Polar coordinates (Sect. 11.3)
Review: Arc-length of a curve.
Polar coordinates denition.
Transformation rules Polar-Cartesian.
Examples of curves in polar coordinates.
Review: Arc-length of a curve
Denition
A curve on the plane is given in parametric for
Ratio test (Sect. 10.5)
The ratio test.
Using the ratio test.
Few more examples.
Comment: The root test.
The ratio test
Remark: The ratio test is a way to determine whether a series
converges or not.
Theorem
an+1
= exists.
n an
Let cfw_an be a positive s
Alternating series and absolute convergence (Sect. 10.6)
Alternating series.
Absolute and conditional convergence.
Absolute convergence test.
Few examples.
Alternating series
Denition
An innite series
an is an alternating series i holds either
an = (1)n |
Area of regions in polar coordinates (Sect. 11.5)
Review: Few curves in polar coordinates.
Formula for the area or regions in polar coordinates.
Calculating areas in polar coordinates.
Transformation rules Polar-Cartesian.
Denition
P = ( r, 0 ) = (x,y)
Th
Comparison tests (Sect. 10.4)
Review: Direct comparison test for integrals.
Direct comparison test for series.
Review: Limit comparison test for integrals.
Limit comparison test for series.
Few examples.
Comparison tests (Sect. 10.4)
Review: Direct compar
Arc-length of a curve on the plane (Sect. 11.2)
Review: Parametric curves on the plane.
The slope of tangent lines to curves.
The arc-length of a curve.
The arc-length function and dierential.
Review: Parametric curves on the plane
Denition
A curve on the
Convergence of Taylor Series (Sect. 10.9)
Review: Taylor series and polynomials.
The Taylor Theorem.
Using the Taylor series.
Estimating the remainder.
Review: Taylor series and polynomials
Denition
The Taylor series and Taylor polynomial order n centered
Binomial functions and Taylor series (Sect. 10.10)
Review: The Taylor Theorem.
The binomial function.
Evaluating non-elementary integrals.
The Euler identity.
Taylor series table.
Review: The Taylor Theorem
Recall: If f : D R is innitely dierentiable, and
Parametrizations of curves on a plane (Sect. 11.1)
Review: Curves on the plane.
Parametric equations of a curve.
Examples of curves on the plane.
The cycloid.
Review: Curves on the plane
Remarks:
Curves on a plane can be described by the set of points (x,
Graphing in polar coordinates (Sect. 11.4)
Review: Polar coordinates.
Review: Transforming back to Cartesian.
Computing the slope of tangent lines.
Using symmetry to graph curves.
Examples:
Circles in polar coordinates.
Graphing the Cardiod.
Graphing the
Review for Final Exam.
10 or 14 problems.
No multiple choice questions.
No notes, no books, no calculators.
Problems similar to homeworks.
Exam covers:
Sections
Sections
Sections
Sections
Sections
6.1, 6.3, 6.5.
7.1-7.7.
8.1-8.5, 8.7.
10.1-10.10.
11.1-11.
Taylor Series (Sect. 10.8)
Review: Power series dene functions.
Functions dene power series.
Taylor series of a function.
Taylor polynomials of a function.
Review: Power series dene functions
Remarks:
Power series dene functions on domains where the serie
Power series (Sect. 10.7)
Power series denition and examples.
The radius of convergence.
The ratio test for power series.
Term by term derivation and integration.
Power series denition and examples
Denition
A power series centered at x0 is the function y
Review for Exam 3.
5 or 6 problems.
Exam covers: 10.2-10.10, 11.1-11.5.
Innite series (10.2).
The integral test (10.3).
Comparison tests (10.4).
The ratio test (10.5).
Alternating series (10.6).
Power series (10.7).
Taylor and Maclaurin series (10.8).
Con
Engineering Strategies and Practices Lecture Notes
September 15 2015
Design Process:
-Involves critical thinking
-Managing information
-Reduces error (due diligence)
Obligation to deeply investigate the problem from multiple perspectives
Finding all relev