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1Limits£¬ continuous and differentiable functions11.1Limits of sequences1 1.2Limits of functions3 1.3Squeeze Theorem9 1.4Continuity of functions11 1.5Differentiable functions16 Module one miscellaneous questions18 2Rolle¬ðs theorem and Mean Value Theorem222.1Rolle¬ðs theorem22 2.2Mean Value Theorem (MVT)24 Module two miscellaneous questions29 3Fundamental Theorem of Calculus and improper integrals343.1Fundamental Theorem of Calculus34 3.2Improper integrations of the form ∫∞af(x)dx35 3.3Approximation of improper integrals38 Module three miscellaneous questions42 4Infinite series and power series454.1Sum of infinite geometric series45 4.2Test for Divergence47 4.3Integral Test48 4.4Comparison Test51 4.5Limit Comparison Test54 4.6Alternating Series Test56 4.7Absolute and conditional convergence59 4.8Ratio Test61 4.9Telescoping series62 4.10Power series64 Module four miscellaneous questions67 5Taylor and Maclaurin series705.1Revision: power series and its convergence70 5.2Taylor and Maclaurin series76 5.3Taylor and Maclaurin polynomials82 5.4Taylor and Maclaurin series with their “friends”88 Module five miscellaneous questions99 6Differential equation1096.1Introduction to differential equation109 6.2Solving the first order differential equations by graphical method1116.3Solving the first order differential equations by numerical method121 6.4Solving the first order differential equations by analytic method124 6.5The first order differential equation and the Taylor or Maclaurin series149 Module six miscellaneous questions156 Answers of exercises165