Calc 1 Review: Taylor’s Thm for f : R → R. Suppose f ∈ C^{k}, Let p_{k}(x) = f(a) + f′(a)(x − a) + ... + \frac{f^{(k)}(a)}{k!}(x − a)^{k} Then f(x) = p_{k}(x) + R_{k}(x, a) where lim_{x→a} \frac{R_{k}(x,a)}{(x−a)^{k}} = 0. Prop 1.2: If f^{(k+1)} exists, then there exists c between a and x such that R_{k}(x, a) = \frac{f^{(k+1)}(c)}{(k+1)!}(x − a)^{k+1} Estimate ln(2) using degree 3 Taylor polynomial for f(x) = ln(x) about a = 1 f(x) = ln(x) f′(x) = x^{−1} f′′(x) = −x^{−2} f′′′(x) = 2x^{−3} p_{3}(x) =

Thus ln(2) ∼ p_{3}(2) =

ln(x) = p_{3}(x) + R_{3}(x − 1) f^{(4)}(x) = −6x^{−4} = −\frac{6}{x^{4}} R_{3}(x − 1) = \frac{f^{(4)}(c)}{(4)!}(x − 1)^{4} =

where c is btwn 1 and 2.

ln(2) = p_{3}(2) + R_{3}(2, 1) = where c ∈ (1, 2)

Multivariable version: Taylor’s Thm for f : R → R. Suppose f ∈ C^{k}, Let p_{k}(x) = f(a) + Σ_{i=1}^{n}f_{x_{i}}(a)(x_{i} − a_{i}) + ...

+\frac{1}{k!}Σ_{i_{1},...,i_{k}=1}^{n}f_{i_{1}...i_{k}}(a)(x_{i_{1}} − a_{i_{1}})...(x_{i_{k}} − a_{i_{k}})

Then f(x) = p_{k}(x) + R_{k}(x, a) where lim_{x→a} \frac{R_{k}(x,a)}{||x−a||^{k}} = 0. If f : R^{n} → R ∈ C^{2}, then R_{1}(x, a) = \frac{1}{2}Σ_{i,j=1}^{n}f_{x_{i}x_{j}}(c)(x_{i} − a_{i})(x_{j} − a_{j}) for some c on the line segment joining a and x. If f : R^{n} → R ∈ C^{k+1}, then R_{k}(x, a) = \frac{1}{(k+1)!}Σ_{i_{1},...,i_{k+1}=1}^{n}f_{x_{i_{1}} ...x_{i_{k+1}}} (c)(x_{i_{1}}−a_{i_{1}})...(x_{i_{k+1}}−a_{i_{k+1}}) for some c on the line segment joining a and x.