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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §10.3 Series: Power & Taylor

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review § Any QUESTIONS About §10.2 Convergence Tests Any QUESTIONS About HomeWork §10.2 → HW

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 3 Bruce Mayer, PE Chabot College Mathematics §10.3 Learning Goals Find the radius and interval of convergence for a power series Study term-by-term differentiation and integration of power series Explore Taylor series representation of functions

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 4 Bruce Mayer, PE Chabot College Mathematics Power Series General Power Series: A form of a GENERALIZED POLYNOMIAL Power Series Convergence Behavior Exclusively ONE of the following holds True a)Converges ONLY for x = 0 (Trival Case) b)Converges for ALL x c)Has a Finite “Radius of Convergence”, R

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 5 Bruce Mayer, PE Chabot College Mathematics Radius of Convergence For the General Power Series Unless a power series converges at any real number, a number R > 0 exists such that the series CONverges absolutely for each x such that | x | < R and DIverges for any other x Thus the “Interval of Convergence”

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 6 Bruce Mayer, PE Chabot College Mathematics Example Radius of Conv. Find R for the Series: Radius of Convergence Interval of Convergence SOLUTION Use the Ratio Test

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 7 Bruce Mayer, PE Chabot College Mathematics Example Radius of Conv. Continue with Limit Evaluation: Thus R = 4 The Interval of Convergence Thus This Series Converges

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 8 Bruce Mayer, PE Chabot College Mathematics Functions as Power Series Many Functions can be represented as Infinitely Long PolyNomials Consider this Function and Domain Recall one of The Geometric Series Thus

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example Fcn by Pwr Series Write as a Power Series → Also Find the Radius of Convergence SOLUTION: Start with the GeoMetric Series First Cast the Fcn into the Form

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 10 Bruce Mayer, PE Chabot College Mathematics Example Fcn by Pwr Series Using Algebraic Processes on the Fcn Thus by the Geometric Series Then the Function by Power Series

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 11 Bruce Mayer, PE Chabot College Mathematics Example Fcn by Pwr Series Now find the Radius of Convergence by the Ratio Test

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example Fcn by Pwr Series Thus for Convergence So the Interval of Convergence: And also the Radius of Convergence

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 13 Bruce Mayer, PE Chabot College Mathematics Pwr Series Derivatives & Integrals Consider a Convergent Power Series And an Associated Function If f ( x ) is differentiable over − R < x < R, then

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 14 Bruce Mayer, PE Chabot College Mathematics Pwr Series Derivatives & Integrals If f ( x ) is Integrable over − R < x < R, then

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 15 Bruce Mayer, PE Chabot College Mathematics Pwr Series Derivatives & Integrals Thus the Derivative of a Power-Series Function Thus the AntiDerivative of a Power-Series Function

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 16 Bruce Mayer, PE Chabot College Mathematics Example Find Fcn by Integ Find a Power Series Equivalent for SOLUTION: First take: Recognize from Before

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 17 Bruce Mayer, PE Chabot College Mathematics Example Find Fcn by Integ Recover the Original Fcn by taking the AntiDerivative of the Just Determined Derivative

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 18 Bruce Mayer, PE Chabot College Mathematics Example Find Fcn by Integ Then To Find C use the original Function Use f (0) = 0 in Power Series fcn Then the Final Power Series Fcn

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 19 Bruce Mayer, PE Chabot College Mathematics Taylor Series Consider some general Function, f ( x ), that might be Represented by a Power Series Thus need to find CoEfficients, a n, such that the Power Series Converges to f ( x ) over some interval. Stated Mathematically Need a n so that:

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 20 Bruce Mayer, PE Chabot College Mathematics Taylor Series If x = 0 and if f (0) is KNOWN then a 0 done, 1→∞ to go…. Next Differentiate Term-by-Term Now if the First Derivative (the Slope) is KNOWN when x = 0, then

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 21 Bruce Mayer, PE Chabot College Mathematics Taylor Series Again Differentiate Term-by-Term Now if the 2 nd Derivative (the Curvature) is KNOWN when x = 0, then

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 22 Bruce Mayer, PE Chabot College Mathematics Taylor Series Another Differentiation Again if the 3 rd Derivative is KNOWN at x = 0 Recognizing the Pattern:

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 23 Bruce Mayer, PE Chabot College Mathematics Taylor Series Thus to Construct a Taylor (Power) Series about an interval “Centered” at x = 0 for the Function f ( x ) Find the Values of ALL the Derivatives of f ( x ) when f ( x ) = 0 Calculate the Values of the Taylor Series CoEfficients by Finally Construct the Power Series from the CoEfficients

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example Taylor Series for ln(e+x) Calculate the Derivatives Find the Values of the Derivatives at 0

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 25 Bruce Mayer, PE Chabot College Mathematics Example Taylor Series for ln(e+x) Generally Then the CoEfficients The 1 st four CoEfficients

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example Taylor Series for ln(e+x) Then the Taylor Series

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 27 Bruce Mayer, PE Chabot College Mathematics Taylor Series at x ≠ 0 The Taylor Series “Expansion” can Occur at “Center” Values other than 0 Consider a function stated in a series centered at b, that is: Now the the Radius of Convergence for the function is the SAME as before:

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 28 Bruce Mayer, PE Chabot College Mathematics Taylor Series at x ≠ 0 To find the CoEfficients need ( x − b ) = 0 which requires x = b, Then the CoEfficient Expression The expansion about non-zero centers is useful for functions (or the derivatives) that are NOT DEFINED when x=0 For Example ln( x ) can NOT be expanded about zero, but it can be about, say, 2

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 29 Bruce Mayer, PE Chabot College Mathematics Example Expand x½ about 4 Expand about b = 4: The 1 st four Taylor CoEfficients

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 30 Bruce Mayer, PE Chabot College Mathematics Example Expand x ½ about 4 SOLUTION: Use the CoEfficients to Construct the Taylor Series centered at b = 4

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 31 Bruce Mayer, PE Chabot College Mathematics Example Expand x ½ about 4 Use the Taylor Series centered at b = 4 to Find the Square Root of 3

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 32 Bruce Mayer, PE Chabot College Mathematics WhiteBoard PPT Work Problems From §10.3 P39 → expand about b = 1 the Function

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 33 Bruce Mayer, PE Chabot College Mathematics All Done for Today Brook Taylor ( )

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 34 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 35 Bruce Mayer, PE Chabot College Mathematics

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 36 Bruce Mayer, PE Chabot College Mathematics P Taylor Series Da1 := diff(ln(x)/x, x) Db2 := diff(Da1, x) Dc3 := diff(Db2, x) Dd4 := diff(Dc3, x)

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 37 Bruce Mayer, PE Chabot College Mathematics P Taylor Series ln(x)/x, x f0 := taylor(ln(x)/x, x = 1, 0) f1 := taylor(ln(x)/x, x = 1, 1) f2 := taylor(ln(x)/x, x = 1, 2)

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 38 Bruce Mayer, PE Chabot College Mathematics P Taylor Series f3 := taylor(ln(x)/x, x = 1, 3) f4 := taylor(ln(x)/x, x = 1, 4) d6 := diff(ln(x)/x, x $ 5)

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 39 Bruce Mayer, PE Chabot College Mathematics P Taylor Series plot(f0, f1, f2, f3, f4, f5, x =0.5..3, GridVisible = TRUE, LineWidth = 0.04*unit::inch, Width = 320*unit::mm, Height = 180*unit::mm, AxesTitleFont = ["sans-serif", 24], TicksLabelFont=["sans-serif", 16])

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 40 Bruce Mayer, PE Chabot College Mathematics

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 41 Bruce Mayer, PE Chabot College Mathematics

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MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 42 Bruce Mayer, PE Chabot College Mathematics

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