taylor |x|

Solution

taylor

∣ x ∣

1 + (x - 1) + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( ∣ x ∣ ) ( 1 )}{1 !} (x - 1) + \frac{\frac{d ^{2}}{d x ^{2}} ( ∣ x ∣ ) ( 1 )}{2 !} (x - 1)^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( ∣ x ∣ ) ( 1 )}{3 !} (x - 1)^{3} + \dots

Evaluate Derivatives

= 1 + \frac{1}{1 !} (x - 1) + \frac{0}{2 !} (x - 1)^{2} + \frac{0}{3 !} (x - 1)^{3} + \frac{0}{4 !} (x - 1)^{4} + \frac{0}{5 !} (x - 1)^{5} + \frac{0}{6 !} (x - 1)^{6} + \frac{0}{7 !} (x - 1)^{7} + \frac{0}{8 !} (x - 1)^{8} + \frac{0}{9 !} (x - 1)^{9} + \frac{0}{10 !} (x - 1)^{10} + \dots

Refine

= 1 + (x - 1) + \dots

\frac{d}{d x} (\frac{1}{6} (e^{3 x} + e^{- 3 x}))

\frac{d x}{d t} = 3 x (5 - x)

\frac{\partial}{\partial x} (\frac{x}{x ^{2} - 2 x})

f (x) = \frac{1}{x ^{2}}, (- 1, 1)

y^{^{'}} = 0.3 y (1 - \frac{y}{2500}) - 120