Which expression represents the Maclaurin polynomial of degree 3 for e^x?

Which expression represents the Maclaurin polynomial of degree 3 for e^x?

• A. 1 + x + x^2/2
• B. 1 + x + x^2/2 + x^3/2
• C. 1 + x + x^2/2 + x^3/6
• D. 1 + x^2/2 + x^3/6

Answer: (C)

Think of the Maclaurin polynomial as the Taylor expansion of e^x around 0, kept only up to degree 3. The power series for e^x around 0 is e^x = 1 + x + x^2/2! + x^3/3! + ..., so keeping terms through x^3 gives 1 + x + x^2/2 + x^3/6. The coefficients come from the derivatives at 0: since every derivative of e^x is e^x, evaluating at 0 gives 1, so the coefficient of x^n is 1/n!. If you omit the x term or misstate the x^3 coefficient, you’re not representing the third-degree truncation of the series.

Think of the Maclaurin polynomial as the Taylor expansion of e^x around 0, kept only up to degree 3. The power series for e^x around 0 is e^x = 1 + x + x^2/2! + x^3/3! + ..., so keeping terms through x^3 gives 1 + x + x^2/2 + x^3/6. The coefficients come from the derivatives at 0: since every derivative of e^x is e^x, evaluating at 0 gives 1, so the coefficient of x^n is 1/n!. If you omit the x term or misstate the x^3 coefficient, you’re not representing the third-degree truncation of the series.