MAT1223

Topics List

Date	Sections	Topics	Prerequisite Skills	Student Learning Outcomes
Week 1	1.3	The Fundamental Theorem of Calculus	Differentiation Rules The Chain Rule Antiderivatives The Definite Integral	Use the Fundamental Theorem of Calculus, Part 1, to evaluate derivatives of integrals. Use the Fundamental Theorem of Calculus, Part 2, to evaluate definite integrals. Explain the relationship between differentiation and integration.
Week 1	1.5	Integration by Substitution	Differentiation Rules Differentials Antiderivatives The Definite Integral The Fundamental Theorem of Calculus	Recognize when to use integration by substitution. Use substitution to evaluate indefinite integrals. Use substitution to evaluate definite integrals.
Week 2	2.1	Area between Curves	Graphing Elementary Functions The Definite Integral The Fundamental Theorem of Calculus Integration by Substitution	Determine the area of a region between two curves by integrating with respect to the independent variable. Find the area of a compound region. Determine the area of a region between two curves by integrating with respect to the dependent variable.
Week 2	2.2	Determining Volumes by Slicing	Areas of basic shapes Volume of a cylinder Graphing elementary functions The Fundamental Theorem of Calculus Integration by Substitution	Determine the volume of a solid by integrating a cross-section (the slicing method). Find the volume of a solid of revolution using the disk method. Find the volume of a solid of revolution with a cavity using the washer method.
Week 3	2.3	Volumes of Revolution, Cylindrical Shells	Graphing elementary functions Determining Volumes by Slicing The Fundamental Theorem of Calculus Integration by Substitution	Calculate the volume of a solid of revolution by using the method of cylindrical shells. Compare the different methods for calculating a volume of revolution.
Week 3	2.4	Arc Length and Surface Area	Differentiation Rules The Fundamental Theorem of Calculus Integration by Substitution	Determine the length of a plane curve between two points. Find the surface area of a solid of revolution.
Week 4	2.5	Physical Applications	Areas of basic shapes Volume of a cylinder Basic Physics (Mass, Force, Work, Newton's Second Law, Hooke's Law) The Fundamental Theorem of Calculus Integration by Substitution	Determine the mass of a one-dimensional object from its linear density function. Determine the mass of a two-dimensional circular object from its radial density function. Calculate the work done by a variable force acting along a line. Calculate the work done in stretching/compressing a spring. Calculate the work done in lifting a rope/cable. Calculate the work done in pumping a liquid from one height to another. Find the hydrostatic force against a submerged vertical plate.
Week 5	2.6	Moments and Center of Mass	Graphing elementary functions The Fundamental Theorem of Calculus Integration by Substitution	Find the center of mass of objects distributed along a line. Find the center of mass of objects distributed in a plane. Locate the center of mass of a thin plate. Use symmetry to help locate the centroid of a thin plate.
Week 5-6	3.1	Integration by Parts	Differentiation Rules Differentials The Fundamental Theorem of Calculus Integration by Substitution	Recognize when to use integration by parts. Use the integration-by-parts formula to evaluate indefinite integrals. Use the integration-by-parts formula to evaluate definite integrals. Use the tabular method to perform integration by parts. Solve problems involving applications of integration using integration by parts.
Week 6	3.2	Trigonometric Integrals	Trigonometric Functions Trigonometric Identities The Fundamental Theorem of Calculus Integration by Substitution Integration by Parts	Evaluate integrals involving products and powers of sin(x) and cos(x). Evaluate integrals involving products and powers of sec(x) and tan(x). Evaluate integrals involving products of sin(ax), sin(bx), cos(ax), and cos(bx). Solve problems involving applications of integration using trigonometric integrals.
Week 6-7	3.3	Trigonometric Substitution	Completing the Square Trigonometric Functions Trigonometric Identities Integration by Substitution Integration by Parts Trigonometric Integrals	Evaluate integrals involving the square root of a sum or difference of two squares. Solve problems involving applications of integration using trigonometric substitution.
Week 7	3.4	Partial Fractions	Factoring Polynomials Completing the Square Long Division of Polynomials Systems of Linear Equations Antiderivatives Integration by Substitution	Integrate a rational function whose denominator is a product of linear and quadratic polynomials. Recognize distinct linear factors in a rational function. Recognize repeated linear factors in a rational function. Recognize distinct irreducible quadratic factors in a rational function. Recognize repeated irreducible quadratic factors in a rational function. Solve problems involving applications of integration using partial fractions.
Week 8	3.7	Improper Integrals	The Fundamental Theorem of Calculus Integration by Substitution Integration by Parts Trigonometric Integrals Trigonometric Substitution Partial Fractions The Limit Laws Limits at Infinity L’Hôpital’s Rule	Recognize improper integrals and determine their convergence or divergence. Evaluate an integral over an infinite interval. Evaluate an integral over a closed interval with an infinite discontinuity within the interval. Use the comparison theorem to determine whether an improper integral is convergent or divergent.
Week 9	5.1	Sequences	The Limit Laws and Squeeze Theorem Limits at Infinity L’Hôpital’s Rule Increasing and Decreasing Functions	Find a formula for the general term of a sequence. Find a recursive definition of a sequence. Determine the convergence or divergence of a given sequence. Find the limit of a convergent sequence. Determine whether a sequence is bounded and/or monotone. Apply the Monotone Convergence Theorem.
Week 10	5.2	Infinite Series	Sigma notation Sequences Partial Fractions	Write an infinite series using sigma notation. Find the nth partial sum of an infinite series. Define the convergence or divergence of an infinite series. Identify a geometric series. Apply the Geometric Series Test. Find the sum of a convergent geometric series. Identify a telescoping series. Find the sum of a telescoping series.
Week 10-11	5.3	The Divergence and Integral Tests	The Limit Laws Limits at Infinity Continuity Increasing and Decreasing Functions L’Hôpital’s Rule Improper Integrals	Use the Divergence Test to determine whether a series diverges. Use the Integral Test to determine whether a series converges or diverges. Use the p-Series Test to determine whether a series converges or diverges. Estimate the sum of a series by finding bounds on its remainder term.
Week 11	5.4	Comparison Tests	Limits at Infinity Increasing and Decreasing Functions L’Hôpital’s Rule The Geometric Series Test The p-Series Test	Use the Direct Comparison Test to determine whether a series converges or diverges. Use the Limit Comparison Test to determine whether a series converges or diverges.
Week 12	5.5	Alternating Series	Limits at Infinity Increasing and Decreasing Functions L’Hôpital’s Rule The Geometric Series Test The p-Series Test Comparison Tests	Use the Alternating Series Test to determine the convergence of an alternating series. Estimate the sum of an alternating series. Explain the meaning of absolute convergence and conditional convergence.
Week 12	5.6	Ratio and Root Tests	Factorials Limits at Infinity L’Hôpital’s Rule	Use the Ratio Test to determine absolute convergence or divergence of a series. Use the Root Test to determine absolute convergence or divergence of a series. Describe a strategy for testing the convergence or divergence of a series.
Week 13	6.1	Power Series and Functions	The Geometric Series Test The Divergence and Integral Tests Comparison Tests Alternating Series Ratio and Root Tests	Identify a power series. Determine the interval of convergence and radius of convergence of a power series. Use a power series to represent certain functions.
Week 14	6.2	Properties of Power Series	Differentiation Rules Antiderivatives The Fundamental Theorem of Calculus Power Series and Functions	Combine power series by addition or subtraction. Multiply two power series together. Differentiate and integrate power series term-by-term. Use differentiation and integration of power series to represent certain functions as power series.
Week 15	6.3	Taylor and Maclaurin Series	Higher-Order Derivatives Power Series and Functions Properties of Power Series	Find a Taylor or Maclaurin series representation of a function. Find the radius of convergence of a Taylor Series or Maclaurin series. Finding a Taylor polynomial of a given order for a function. Use Taylor's Theorem to estimate the remainder for a Taylor series approximation of a given function.