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

2. BMayer@ChabotCollege.edu 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-18 10.2

3. BMayer@ChabotCollege.edu 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

4. BMayer@ChabotCollege.edu 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

5. BMayer@ChabotCollege.edu 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”

6. BMayer@ChabotCollege.edu 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

7. BMayer@ChabotCollege.edu 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

8. BMayer@ChabotCollege.edu 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

9. BMayer@ChabotCollege.edu 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

10. BMayer@ChabotCollege.edu 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

11. BMayer@ChabotCollege.edu 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

12. BMayer@ChabotCollege.edu 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

13. BMayer@ChabotCollege.edu 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

14. BMayer@ChabotCollege.edu 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

15. BMayer@ChabotCollege.edu 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

16. BMayer@ChabotCollege.edu 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

17. BMayer@ChabotCollege.edu 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

18. BMayer@ChabotCollege.edu 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

19. BMayer@ChabotCollege.edu 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:

20. BMayer@ChabotCollege.edu 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

21. BMayer@ChabotCollege.edu 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

22. BMayer@ChabotCollege.edu 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:

23. BMayer@ChabotCollege.edu 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

24. BMayer@ChabotCollege.edu 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

25. BMayer@ChabotCollege.edu 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

26. BMayer@ChabotCollege.edu 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

27. BMayer@ChabotCollege.edu 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:

28. BMayer@ChabotCollege.edu 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

29. BMayer@ChabotCollege.edu 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

30. BMayer@ChabotCollege.edu 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

31. BMayer@ChabotCollege.edu 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

32. BMayer@ChabotCollege.edu 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

33. BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 33 Bruce Mayer, PE Chabot College Mathematics All Done for Today Brook Taylor (1685-1731)

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

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

36. BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 36 Bruce Mayer, PE Chabot College Mathematics P10.3-39 Taylor Series  Da1 := diff(ln(x)/x, x)  Db2 := diff(Da1, x)  Dc3 := diff(Db2, x)  Dd4 := diff(Dc3, x)

37. BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 37 Bruce Mayer, PE Chabot College Mathematics P10.3-39 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)

38. BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 38 Bruce Mayer, PE Chabot College Mathematics P10.3-39 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)

39. BMayer@ChabotCollege.edu MTH16_Lec-19_sec_10-3_Taylor_Series.pptx 39 Bruce Mayer, PE Chabot College Mathematics P10.3-39 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])

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

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

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