4.08g Derivatives: inverse trig and hyperbolic functions

8 \includegraphics[max width=\textwidth, alt={}, center]{5b43cb39-7560-4484-ba6f-17303e986f47-10_369_1531_260_306} The diagram shows the curve \(\mathrm { y } = \frac { 1 } { \sqrt { \mathrm { x } ^ { 2 } + \mathrm { x } + 1 } }\) for \(x \geqslant 0\), together with a set of \(n\) rectangles of unit width. By considering the sum of the areas of these rectangles, show that $$\sum _ { r = 1 } ^ { n } \frac { 1 } { \sqrt { r ^ { 2 } + r + 1 } } < \ln \left( \frac { 1 } { 3 } + \frac { 2 } { 3 } n + \frac { 2 } { 3 } \sqrt { n ^ { 2 } + n + 1 } \right)$$

4

1. Starting from the definitions of cosh and sinh in terms of exponentials, prove that $$\cosh ^ { 2 } x - \sinh ^ { 2 } x = 1 .$$
2. Show that \(\frac { \mathrm { d } } { \mathrm { dx } } \left( \tan ^ { - 1 } ( \sinh x ) \right) = \operatorname { sech } x\).
3. Sketch the graph of \(y = \operatorname { sechx }\), stating the equation of the asymptote.
4. By considering a suitable set of \(n\) rectangles of unit width, use your sketch to show that $$\sum _ { r = 1 } ^ { n } \operatorname { sechr } < \tan ^ { - 1 } ( \operatorname { sinhn } )$$
5. Hence state an upper bound, in terms of \(\pi\), for \(\sum _ { r = 1 } ^ { \infty }\) sech \(r\).

5. Determine

1. \(\int \frac { 1 } { \sqrt { x ^ { 2 } - 3 x + 5 } } \mathrm {~d} x\)
2. \(\int \frac { 1 } { \sqrt { 63 + 4 x - 4 x ^ { 2 } } } \mathrm {~d} x\)

1. In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

1. Show that $$\int _ { 4 } ^ { 4 \sqrt { 3 } } \frac { 8 } { 16 + x ^ { 2 } } d x = p \pi$$ where \(p\) is a rational number to be determined.
2. Determine the exact value of \(k\) for which $$\int _ { \frac { 3 } { 4 } } ^ { k } \frac { 2 } { \sqrt { 9 - 4 x ^ { 2 } } } d x = \frac { \pi } { 12 }$$

4. \(y = \arctan ( \sqrt { } x ) , \quad x > 0,0 < y < \frac { \pi } { 2 }\).

1. Find the value of \(\frac { \mathrm { dy } } { \mathrm { dx } }\) at \(\mathrm { x } = \frac { 1 } { 4 }\).
2. Show that \(2 x ( 1 + x ) \frac { d ^ { 2 } y } { d x ^ { 2 } } + ( 1 + 3 x ) \frac { d y } { d x } = 0\).

8. The curve \(C\) has equation $$y = \ln \left( \frac { \mathrm { e } ^ { x } + 1 } { \mathrm { e } ^ { x } - 1 } \right) , \quad \ln 2 \leqslant x \leqslant \ln 3$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = - \frac { 2 \mathrm { e } ^ { x } } { \mathrm { e } ^ { 2 x } - 1 }$$
2. Find the length of the curve \(C\), giving your answer in the form \(\ln a\), where \(a\) is a rational number.
   (6)

4. The curve \(C\) has equation $$y = \operatorname { arsinh } x + x \sqrt { x ^ { 2 } + 1 } , \quad 0 \leqslant x \leqslant 1$$

1. Show that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 \sqrt { x ^ { 2 } + 1 }\)
2. Hence show that the length of the curve \(C\) is given by $$\int _ { 0 } ^ { 1 } \sqrt { 4 x ^ { 2 } + 5 } d x$$
3. Using the substitution \(x = \frac { \sqrt { 5 } } { 2 } \sinh u\), find the exact length of the curve \(C\), giving your answer in the form \(a + b \ln c\), where \(a , b\) and \(c\) are constants to be found.

1

1. 1. Given that \(\mathrm { f } ( t ) = \arcsin t\), write down an expression for \(\mathrm { f } ^ { \prime } ( t )\) and show that $$\mathrm { f } ^ { \prime \prime } ( t ) = \frac { t } { \left( 1 - t ^ { 2 } \right) ^ { \frac { 3 } { 2 } } }$$
   2. Show that the Maclaurin expansion of the function \(\arcsin \left( x + \frac { 1 } { 2 } \right)\) begins $$\frac { \pi } { 6 } + \frac { 2 } { \sqrt { 3 } } x$$ and find the term in \(x ^ { 2 }\).
2. Sketch the curve with polar equation \(r = \frac { \pi a } { \pi + \theta }\), where \(a > 0\), for \(0 \leqslant \theta < 2 \pi\). Find, in terms of \(a\), the area of the region bounded by the part of the curve for which \(0 \leqslant \theta \leqslant \pi\) and the lines \(\theta = 0\) and \(\theta = \pi\).
3. Find the exact value of the integral $$\int _ { 0 } ^ { \frac { 3 } { 2 } } \frac { 1 } { 9 + 4 x ^ { 2 } } \mathrm {~d} x$$

7 It is given that \(\mathrm { f } ( x ) = \tanh ^ { - 1 } \left( \frac { 1 - x } { 2 + x } \right)\), for \(x > - \frac { 1 } { 2 }\).

1. Show that \(\mathrm { f } ^ { \prime } ( x ) = - \frac { 1 } { 1 + 2 x }\), and find \(\mathrm { f } ^ { \prime \prime } ( x )\).
2. Show that the first three terms of the Maclaurin series for \(\mathrm { f } ( x )\) can be written as \(\ln a + b x + c x ^ { 2 }\), for constants \(a , b\) and \(c\) to be found.

5

1. Prove that, if \(y = \sin ^ { - 1 } x\), then \(\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { \sqrt { 1 - x ^ { 2 } } }\).
2. Find the Maclaurin series for \(\sin ^ { - 1 } x\), up to and including the term in \(x ^ { 3 }\).
3. Use the result of part (ii) and the Maclaurin series for \(\ln ( 1 + x )\) to find the Maclaurin series for \(\left( \sin ^ { - 1 } x \right) \ln ( 1 + x )\), up to and including the term in \(x ^ { 4 }\).

5 It is given that \(y = \tan ^ { - 1 } 2 x\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) and show that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 4 x \left( \frac { \mathrm {~d} y } { \mathrm {~d} x } \right) ^ { 2 } = 0\).
2. Find the Maclaurin series for \(y\) up to and including the term in \(x ^ { 3 }\). Show all your working.
3. The result in part (ii), together with the value \(x = \frac { 1 } { 2 }\), is used to find an estimate for \(\pi\). Show that this estimate is only correct to 1 significant figure.

6 \includegraphics[max width=\textwidth, alt={}, center]{debf6581-25ff-4692-bdfb-154675a3cdb0-3_608_1134_258_504} The diagram shows the curve \(y = \mathrm { f } ( x )\), defined by $$f ( x ) = \begin{cases} x ^ { x } & \text { for } 0 < x \leqslant 1 , \\ 1 & \text { for } x = 0 . \end{cases}$$

1. By first taking logarithms, show that the curve has a stationary point at \(x = \mathrm { e } ^ { - 1 }\). The area under the curve from \(x = 0.5\) to \(x = 1\) is denoted by \(A\).
2. By considering the set of three rectangles shown in the diagram, show that a lower bound for \(A\) is 0.388 .
3. By considering another set of three rectangles, find an upper bound for \(A\), giving 3 decimal places in your answer. The area under the curve from \(x = 0\) to \(x = 0.5\) is denoted by \(B\).
4. Draw a diagram to show rectangles which could be used to find lower and upper bounds for \(B\), using not more than three rectangles for each bound. (You are not required to find the bounds.)

1 It is given that \(\mathrm { f } ( x ) = \tan ^ { - 1 } 2 x\) and \(\mathrm { g } ( x ) = p \tan ^ { - 1 } x\), where \(p\) is a constant. Find the value of \(p\) for which \(\mathrm { f } ^ { \prime } \left( \frac { 1 } { 2 } \right) = \mathrm { g } ^ { \prime } \left( \frac { 1 } { 2 } \right)\).

6

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sinh ^ { - 1 } x \right) = \frac { 1 } { \sqrt { x ^ { 2 } + 1 } }\).
2. Given that \(y = \cosh \left( a \sinh ^ { - 1 } x \right)\), where \(a\) is a constant, show that $$\left( x ^ { 2 } + 1 \right) \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + x \frac { \mathrm {~d} y } { \mathrm {~d} x } - a ^ { 2 } y = 0$$

4 It is given that, for \(n \geqslant 0\), $$I _ { n } = \int _ { 0 } ^ { 1 } x ^ { n } e ^ { - x ^ { 2 } } d x$$

1. Find \(I _ { 1 }\) in terms of c .
2. Show that $$I _ { n + 2 } = \frac { n + 1 } { 2 } I _ { n } - \frac { 1 } { 2 \mathrm { e } }$$
3. Find \(I _ { 5 }\) in terms of \(e\).

1 Given that \(y = \sin ^ { - 1 } \left( x ^ { 2 } \right)\), find \(\frac { d y } { d x }\).

3

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } u } \left( \sinh ^ { - 1 } u \right) = \frac { 1 } { \sqrt { u ^ { 2 } + 1 } }\).
2. Find the equation of the normal to the graph of \(\mathrm { y } = \sinh ^ { - 1 } 2 \mathrm { x }\) at the point where \(x = \sqrt { 6 }\). Give your answer in the form \(\mathrm { y } = \mathrm { mx } + \mathrm { c }\) where \(m\) and \(c\) are given in exact, non-hyperbolic form.

5

1. Given that \(u = \sqrt { 1 - x ^ { 2 } }\), find \(\frac { \mathrm { d } u } { \mathrm {~d} x }\).
2. Use integration by parts to show that $$\int _ { 0 } ^ { \frac { \sqrt { 3 } } { 2 } } \sin ^ { - 1 } x \mathrm {~d} x = a \sqrt { 3 } \pi + b$$ where \(a\) and \(b\) are rational numbers.

1. Show that

$$\int _ { 0 } ^ { \infty } \frac { 8 x - 12 } { \left( 2 x ^ { 2 } + 3 \right) ( x + 1 ) } \mathrm { d } x = \ln k$$ where \(k\) is a rational number to be found.

1. (a)

$$y = \tan ^ { - 1 } x$$ Assuming the derivative of \(\tan x\), prove that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { 1 } { 1 + x ^ { 2 } }$$ $$\mathrm { f } ( x ) = x \tan ^ { - 1 } 4 x$$ (b) Show that $$\int \mathrm { f } ( x ) \mathrm { d } x = A x ^ { 2 } \tan ^ { - 1 } 4 x + B x + C \tan ^ { - 1 } 4 x + k$$ where \(k\) is an arbitrary constant and \(A , B\) and \(C\) are constants to be determined.
(c) Hence find, in exact form, the mean value of \(\mathrm { f } ( x )\) over the interval \(\left[ 0 , \frac { \sqrt { 3 } } { 4 } \right]\)

6

1. Given that \(y = ( x - 2 ) \sqrt { 5 + 4 x - x ^ { 2 } } + 9 \sin ^ { - 1 } \left( \frac { x - 2 } { 3 } \right)\), show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = k \sqrt { 5 + 4 x - x ^ { 2 } }$$ where \(k\) is an integer.
2. Hence show that $$\int _ { 2 } ^ { \frac { 7 } { 2 } } \sqrt { 5 + 4 x - x ^ { 2 } } \mathrm {~d} x = p \sqrt { 3 } + q \pi$$ where \(p\) and \(q\) are rational numbers.
   [0pt] [3 marks]

4 You are given that \(y = \tan ^ { - 1 } \sqrt { 2 x }\).

1. Find \(\frac { \mathrm { d } y } { \mathrm {~d} x }\).
2. Show that \(\int _ { \frac { 1 } { 6 } } ^ { \frac { 1 } { 2 } } \frac { \sqrt { x } } { \left( x + 2 x ^ { 2 } \right) } \mathrm { d } x = k \pi\) where \(k\) is a number to be determined in exact form.

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

1. 1. Determine \(f ^ { \prime \prime } ( x )\).
   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 \(\mathrm { f } ( \mathrm { x } )\), find an approximation to \(\int _ { 0 } ^ { \frac { 1 } { 2 } } \mathrm { f } ( x ) \mathrm { d } x\). Give your answer correct to 6 decimal places.
2. 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.

1

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 \(x ^ { 5 }\).
2. Find, in exact form, the value of the following integral. $$\int _ { 0 } ^ { \frac { 3 } { 4 } } \frac { 1 } { \sqrt { 3 - 4 x ^ { 2 } } } \mathrm {~d} x$$
3. 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}

13

1. (a) Given that \(x \geqslant 1\), show that \(\sec ^ { - 1 } x = \cos ^ { - 1 } \left( \frac { 1 } { x } \right)\), and deduce that \(\frac { \mathrm { d } } { \mathrm { d } x } \left( \sec ^ { - 1 } x \right) = \frac { 1 } { x \sqrt { x ^ { 2 } - 1 } }\).
   (b) Use integration by parts to determine \(\int \sec ^ { - 1 } x \mathrm {~d} x\).
2. \includegraphics[max width=\textwidth, alt={}, center]{5d526fd9-72f8-42b1-b156-fd4a0c764c82-4_670_1029_1073_596} The diagram shows the curve \(S\) with equation \(y = \sec ^ { - 1 } x\) for \(x \geqslant 1\). The line \(L\), with gradient \(\frac { 1 } { \sqrt { 2 } }\), is the tangent to \(S\) at the point \(P\) and cuts the \(x\)-axis at the point \(Q\). The point \(I\) has coordinates \(( 1,0 )\).
   (a) Determine the exact coordinates of \(P\) and \(Q\).
   (b) The region \(R\), shaded on the diagram, is bounded by the line segments \(P Q\) and \(Q I\) and the \(\operatorname { arc } I P\) of \(S\). Show that \(R\) has area $$\ln ( 1 + \sqrt { 2 } ) - \frac { \pi ( 8 - \pi ) \sqrt { 2 } } { 32 } .$$ {www.cie.org.uk} after the live examination series. }