Calculus II

At the conclusion of this course, the student should be able to:

• compute finite Riemann sums and use to estimate area
• form limits of Riemann sums and write the corresponding definite integral
• recognize and apply the Fundamental Theorem of Calculus
• evaluate integrals involving exponential functions to any base
• evaluate integrals involving basic trigonometric functions and integrals whose solutions require inverse trigonometric functions
• choose an appropriate method and apply the following techniques to find antiderivatives and evaluate definite integrals:
• integration by parts
• trigonometric and rationalizing substitution
• completing the square for integrals involving quadratic expressions
• partial fractions
• integrals of products of trigonometric functions
• apply integration to problems involving areas, volumes, arc length, work, velocity and acceleration
• be able to determine the convergence or divergence of improper integrals either directly, or by using the comparison test
• determine if a given sequence converges or diverges
• determine if a sequence is bounded and/or monotonic
• determine the sum of a geometric series
• be able to choose an appropriate test and determine series convergence/divergence using:
• integral test
• simple comparison test
• limit comparison test
• ratio test
• root test (optional)
• alternating series test
• distinguish and apply concepts of absolute and conditional convergence of a series
• determine the radius and interval of convergence of a power series
• approximate a differentiable function by a Taylor polynomial, determine the remainder term, and compute the error in using the approximation
• find a Taylor or Maclaurin series representing specified functions by:
• "direct" computation
• means of substitution, differentiation or integration of related power series
• find the area of a region bounded by the graph of a polar equation or parametric equations
• find the lengths of curves in polar coordinates or in parametric form
• solve first order differential equations by the method of separation of variables; apply to growth and decay problems