Mathematics

Problem

Let

f

be the function defined for all real

x

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

. (a) Compute

f(0)

f'(0)

f''(0)

f'''(0)

, and

f^{(4)}(0)

. Show all algebraic work leading to each value. (b) Using the values from part (a), write the fourth-degree Maclaurin polynomial

P_4(x)

for

f

. (c) Use

P_4

to approximate

f(0.1)

. (d) Explain whether the approximation in part (c) is likely to be an overestimate or an underestimate, citing an appropriate property of

f

or its derivatives.