Techniques of integration; applications of definite integrals; polar and parametric equations; sequences and infinite series; introduction to differential equations and to vectors.
80.000-90.000 Total Hours
Total Hours
160.000-180.000 Total Hours
80.000-90.000 Total Hours
Prerequisite: MATH 150 Prerequisite or Corequisite: None Concurrent Corequisite: None Course Advisories: None Limitation on Enrollment: None
Course Objectives:
Apply techniques of integration to indefinite, definite, and improper integrals.
Use integrals to solve problems in the physical and mathematical sciences and other disciplines.
Analyze behavior of curves given in parametric form, including applications to differential and integral calculus.
Analyze behavior of curves given in polar form, including applications to differential and integral calculus.
Calculate the Taylor polynomials and series of a function about a point, including error estimates, and use them to solve problems in the physical and mathematical sciences.
Apply beginning techniques of differential equations to appropriate situations.
Use vector algebra to solve geometric and algebraic problems.
Compute areas and lengths using polar coordinates.
Apply integration to areas, volumes, work, and arclength.
Apply convergence tests to sequences and series
Represent functions as power series
Student Learning Outcomes
MATH160 SLO1 - Recognize and apply appropriate techniques to evaluate definite, indefinite, and improper integrals.
MATH160 SLO2 - Approximate definite integrals or a function’s value by appropriate numerical methods, including error estimation.
MATH160 SLO3 - Use definite integrals for geometry, physics, and other applications.
MATH160 SLO4 - Use techniques of calculus to analyze polar and parametric curves.
MATH160 SLO5 - Express functions as power series and analyze convergence of infinite series..
MATH160 SLO6 - Solve separable and linear differential equations.
Techniques of integration
Substitution
Integration by parts
Tables of Integrals
Partial Fraction Decomposition
Trigonometric Substitution
Trigonometric Integrals
Numerical Approximations of definite integrals including the trapezoidal rule and Simpson's rule.
Improper Integrals
Applications of the integral
Applications to Geometry
Areas between Curves (Review)
Volume, volume of a solid of revolution.
Arc Length
Surface Area of Solids of Revolution
Applications to Physics
Work
Hydrostatic Force
Moments and Centers of Mass
Applications to Economics (optional)
Probability Distributions (optional)
Differential Equations
Introduction to first order ODE's, slope fields, growth and decay.
Linear and separable first order differential equations.
Applications of first and second order equations
Sequences and infinite series
Convergence of Sequences
Convergence of series using geometric series, the comparison tests, the alternating series test, the root test, and the ratio test.
Radius and interval of convergence of a power series.
Differentiation and integration of power series
Taylor Series
Radius of Convergence
Error in Taylor Approximations
Binomial Series
New Series by Substitution, Differentiation and Integration
Deriving Euler's Formula
Geometric Series
Fourier Series (optional)
Vectors
Vector Arithmetic
Applications to force, work and torque
Parametric Equations and Polar Coordinates
Calculus with curves in parametric form
Polar Coordinates
Areas and Lengths
Calculus with curves in polar form
Conic Sections
Methods of Instruction
Directed Study
Lecture
Projects
Lecture is the primary activity, along with student problem-solving. Problem-solving will frequently require the use of graphing calculators or computer software. Students are also expected to work outside of class on assigned exercises and supplemental reading from the text.
Find the areas within the following polar curves: r=1+cos(theta), r=2-sin(theta), r=1-sin(theta)
A. Appropriate Readings: Students are required to read assigned chapters in texts.
B. Writing Assignments: Students must work assigned mathematical problems requiring the manipulation of abstract symbols, and give a clear and logical written presentation of their solutions..
C. Appropriate Outside Assignments: Students will be expected to spend a sufficient amount of time outside of class to practice techniques taught during class time, read assigned materials, and complete frequent homework assignments.
D. Appropriate Assignments that Demonstrate Critical Thinking: Students must demonstrate mathematical skills which involve analyzing information, recognizing concepts in new contexts, and drawing analogies. They must also analyze logical arguments for validity and write proofs of their own using both inductive and deductive reasoning within a logical system.
A student's grade will be based on multiple measures of performance in the solving of problems, preparation and analysis of graphs, and analysis of logical arguments. Such measures will include at least three exams and a comprehensive final examination requiring demonstrations of problem-solving skills. In addition, instructors may make use of quizzes, written homework assignments, or other appropriate means to judge a student's dexterity with mathematical skills and familiarity with mathematical vocabulary. Calculator (or computer) use is incorporated in the course, but students are expected to perform differentiation and some integration "by hand" and/or using tables.
Calculus with Early TranscendentalsRogawski, Freeman Publishing, 2019
TI-84 Graphing Calculator, Maple, or equivalent computer algebra system