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The video uses these three conditions because a generic quadratic polynomial has exactly three unknown coefficients. By requiring the approximation to match the function value, the first derivative (slope), and the second derivative (concavity) at the expansion point, the lecture creates a system of three equations that determines the three coefficients uniquely within the worked setup.

Conditions: The approximating polynomial is quadratic.; The matching point is $x=0$.; The target function and polynomial can be differentiated at least twice at that point.

The video uses these three conditions because a generic quadratic polynomial has exactly three unknown coefficients. By requiring the approximation to match the function value, the first derivative (slope), and the second derivative (concavity) at the expansion point, the lecture creates a system of three equations that determines the three coefficients uniquely within the worked setup.

Conditions: The approximating polynomial is quadratic.; The matching point is $x=0$.; The target function and polynomial can be differentiated at least twice at that point.

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 $c_0+c_1x+c_2x^2$.