1 / 27

MATH 119 MIDTERM REVIEW

MATH 119 MIDTERM REVIEW. 2010 Outreach Trip. Summary Date Aug 20 – Sept 4 Location Cusco, Peru # Students 22 Project Cost $16,000. Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects

oihane
Download Presentation

MATH 119 MIDTERM REVIEW

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. MATH 119 MIDTERM REVIEW

  2. 2010 Outreach Trip Summary Date Aug 20 – Sept 4 Location Cusco, Peru # Students 22 Project Cost $16,000 Building Projects Kindergarten Classroom provides free education Sewing Workshop enables better job prospects ELT Classroom enables better job prospects More info @ studentsofferingsupport.ca/blog

  3. Tutor: Maysum Panju • 3B Computational Mathematics • Lots of tutoring experience • Interests: • Reading • Pokémon • Calculus

  4. Outline • Approximating Things • Newton’s Method, Fixed-Point Iteration • Guessing with Polynomials • Interpolating Polynomials, Taylor Polynomials, Taylor’s Remainder Theorem • Very Big Sums • Infinite Series, Convergence Tests, Power Series • Questions?

  5. Writing Solutions A good solution includes… • An introductory statement: what you are given and what you have to show/find; • A concluding statement: summarize the conclusion briefly; • Justifications of the main steps: refer to definitions, rules, and known properties; • Some sentences of guidance for the reader, e.g. how you are going to solve the problem.

  6. Newton’s Method • An iterativemethod for finding roots of a function. • Guess a root. • Find the tangent line there. • Find the x-intercept of the tangent line. • This is your new guess! • Repeat. • Formulaically:

  7. Fixed Point Method • An iterative method for finding a solution to an equation of the form • Guess a solution • Compute • This is your new guess! • Repeat. • Won’t always converge! To be safe, require that on the interval you are working in,

  8. Fixed Point Example Suppose you want to find a root of Which of the following choices of will give a good fixed point iteration?

  9. Pictures Newton Method Fixed Point Iteration

  10. Interpolating Polynomials • Given data points Find a polynomial that goes through all of them. (Called an “interpolant”.) • Why? Polynomials are “easy”. • Fact: Given n points, there is a unique polynomial of degree at most n – 1passing through them. • Fact: Polynomials frequently give bad approximations, especially far from the data.

  11. Newton Interpolation • Find a polynomial that goes through the points which are equally spaced with a distance of h units betweenand • Construct a table of differences. • Fit into the polynomial template.

  12. Interpolation Example • Suppose your data points are What is an estimate for f (2.25)?

  13. Lagrange Linear Interpolation • When guessing the function betweentwo known points… • Assume the function is just a line that connects the two dots. • Such a simple concept!!!! (Such an annoying formula!) • Given two points and , set

  14. This time, be linear. • Suppose your data points are What is an estimate for f (2.25)?

  15. Break Time!

  16. Taylor Polynomial • Given information about a curve at a specific point, approximate it for the surrounding area. • Use values of higher derivatives: When a = 0, this is called a Maclaurin Polynomial.

  17. Taylor Polynomial

  18. Taylor’s Inequality • An approximation is only as good as the error bound. • Error function: where is an upper bound forbetween x and a

  19. Taylor ApproximationExample • Compute using a quadratic Taylor polynomial. Bound the error.

  20. Taylor Series to Know

  21. Taylor Series Exercise • Find Taylor Polynomials (around x=0) for…

  22. Infinite Series • Sums of infinitely many terms. • Example: Taylor Series! • Two main questions: • Does the series converge? …. Hard • What does the series converge to? … VERY Hard • We focus on (1) … in general, work with infinite series is an art, not a science.

  23. Convergence Tests • Divergence test: • A series can only converge if the terms get small. • RatioTest: • A series can only converge if the terms keep getting smaller.

  24. Convergence Tests • Integral test: • A series converges exactly when the equivalent integral does. • P series test: • Sums of powers converge only for small powers.

  25. Convergence Tests • Alternating Series test: • An alternating series converges if the terms keep getting smaller. if • The error in this case is

  26. Infinite Series Practice • Which converge?

  27. Questions and Practice Problems

More Related