Calculus II Honours Supplement

Upon completion of MATH 1221 the student should be able to:

• Demonstrate understanding of upper and lower Riemann sums and refinements, and how they relate to the definition of a definite integral
• Use the definition of a definite (Riemann) integral to prove properties of the definite integral
• Prove the Mean Value Theorem for integrals
• Prove the Fundamental Theorem of Calculus
• Prove the substitution rule for integration
• Define the logarithm as an integral and use properties of integrals to derive properties of logarithms
• Use the integral definition of the logarithm to define the exponential function and derive its properties
• Use tables of integrals to assist in determining definite and indefinite integrals
• Approximate definite integrals using numerical methods
• Solve problems involving further applications of definite integrals such as, but not limited to, moments, center of mass, probability, hydrostatic force and pressure, logistic growth, centroids and the theorem of Pappus
• State the definition of the limit of a sequence and use it to evaluate the limit of sequences
• Use the definition of the limit of a sequence to prove properties about limits of sequences
• Understand the role of the completeness property in convergence of bounded monotonic sequences
• State the definition for convergence of a series and use it to show convergence of different series
• Use the definition of a convergent series to prove properties of convergent series
• Justify the validity of the following convergence tests: comparison test, limit comparison test, integral test, alternating series test, ratio test, root test
• State the definition of absolute convergence and conditional convergence of a series
• Prove that the absolute convergence of a series implies convergence of a series
• Estimate the value of a convergent sequence using error bounds for partial sums
• State the definition of a power series
• Prove properties of power series
• State the definition of a Taylor series
• Prove the remainder theorems for Taylor series
• Derive the binomial series and solve problems using a binomial approximation