1.07k Differentiate trig: sin(kx), cos(kx), tan(kx)

8 Show that $$\int _ { 0 } ^ { \frac { \pi } { 2 } } ( x \sin 4 x ) \mathrm { d } x = - \frac { \pi } { 8 }$$

6. $$y = \ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right) \quad - \frac { 1 } { 2 } < x < \frac { 1 } { 2 }$$

1. Show that $$\frac { \mathrm { d } y } { \mathrm {~d} x } = 2 - 3 \tan 3 x$$
2. Determine \(\frac { \mathrm { d } ^ { 4 } y } { \mathrm {~d} x ^ { 4 } }\)
3. Hence determine the first 3 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln \left( \mathrm { e } ^ { 2 x } \cos 3 x \right)\), giving each coefficient in simplest form.
4. Use the Maclaurin series expansion for \(\ln ( 1 + x )\) to write down the first 4 non-zero terms in ascending powers of \(x\) of the Maclaurin series expansion of \(\ln ( 1 + k x )\), where \(k\) is a constant.
5. Hence determine the value of \(k\) for which $$\lim _ { x \rightarrow 0 } \left( \frac { 1 } { x ^ { 2 } } \ln \frac { \mathrm { e } ^ { 2 x } \cos 3 x } { 1 + k x } \right)$$ exists.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{ac5bf967-8b97-4bf3-991f-28c3ec7a25da-4_622_896_957_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{1c957cfe-bead-41d9-8985-479e876e1616-4_620_896_959_333} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled C has coordinates ( \(0.3,0.04\) ). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + \mathrm { g }\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

10 A curve has equation $$y = \mathrm { e } ^ { a x } \cos b x$$ where \(a\) and \(b\) are constants.

1. Show that, at any stationary points on the curve, \(\tan b x = \frac { a } { b }\).
2. \includegraphics[max width=\textwidth, alt={}, center]{8a0a6e46-99cf-4217-93ad-5ed6e9d7c4ef-4_624_897_1062_342} Values of related quantities \(x\) and \(y\) were measured in an experiment and plotted on a graph of \(y\) against \(x\), as shown in the diagram. Two of the points, labelled \(A\) and \(B\), have coordinates \(( 0,1 )\) and \(( 0.2 , - 0.8 )\) respectively. A third point labelled \(C\) has coordinates \(( 0.3,0.04 )\). Attempts were then made to find the equation of a curve which fitted closely to these three points, and two models were proposed. In the first model the equation is \(y = \mathrm { e } ^ { - x } \cos 15 x\).
   In the second model the equation is \(y = f \cos ( \lambda x ) + g\), where the constants \(f , \lambda\), and \(g\) are chosen to give a maximum precisely at the point \(A ( 0,1 )\) and a minimum precisely at the point \(B ( 0.2 , - 0.8 )\). By calculating suitable values evaluate the suitability of the two models.

10

1. Show that \(\frac { \mathrm { d } } { \mathrm { d } x } ( \sin 3 x - 3 x \cos 3 x ) = 9 x \sin 3 x\). The curve shown in the figure below is part of the graph of the function \(y = x \sin 3 x\). \includegraphics[max width=\textwidth, alt={}, center]{3e4281d1-dbad-46a2-bbb7-97706bda2dfa-3_508_1136_1939_466}
2. Show that \(\int _ { 0 } ^ { \frac { 2 \pi } { 3 } } | x \sin 3 x | \mathrm { d } x = \frac { 4 \pi } { 9 }\).

\includegraphics{figure_10} The diagram shows part of the curve \(y = \frac{8}{\sqrt{(3x + 4)}}\). The curve intersects the \(y\)-axis at \(A(0, 4)\). The normal to the curve at \(A\) intersects the line \(x = 4\) at the point \(B\).

1. Find the coordinates of \(B\). [5]
2. Show, with all necessary working, that the areas of the regions marked \(P\) and \(Q\) are equal. [6]

The function \(\text{f}\) is defined by \(\text{f}(x) = \tan^2\left(\frac{1}{2}x\right)\) for \(0 \leqslant x < \pi\).

1. Find the exact value of \(\text{f}'\left(\frac{\pi}{3}\right)\). [3]

1. By differentiating \(\frac{\cos x}{\sin x}\), show that if \(y = \cot x\) then \(\frac{dy}{dx} = -\cosec^2 x\). [3]
2. Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cosec^2 x \, dx = \sqrt{3}\). [2]

By using appropriate trigonometrical identities, find the exact value of

1. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \cot^2 x \, dx\), [3]
2. \(\int_{\frac{\pi}{6}}^{\frac{\pi}{2}} \frac{1}{1 - \cos 2x} \, dx\). [3]

A curve has equation \(y = 2\sin 2x - 5\cos 2x + 6\) and is defined for \(0 \leq x \leq \pi\). Find the \(x\)-coordinates of the stationary points of the curve, giving your answers correct to 3 significant figures. [6]

\includegraphics{figure_7} The diagram shows the curve with equation \(y = \sin 2x + 3\cos 2x\) for \(0 \leqslant x \leqslant \pi\). At the points \(P\) and \(Q\) on the curve, the gradient of the curve is 3.

1. Find an expression for \(\frac{dy}{dx}\). [2]
2. By first expressing \(\frac{dy}{dx}\) in the form \(R\cos(2x + \alpha)\), where \(R > 0\) and \(0 < \alpha < \frac{1}{2}\pi\), find the \(x\)-coordinates of \(P\) and \(Q\), giving your answers correct to 4 significant figures. [8]

Find the exact coordinates of the stationary point of the curve \(y = e^{2x} \sin 2x\) for \(0 \leqslant x < \frac{1}{2}\pi\). [5]

\includegraphics{figure_11} The diagram shows part of the curve \(y = (x + 1)^2 + (x + 1)^{-1}\) and the line \(x = 1\). The point \(A\) is the minimum point on the curve.

1. Show that the \(x\)-coordinate of \(A\) satisfies the equation \(2(x + 1)^3 = 1\) and find the exact value of \(\frac{\mathrm{d}^2 y}{\mathrm{d}x^2}\) at \(A\). [5]
2. Find, showing all necessary working, the volume obtained when the shaded region is rotated through \(360°\) about the \(x\)-axis. [6]

A curve has equation \(y = \frac{3 \cos x}{2 + \sin x}\), for \(-\frac{1}{2}\pi \leqslant x \leqslant \frac{1}{2}\pi\).

1. Find the exact coordinates of the stationary point of the curve. [6]
2. The constant \(a\) is such that \(\int_0^a \frac{3 \cos x}{2 + \sin x} \, dx = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures. [4]

\includegraphics{figure_7} The diagram shows the curve \(y = 5\sin^2 x \cos^3 x\) for \(0 \leqslant x \leqslant \frac{1}{2}\pi\), and its maximum point \(M\). The shaded region \(R\) is bounded by the curve and the \(x\)-axis.

1. Find the \(x\)-coordinate of \(M\), giving your answer correct to 3 decimal places. [5]
2. Using the substitution \(u = \sin x\) and showing all necessary working, find the exact area of \(R\). [4]

Given that \(y = x \sin x\), find \(\frac{d^2y}{dx^2}\) and \(\frac{d^4y}{dx^4}\), simplifying your results as far as possible, and show that $$\frac{d^6y}{dx^6} = -x \sin x + 6 \cos x.$$ [3] Use induction to establish an expression for \(\frac{d^{2n}y}{dx^{2n}}\), where \(n\) is a positive integer. [5]

A substance is decaying in such a way that its mass, \(m\) kg, at a time \(t\) years from now is given by the formula $$m = 240e^{-0.04t}.$$

1. Find the time taken for the substance to halve its mass. [3]
2. Find the value of \(t\) for which the mass is decreasing at a rate of 2.1 kg per year. [4]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(x^2 e^{-x}\) [3]
3. \(\frac{\sin x}{3 + 2\cos x}\) [4]

Differentiate each of the following with respect to \(x\) and simplify your answers.

1. \(\cot x^2\) [2]
2. \(\frac{\sin x}{3 + 2\cos x}\) [4]

\includegraphics{figure_1} Figure 1 shows a sketch of part of the curve with equation $$y = \sin (\cos x).$$ The curve cuts the \(x\)-axis at the points \(A\) and \(C\) and the \(y\)-axis at the point \(B\).

1. Find the coordinates of the points \(A\), \(B\) and \(C\). [3]
2. Prove that \(B\) is a stationary point. [2]

Given that the region \(OCB\) is convex,

1. show that, for \(0 \leq x \leq \frac{\pi}{2}\), $$\sin (\cos x) \leq \cos x$$ and $$(1 - \frac{2}{\pi} x) \sin 1 \leq \sin (\cos x)$$ and state in each case the value or values of \(x\) for which equality is achieved. [6]
2. Hence show that $$\frac{\pi}{4} \sin 1 < \int_0^{\frac{\pi}{2}} \sin(\cos x) \, dx < 1.$$ [4]

\includegraphics{figure_1} Figure 1 shows a sketch of the curve \(C\) with equation $$y = \cos x \ln(\sec x), \quad -\frac{\pi}{2} < x < \frac{\pi}{2}$$ The points \(A\) and \(B\) are maximum points on \(C\).

1. Find the coordinates of \(B\) in terms of e. [5]

The finite region \(R\) lies between \(C\) and the line \(AB\).

1. Show that the area of \(R\) is $$\frac{2}{e} \arccos \left(\frac{1}{e}\right) + 2\ln \left(e + \sqrt{(e^2 - 1)}\right) - \frac{4}{e} \sqrt{(e^2 - 1)}.$$ [arccos \(x\) is an alternative notation for \(\cos^{-1} x\)] [8]

The diagram shows part of the graph of \(y = x^2\). The normal to the curve at the point \(A(1, 1)\) meets the curve again at \(B\). Angle \(AOB\) is denoted by \(\alpha\). \includegraphics{figure_4}

1. Determine the coordinates of \(B\). [6]
2. Hence determine the exact value of \(\tan \alpha\). [3]