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+ Approximating Functions f(x) = sin(x) f(0) = 1, f’(0) = 1, f’’(0) = 0, f’’’(0) = -1,… What is f(1)? i.e . What is sin(1)?
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+ Famous Dead People James Gregory (1671) Brook Taylor (1712) Colin Maclaurin (1698-1746) Joseph-Louis Lagrange (1736-1813) Augustin-Louis Cauchy (1789-1857)
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+ Approximations Linear Approximation f (x) = f (a) + f ʹ′(a)(x − a) + R1 (x)(x − a) R1 (x)(x − a) = f (x) − f (a) − f ʹ′(x − a) Quadratic Approximation f ʹ′ (a)€ f (x) = f (a) + f ʹ′(a)(x − a) + (x − a) 2 + R2 (x)(x − a) 2 2 f ʹ′ (a) 2 R2 (x)(x − a) = f (x) − f (a) − f ʹ′(a)(x − a) − (x − a) 2 2€
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+ Taylor’s Theorem Letk≥1 be an integer and f : R →R be k times differentiable at a ∈ R . Then there exists a function R : R →R such that k f ʹ′ (a) f k (a)f (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k + Rk (x)(x − a) k 2! k! € € Note: Taylor Polynomial of degree k is: € f ʹ′ (a) f k (a) Pk (x) = f (a) − f ʹ′(a)(x − a) + (x − a) 2 + ...+ (x − a) k 2! k!
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+ Works for Linear Approximations f (x) = c 0 + c1 (x) f (a) = c 0 + c1 (a) f ʹ′(a) = c1 f (x)(a)fʹ′= c + c (x − a) f = f (a) = c (a) + 0 11 1€ f (x) = f (a) + c1€ € € a) (x − € f (x) = c 0 + c1 (a) + c1 (x − a) = c 0 + c1 (x) €€ €
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+ Works for Quadratic Approximations f (x) = c 0 + c1 (x) + c 2 (x 2 ) f (a) = c 0 + c1 (a) + c 2 (a 2 ) f ʹ′(a) = c1 + 2c 2 (a) f ʹ′ (a) = 2c 2 € = c + c (a) + c (a 2 ) + [c + 2c (a)](x − a) + 2c 2 [x − a]2 = € f (x) 0 1 2 1 2 € 2 c 0€ c1(a) + c 2 (a 2 ) + c1(x − a) + 2c 2 (x − a) + c 2 (x) 2 − c 2 (2ax) + c 2 (a) 2 = + f (x) = c 0 + c1 (x) + c 2 (x 2 )€
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+ Implications Any smooth functions with all the same derivatives at a point MUST be the same function!
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+ Proof: If f and g are smooth functions that agree over some interval, they MUST be the same function Let f and g be two smooth functions that agree for some open interval (a,b), but not over all of R 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 Without loss of generality, we will form S, the set of all x>a, such that f(x)≠0 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. c, being a greatest lower bound of S, is also an element of S, since S is closed Now we see that h=0 on (a,c), but h≠0 at c. So, h is discontinuous at c, but then h cannot be smooth Thus we have reached a contradiction, and so f and g must agree everywhere!
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+ Suppose f(x) can be rewritten as a power series… f (x) = c 0 + c1 (x − a) + c 2 (x − a) 2 + ...+ c n (x − a) n c 0 = f (a) f ʹ′(x) = c1 + 2c 2 (x − a) + 3c 3 (x f − a) 2 + ...+ nc n (x − a) n −1 k (a) ck =€ k! c1 = f ʹ′(a)€ f ʹ′ (x) = 2c 2 + 3∗2c 3 (x − € + 4 ∗ 3c 4 (x − a) 2 + ...+ n ∗(n −1)c n (x − a) n −2 a)€ f ʹ′ (a) c2 =€ 2!€ f k (a) ck = k!€
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+ Entirety (Analytic Functions) A function f(x) is said to be entire if it is equal to its Taylor Series everywhere Entire Not Entire sin(x) log(1+x)
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+ Proof: sin(x) is entire Maclaurin Series sin(0)=1 sin’(0)=0 ∞ sin’(0)=-1 (−1) n 2n +1 sin(x) = ∑ x sin’(0)=0 n =0 (2n +1)! sin’(0)=1 sin’(0)=0 sin’(0)=-1 … etc. €
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+ Proof: sin(x) is entire ∞ (−1) n 2n +1 sin(x) = ∑ x n =0 (2n +1)! Lagrange formula for the remainder: Let f : R →R be k+1 times differentiable on (a,x) and continuous on [a,x]. Then f k +1 (z) k +1 Rk (x) = (x − a) (k +1)!€ for some z in (x,a)€
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+ Proof: sin(x) is entire First, sin(x) is continuous and infinitely differentiable over all of R If we look at the Taylor Polynomial of degree k f k +1 (z) k +1 Rk (x) = (x − a) (k +1)! Note though f k +1 (z) ≤ 1 for all z in R k +1 (x − a) Rk (x) ≤€ (k +1)! €€
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+ Proof: sin(x) is entire However, as k goes to infinity, we see Rk (x) ≤ 0 Applyingthe Squeeze Theorem to our original € equation, we obtain that as k goes to infinity f (x) = Tk (x) and thus sin(x) is complete€
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+ Maclaurin Series Examples ∞ ∞ xn xn log(1 − x) = −∑ log(1+ x) = ∑ (−1) n +1 n =1 n! n =1 n! ∞ ∞ 1 (−1) n (2n)! = ∑ xn 1+ x = ∑ xn 1 − x n =0 n =0 (1 − 2n)(n!) 2 (4) n ∞€ xn€ ex = ∑ n =0 n! ∞€ ∞ € (−1) n (−1) n 2n sin(x) = ∑ x 2n +1 cos(x) = ∑ x n =0 (2n +1)! n =0 (2n)!€ ix Note: e = cos(x) + isin(x)€ €
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+ Applications Physics Special Relativity Equation Fermat’s Principle (Optics) Resistivity of Wires Electric Dipoles Periods of Pendulums Surveying (Curvature of the Eart)
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+ Special Relativity m0 m= KE = mc 2 − m0c 2 1− v2 c2 m0c 2 ⎡⎛ v 2 ⎞ −1/ 2 ⎤ KE = − m0c 2 = m0c 2 ⎢⎜1 − 2 ⎟ −1⎥ 2 1−v c 2 € ⎢⎝ c ⎠ ⎣ ⎥ ⎦€ If v ≤ 100 m/s€ Then according to Taylor’s Inequality 1 3m0c 2 100 4 −10 R1 (x) ≤ 2 2 4 < (4.17 × 10 )m0 2 4(1 −100 /c ) c
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