Taylor series


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Taylor series

  1. 1. Taylor Series<br />John Weiss<br />
  2. 2. Approximating Functions<br />f(0)= 4<br />What is f(1)?<br />f(x) = 4?<br />f(1) = 4?<br />
  3. 3. Approximating Functions<br />f(0)= 4, f’(0)= -1<br />What is f(1)?<br />f(x) = 4 - x?<br />f(1) = 3?<br />
  4. 4. Approximating Functions<br />f(0)= 4, f’(0)= -1, f’’(0)= 2<br />What is f(1)?<br />f(x) = 4 – x + x2? <br />(same concavity)<br />f(1) = 4?<br />
  5. 5. Approximating Functions<br />f(x) = sin(x)<br />What is f(1)?<br />f(0) = 0, f’(0) = 1<br />f(x) = 0 + x?<br />f(1) = 1?<br />
  6. 6. Approximating Functions<br />f(x) = sin(x)<br />f(0) = 1, f’(0) = 1, f’’(0) = 0, f’’’(0) = -1,…<br />What is f(1)? i.e . What is sin(1)?<br />
  7. 7. Famous Mathematicians<br />James Gregory (1671)<br />Brook Taylor (1712)<br />Colin Maclaurin (1698-1746)<br />Joseph-Louis Lagrange (1736-1813)<br />Augustin-Louis Cauchy (1789-1857)<br />
  8. 8. Approximations<br />Linear Approximation<br />Quadratic Approximation<br />
  9. 9. Taylor’s Theorem<br />Let k≥1 be an integer and be k times differentiable at .<br />Then there exists a function such that<br />Note: Taylor Polynomial of degree k is:<br />
  10. 10. Works for Linear Approximations<br />
  11. 11. Works for Quadratic Approximations<br />
  12. 12. f(x) = sin(x)Degree 1<br />
  13. 13. f(x) = sin(x)Degree 3<br />
  14. 14. f(x) = sin(x)Degree 5<br />
  15. 15. f(x) = sin(x)Degree 7<br />
  16. 16. f(x) = sin(x)Degree 11<br />
  17. 17. Implications<br />If fand g have the same value and all of the same derivatives at a point, then they must be the same function!<br />
  18. 18. Proof: If f and g are smooth functions that agree over some interval, they MUST be the same function<br />Let f and g be two smooth functions that agree for some open interval (a,b), but not over all of R<br />Define h as the difference, f – g, and note that h is smooth, being the difference of two smooth functions. Also h=0 on (a,b), but h≠0 at other points in R<br />Without loss of generality, we will form S, the set of all x>a, such that f(x)≠0 <br />Note that a is a lower bound for this set, S, and being a subset of R, S is complete so S has a real greatest lower bound, call it c.<br />c, being a greatest lower bound of S, is also an element of S, since S is closed<br />Now we see that h=0 on (a,c), but h≠0 at c. So, h is discontinuous at c, so then h cannot be smooth<br />Thus we have reached a contradiction, and so f and g must agree everywhere!<br />
  19. 19. Suppose f(x) can be rewritten as a power series…<br />
  20. 20. Entirety (Analytic Functions)<br />A function f(x) is said to be entire if it is equal to its Taylor Series everywhere<br />Entire<br />sin(x)<br />Not Entire<br />log(1+x)<br />
  21. 21. Proof: sin(x) is entire<br />Maclaurin Series<br />sin(0)=0<br />sin’(0)=1<br />sin’’(0)=0<br />sin’’’(0)=-1<br />sin’’’’(0)=0<br />sin’’’’’(0)=1<br />sin’’’’’’(0)=0<br />… etc.<br />
  22. 22. Proof: sin(x) is entire<br />Lagrange formula for the remainder:<br />Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then <br /> for some z in (a,x)<br />
  23. 23. Proof: sin(x) is entire <br />First, sin(x) is continuous and infinitely differentiable over all of R<br />If we look at the Taylor Polynomial of degree k<br />Note though for all z in R<br />
  24. 24. Proof: sin(x) is entire<br />However, as k goes to infinity, we see<br />Applying the Squeeze Theorem to our original equation, we obtain that as k goes to infinity<br />and thus sin(x) is entire since it is equal to its Taylor series<br />
  25. 25. Maclaurin Series Examples<br />Note: <br />
  26. 26. Applications<br />Physics<br />Special Relativity Equation<br />Fermat’s Principle (Optics)<br />Resistivity of Wires<br />Electric Dipoles<br />Periods of Pendulums<br />Surveying (Curvature of the Earth)<br />
  27. 27. Special Relativity<br />Let . If v ≤ 100 m/s<br />Then according to Taylor’s Inequality (Lagrange)<br />
  28. 28. Lagrange Remainder<br />Lagrange formula for the remainder:<br />Let be k+1 times differentiable on (a,x) and continuous on [a,x]. Then <br /> for some z in (a,x)<br />
  29. 29. Special Relativity<br />Let . If v ≤ 100 m/s<br />Then according to Taylor’s Inequality (Lagrange)<br />
  30. 30. The End<br />