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The coefficients are found sequentially by applying the three matching conditions at $x=0$. First, substituting $x=0$ into the polynomial and the function gives $c_0=1$.

Conditions: The approximation is centered at $x=0$.; The target function is $e^x$.; The approximating polynomial is $c_0+c_1x+c_2x^2$.

The coefficients are found sequentially by applying the three matching conditions at $x=0$. First, substituting $x=0$ into the polynomial and the function gives $c_0=1$.

Conditions: The approximation is centered at $x=0$.; The target function is $e^x$.; The approximating polynomial is $c_0+c_1x+c_2x^2$.