Math · Calculus · Worked example

Approximate e^0.2 with a quadratic

Build the degree-two approximation at zero and bound its error.

Find the coefficients

Every derivative of eˣ is eˣ, so each derivative at zero is 1.

P_{2} (x) = 1 + x + \frac{x^{2}}{2}

Substitute 0.2 in the polynomial.

P_{2} (0.2) = 1 + 0.2 + 0.02 = 1.22

On [0,0.2], the third derivative is at most e^0.2<1.23.

∣ e^{0.2} - 1.22 ∣ < \frac{1.23 {(0.2)}^{3}}{6} = 0.00164

The quadratic gives 1.22 with error less than 0.00164.

Find the cubic Taylor polynomial of sin x at zero.

Show the answer and explanation

The first four derivatives at zero are 0,1,0,−1; divide by the corresponding factorials.

Change an input and predict the result before checking it.

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