[Maximum number: 7]

The function f is defined by $f(x)=\mathrm{e}^{-x} \cos x+x-1$ .
By finding a suitable number of derivatives of f, determine the first non-zero term in its Maclaurin series.

(a)

Find the first three terms of the Maclaurin series for $\ln \left(1+\mathrm{e}^{x}\right)$ .

[ 6 ]

(b)

Hence, or otherwise, determine the value of $\lim _{x \rightarrow 0} \frac{2 \ln \left(1+\mathrm{e}^{x}\right)-x-\ln 4}{x^{2}}$ .

[ 4 ]

[Maximum number: 10]

In this question, you will be investigating the family of functions of the form $f(x)=x^{n} \mathrm{e}^{-x}$ .
Consider the family of functions $f_{n}(x)=x^{n} \mathrm{e}^{-x}$ , where $x \geq 0$ and $n \in \mathbb{Z}^{+}$ .
When n=1, the function $f_{1}(x)=x \mathrm{e}^{-x}$ , where $x \geq 0$ .

(a)

Show that the area of the region bounded by the graph $y=f_{1}(x)$ , the x-axis and the line x=b, where b>0, is given by $\frac{\mathrm{e}^{b}-b-1}{\mathrm{e}^{b}}$ .

You may assume that the total area, $A_{n}$ , of the region between the graph $y=f_{n}(x)$ and the x-axis can be written as $A_{n}=\int_{0}^{\infty} f_{n}(x) \mathrm{d} x$ and is given by $\lim _{b \rightarrow \infty} \int_{0}^{b} f_{n}(x) \mathrm{d} x$ .

[ 6 ]

(b)

(i)

Use l'Hôpital's rule to find $\lim _{b \rightarrow \infty} \frac{\mathrm{e}^{b}-b-1}{\mathrm{e}^{b}}$ . You may assume that the condition for applying l'Hôpital's rule has been met.

[ 2 ]

(ii)

Hence write down the value of $A_{1}$ .

You are given that $A_{2}=2$ and $A_{3}=6$ .

[ 1 ]

(c)

Suggest an expression for $A_{n}$ in terms of n, where $n \in \mathbb{Z}^{+}$ .

[ 1 ]

[Maximum number: 6]

In this question you will investigate series of the form

\sum_{i=1}^{n} \boldsymbol{i}^{q}=1^{q}+2^{q}+3^{q}+\ldots+n^{q} \text { where } n, q \in \mathbb{Z}^{+}

and use various methods to find polynomials, in terms of n, for such series.
When q=1, the above series is arithmetic.

(a)

(i)

Show that $\lim _{x \rightarrow 1} f_{1}(x)$ is in indeterminate form.

[ 1 ]

(ii)

Hence, by applying l'Hôpital's rule, show that $\lim _{x \rightarrow 1} f_{1}(x)=\frac{1}{2} n(n+1)$ .

[ 5 ]

[Maximum number: 7]

Use l'Hôpital's rule to determine the value of

\lim _{x \rightarrow 0} \frac{\sin ^{2} x}{x \ln (1+x)}

[Maximum number: 6]

Use L'Hôpital's Rule to find $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{x}-1-x \cos x}{\sin ^{2} x}$ .

[Maximum number: 5]

Find $\lim _{x \rightarrow \frac{1}{2}}\left(\frac{\left(\frac{1}{4}-x^{2}\right)}{\cot \pi x}\right)$ .

[Maximum number: 7]

Find $\lim _{x \rightarrow 0}\left(\frac{1-\cos x^{6}}{x^{12}}\right)$ .

[Maximum number: 8]

Given that $\frac{\mathrm{d} y}{\mathrm{~d} x}-2 y^{2}=\mathrm{e}^{x}$ and y=1 when x=0, use Euler's method with a step length of 0.1 to find an approximation for the value of y when x=0.4. Give all intermediate values with maximum possible accuracy.

[Maximum number: 17]

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

(a)

By finding a suitable number of derivatives of f, determine the Maclaurin series for f(x) as far as the term in $x^{3}$ .

[ 7 ]

(b)

Hence, or otherwise, determine the exact value of $\lim _{x \rightarrow 0} \frac{\mathrm{e}^{x} \sin x-x-x^{2}}{x^{3}}$ .

[ 3 ]

(c)

The Maclaurin series is to be used to find an approximate value for f(0.5).

[ 7 ]

(i)

Use the Lagrange form of the error term to find an upper bound for the absolute value of the error in this approximation.

(ii)

Deduce from the Lagrange error term whether the approximation will be greater than or less than the actual value of f(0.5).

[ 7 ]

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