## Exam-Style Question on Maclaurin Series## A mathematics exam-style question with a worked solution that can be revealed gradually |

Question id: 632. This question is similar to one that appeared on an IB AA Higher paper in 2022. The use of a calculator is not allowed.

The function \(f\) is defined by \(f(x) = e^x \sin{x} \), where \(x \in \mathbb{R}\).

(a) Find the Maclaurin series for \(f(x)\) up to and including the \(x^3\) term.

(b) Hence, find an approximate value for \( \int_0^1 e^{x^2}\sin(x^2)dx\).

The function \(g\) is defined by \(g(x) = e^x\cos{x}\), where \(x \in \mathbb{R}\).

(c) Show that \(g(x)\) satisfies the equation \(g''(x) = 2(g'(x) — g(x)) \).

(d) Show that \(g(x)\) also satisfies the equation \(g^{(4)}(x) = 2(g'''(x) — g''(x)) \).

(e) Using these results, find the Maclaurin series for \(g(x)\) up to and including the \(x^4 \) term.

(f) Hence determine the value of \( \lim \limits_{x \to 0} \frac{e^x \cos{x} -1-x}{x^3} \)

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