Let be the function defined for all real by .

MATHEMATICS · Multi-part · IB Diploma Programme · Mathematics: Analysis and Approaches · Calculus · Calculus

Problem

Let

f

be the function defined for all real

x

f(x)=e^{2x}\sin x

. (a) Write the first three nonzero terms of the Maclaurin series for

e^{2x}

and the first three nonzero terms of the Maclaurin series for

\sin x

. Clearly indicate the degree of each term. (b) Using your expansions from part (a), determine the coefficient of

x^{5}

in the Maclaurin series of

f

. Show all algebraic steps. (c) Hence calculate

f^{(5)}(0)

. State your final answer as an exact value.