Course Description
This course is a supplement to MATH 1220-Calculus II with an emphasis on proving theorems from MATH 1220. Topics include the definite integral, properties of integrals, applications of integration, sequences and series, and Taylor series.
Course Content
- The Definite Integral
- Applications of Integration
- Sequences and Series
- Taylor Series
Learning Activities
Seminar, assignments, group activities
Means of Assessment
Assessments will be in accordance with the Douglas College Evaluation Policy. The instructor will present a written course outline with specific evaluation criteria at the beginning of the semester. Evaluation will be based on the following:
Tests: 0 – 70%
Assignments: 0 – 80%
Group Work: 0 – 10%
Attendance: 0 – 5%
Final examination: 20 – 40%
Total: 100%
Learning Outcomes
Upon successful completion of the course, students will be able to:
- explain how upper and lower Riemann sums and refinements 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.
- prove properties of improper integrals.
- approximate definite integrals using numerical methods and find error bounds.
- 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.
- utilize 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, and 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.
Textbook Materials
Textbooks and materials are to be purchased by students. A list of required and textbooks and materials is provided for students at the beginning of the semester. Example texts may include:
Calculus, Michael Spivak (3rd edition or later).