math 119 midterm review
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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 - PowerPoint PPT PresentationTRANSCRIPT
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MATH 119 MIDTERM REVIEW
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2010 Outreach TripSummaryDate Aug 20 – Sept 4Location Cusco, Peru# Students 22Project Cost $16,000
Building ProjectsKindergarten Classroom provides free
educationSewing Workshopenables better job prospectsELT Classroom enables better job prospectsMore info @
studentsofferingsupport.ca/blog
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Tutor: Maysum Panju
• 3B Computational Mathematics
• Lots of tutoring experience
• Interests: – Reading– Pokémon– Calculus
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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?
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Writing Solutions
• 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.
A good solution includes…
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Newton’s Method
• An iterative method 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:
6
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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,
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Fixed Point Example
Suppose you want to find a root of
Which of the following choices of will give a good fixed point iteration?
• • •
•
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Pictures
Newton Method Fixed Point Iteration
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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 – 1 passing through them.
• Fact: Polynomials frequently give bad approximations, especially far from the data.
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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.
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Interpolation Example
• Suppose your data points are
What is an estimate for f (2.25)?
x 1 1.5 2 2.5f(x) 2 0 2 7
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Lagrange Linear Interpolation
• When guessing the function between two 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
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This time, be linear.
• Suppose your data points are
What is an estimate for f (2.25)?
x 1 1.5 2 2.5f(x) 2 0 2 7
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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.
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Taylor’s Inequality
• An approximation is only as good as the error bound.
• Error function:
where is an upper bound forbetween x and a
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Taylor ApproximationExample
• Compute using a quadratic Taylor polynomial. Bound the error.
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Taylor Series to Know
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Taylor Series Exercise
• Find Taylor Polynomials (around x=0) for…
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Infinite Series
• Sums of infinitely many terms.– Example: Taylor Series!
• Two main questions:1) Does the series converge? …. Hard2) What does the series converge to? … VERY Hard
• We focus on (1) … in general, work with infinite series is an art, not a science.
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Convergence Tests
• Divergence test: – A series can only converge if the terms get small.
• Ratio Test:– A series can only converge if the terms
keep getting smaller.
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Convergence Tests
• Integral test: – A series converges exactly when the equivalent
integral does.
• P series test:– Sums of powers converge only for small powers.
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Convergence Tests
• Alternating Series test: – An alternating series converges if the terms keep
getting smaller.
if
– The error in this case is
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Infinite Series Practice
• Which converge?
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Questions and Practice Problems