Taylor and MacLaurin Series

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# Taylor and MacLaurin Series - PowerPoint PPT Presentation

Taylor and MacLaurin Series. Lesson 9.7. Centered at c or expanded about c. Taylor &amp; Maclaurin Polynomials. Consider a function f(x) that can be differentiated n times on some interval I Our goal: find a polynomial function M(x) which approximates f

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### Taylor and MacLaurin Series

Lesson 9.7

Centered at c or expanded about c

Taylor & Maclaurin Polynomials
• Consider a function f(x) that can be differentiated n times on some interval I
• Our goal: find a polynomial function M(x)
• which approximates f
• at a number c in its domain
• Initial requirements
• M(c) = f(c)
• M '(c) = f '(c)
Linear Approximations
• The tangent line is a good approximation of f(x) for x near a

True value f(x)

Approx. value of f(x)

f'(a) (x – a)

(x – a)

f(a)

a

x

Linear Approximations
• Taylor polynomial degree 1
• Approximating f(x) for x near 0
• Consider
• How close are these?
• f(.05)
• f(0.4)

View Geogebra demo

• For a more accurate approximation to f(x) = cos x for x near 0
• We determine
• At x = 0 we must have
• The functions to agree
• The first and second derivatives to agree
• So
• Now how close are these?

View Geogebra demo

Taylor Polynomial Degree 2
• In general we find the approximation off(x) for x near 0
• Try for a different function
• f(x) = sin(x)
• Let x = 0.3
Higher Degree Taylor Polynomial
• For approximating f(x) for x near 0
• Note for f(x) = sin x, Taylor Polynomial of degree 7

View Geogebra demo

Improved Approximating
• We can choose some other value for x, say x = c
• Then for f(x) = sin(x – c) the nth degree Taylor polynomial at x = c
• Try for c =  / 3
Assignment
• Lesson 9.7A
• Page 658
• Exercises 1 – 4 all5 - 29 odd