What is taylor e^{-2x^2} ?

The answer to taylor e^{-2x^2} is 1-2x^2+2x^4-4/3 x^6+2/3 x^8+\ldots

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taylor e^{-2x^2}

taylor

e^{- 2 x^{2}}

1 - 2 x^{2} + 2 x^{4} - \frac{4}{3} x^{6} + \frac{2}{3} x^{8} + \dots

Solution steps

Apply the Taylor Formula

= 1 + \frac{\frac{d}{d x} ( e ^{- 2 x^{2}} ) ( 0 )}{1 !} x + \frac{\frac{d ^{2}}{d x ^{2}} ( e ^{- 2 x^{2}} ) ( 0 )}{2 !} x^{2} + \frac{\frac{d ^{3}}{d x ^{3}} ( e ^{- 2 x^{2}} ) ( 0 )}{3 !} x^{3} + \dots

Evaluate Derivatives

= 1 + \frac{0}{1 !} x + \frac{- 4}{2 !} x^{2} + \frac{0}{3 !} x^{3} + \frac{48}{4 !} x^{4} + \frac{0}{5 !} x^{5} + \frac{- 960}{6 !} x^{6} + \frac{0}{7 !} x^{7} + \frac{26880}{8 !} x^{8} + \dots

Refine

= 1 - 2 x^{2} + 2 x^{4} - \frac{4}{3} x^{6} + \frac{2}{3} x^{8} + \dots

x = - y^{2} + 2, x = y^{2} - 4

\int_{- 2}^{3} \frac{42}{x ^{4}} d x

\frac{d}{d x} (8 x - \frac{3}{x})

\frac{d}{d t} (5^{2 t})

\frac{d}{d x} (a e^{2 x})

What is taylor e^{-2x^2} ?
The answer to taylor e^{-2x^2} is 1-2x^2+2x^4-4/3 x^6+2/3 x^8+\ldots