounded integrands7.4 Problems7.5 Sequences and convergence8.1 Infinite series and geometric series8.2 Convergence tools for power series8.3 Power series: radius and interval of convergence8.4 Operations on power series8.5 Problems8.6 Taylor and Maclaurin polynomials9.1 Taylor’s theorem and approximation error9.2 Taylor and Maclaurin series9.3 Problems9.4 Course contents