MATH 119 MIDTERM REVIEW

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Tutor: Maysum Panju

• 3B Computational Mathematics

• Lots of tutoring experience

• Interests: – Reading– Pokémon– Calculus

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?

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…

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:

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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?

• • •

•

Pictures

Newton Method Fixed Point Iteration

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.

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.

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

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

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

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.

Taylor Polynomial

Taylor’s Inequality

• An approximation is only as good as the error bound.

• Error function:

where is an upper bound forbetween x and a

Taylor ApproximationExample

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

Taylor Series to Know

Taylor Series Exercise

• Find Taylor Polynomials (around x=0) for…

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.

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.

Convergence Tests

• Integral test: – A series converges exactly when the equivalent

integral does.

• P series test:– Sums of powers converge only for small powers.

Convergence Tests

• Alternating Series test: – An alternating series converges if the terms keep

getting smaller.

if

– The error in this case is

Infinite Series Practice

• Which converge?

Questions and Practice Problems