Maclaurin series for inverse trigonometric functions

OCR MEI FP2 2014 June Q1

19 marks Standard +0.8

1

Given that $\mathrm { f } ( x ) = \arccos x$,
1. sketch the graph of $y = \mathrm { f } ( x )$,
2. show that $\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }$,
3. obtain the Maclaurin series for $\mathrm { f } ( x )$ as far as the term in $x ^ { 3 }$.
A curve has polar equation $r = \theta + \sin \theta , \theta \geqslant 0$.
1. By considering $\frac { \mathrm { d } r } { \mathrm {~d} \theta }$ show that $r$ increases as $\theta$ increases. Sketch the curve for $0 \leqslant \theta \leqslant 4 \pi$.
2. You are given that $\sin \theta \approx \theta$ for small $\theta$. Find in terms of $\alpha$ the approximate area bounded by the curve and the lines $\theta = 0$ and $\theta = \alpha$, where $\alpha$ is small.

OCR Further Pure Core 2 2020 November Q10

10 marks Standard +0.8

10 Let $\mathrm { f } ( x ) = \sin ^ { - 1 } ( x )$.

1. Determine $\mathrm { f } ^ { \prime \prime } ( x )$.
2. Determine the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$.
3. By considering the first two non-zero terms of the Maclaurin expansion for $\mathrm { f } ( x )$, find an approximation to $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer correct to 6 decimal places.
By writing $\mathrm { f } ( x )$ as $\sin ^ { - 1 } ( x ) \times 1$, determine the value of $\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x$. Give your answer in exact form.

Edexcel CP2 2021 June Q3

6 marks Standard +0.8

3. $$f ( x ) = \arcsin x \quad - 1 \leqslant x \leqslant 1$$

Determine the first two non-zero terms, in ascending powers of $x$, of the Maclaurin series for $\mathrm { f } ( x )$, giving each coefficient in its simplest form.
Substitute $x = \frac { 1 } { 2 }$ into the answer to part (a) and hence find an approximate value for $\pi$ Give your answer in the form $\frac { p } { q }$ where $p$ and $q$ are integers to be determined.

OCR MEI FP2 2016 June Q1

Standard +0.3

1

1. Given that $\mathrm { f } ( x ) = \arctan x$, write down an expression for $\mathrm { f } ^ { \prime } ( x )$. Assuming that $x$ is small, use a binomial expansion to express $\mathrm { f } ^ { \prime } ( x )$ in ascending powers of $x$ as far as the term in $x ^ { 4 }$.
2. Hence express $\arctan x$ in ascending powers of $x$ as far as the term in $x ^ { 5 }$.
Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
A curve has polar equation $r = \frac { a } { \sqrt { \theta } }$ where $a > 0$.
1. Sketch the curve for $\frac { \pi } { 4 } \leqslant \theta \leqslant 2 \pi$.
2. State what happens to $r$ as $\theta$ tends to zero.
3. Find the area of the region enclosed by the part of the curve sketched in part (i) and the lines $\theta = \frac { \pi } { 4 }$ and $\theta = 2 \pi$. Give your answer in an exact simplified form.
  1. (i) Express $2 \sin \frac { 1 } { 2 } \theta \left( \sin \frac { 1 } { 2 } \theta - \mathrm { j } \cos \frac { 1 } { 2 } \theta \right)$ in terms of $z$ where $z = \cos \theta + \mathrm { j } \sin \theta$.
    (ii) The series $C$ and $S$ are defined as follows. $$\begin{aligned} C & = 1 - \binom { n } { 1 } \cos \theta + \binom { n } { 2 } \cos 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \cos n \theta \\ S & = - \binom { n } { 1 } \sin \theta + \binom { n } { 2 } \sin 2 \theta - \ldots + ( - 1 ) ^ { n } \binom { n } { n } \sin n \theta \end{aligned}$$ Show that $$C + \mathrm { j } S = \left\{ - 2 \mathrm { j } \sin \frac { 1 } { 2 } \theta \left( \cos \frac { 1 } { 2 } \theta + \mathrm { j } \sin \frac { 1 } { 2 } \theta \right) \right\} ^ { n } .$$ Hence show that, for even values of $n$, $$\frac { C } { S } = \cot \left( \frac { 1 } { 2 } n \theta \right)$$
  2. Write the complex number $z = \sqrt { 6 } + \mathrm { j } \sqrt { 2 }$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$, expressing $r$ and $\theta$ as simply as possible. Hence find the cube roots of $z$ in the form $r \mathrm { e } ^ { \mathrm { j } \theta }$. Show the points representing $z$ and its cube roots on an Argand diagram.
    1. Find the eigenvalues and eigenvectors of the matrix $\mathbf { M }$, where $$\mathbf { M } = \left( \begin{array} { l l } \frac { 1 } { 2 } & \frac { 1 } { 2 } \\ \frac { 2 } { 3 } & \frac { 1 } { 3 } \end{array} \right)$$ Hence express $\mathbf { M }$ in the form $\mathbf { P D P } ^ { - 1 }$ where $\mathbf { D }$ is a diagonal matrix.
    2. Write down an equation for $\mathbf { M } ^ { n }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$. Hence obtain expressions for the elements of $\mathbf { M } ^ { n }$.
      Show that $\mathbf { M } ^ { n }$ tends to a limit as $n$ tends to infinity. Find that limit.
    3. Express $\mathbf { M } ^ { - 1 }$ in terms of the matrices $\mathbf { P }$ and $\mathbf { D }$. Hence determine whether or not $\left( \mathbf { M } ^ { - 1 } \right) ^ { n }$ tends to a limit as $n$ tends to infinity. Section B (18 marks)
      1. Given that $y = \cosh x$, use the definition of $\cosh x$ in terms of exponential functions to prove that $$x = \pm \ln \left( y + \sqrt { y ^ { 2 } - 1 } \right) .$$
      2. Solve the equation $$\cosh x + \cosh 2 x = 5$$ giving the roots in an exact logarithmic form.
      3. Sketch the curve with equation $y = \cosh x + \cosh 2 x$. Show on your sketch the line $y = 5$. Find the area of the finite region bounded by the curve and the line $y = 5$. Give your answer in an exact form that does not involve hyperbolic functions. \section*{END OF QUESTION PAPER}

OCR MEI FP2 2009 June Q3

19 marks Standard +0.8

1. Sketch the graph of $y = \arcsin x$ for $-1 \leq x \leq 1$. [1] Find $\frac{dy}{dx}$, justifying the sign of your answer by reference to your sketch. [4]
2. Find the exact value of the integral $\int_0^1 \frac{1}{\sqrt{2 - x^2}} dx$. [3]
The infinite series $C$ and $S$ are defined as follows. $$C = \cos \theta + \frac{1}{3}\cos 3\theta + \frac{1}{5}\cos 5\theta + \ldots$$ $$S = \sin \theta + \frac{1}{3}\sin 3\theta + \frac{1}{5}\sin 5\theta + \ldots$$ By considering $C + jS$, show that $$C = \frac{3\cos \theta}{5 - 3\cos 2\theta},$$ and find a similar expression for $S$. [11]