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

A car, starting from rest, is driven along a horizontal track. The velocity of the car, \(v \text{m s}^{-1}\), at time \(t\) seconds, is modelled by the equation $$v = 0.48t^2 - 0.024t^3 \text{ for } 0 \leq t \leq 15$$

1. Find the distance the car travels during the first 10 seconds of its journey. [3 marks]
2. Find the maximum speed of the car. Give your answer to three significant figures. [4 marks]
3. Deduce the range of values of \(t\) for which the car is modelled as decelerating. [2 marks]

A curve has equation \(y = 2x \cos 3x + (3x^2 - 4) \sin 3x\)

1. Find \(\frac{dy}{dx}\), giving your answer in the form \((mx^2 + n) \cos 3x\), where \(m\) and \(n\) are integers. [4 marks]
2. Show that the \(x\)-coordinates of the points of inflection of the curve satisfy the equation $$\cot 3x = \frac{9x^2 - 10}{6x}$$ [4 marks]

The function g is defined by $$g(x) = \mathrm{e}^{\sin x} \quad (0 \leq x \leq 2\pi)$$ The diagram below shows the graph of \(y = g(x)\) \includegraphics{figure_8}

1. Find the \(x\)-coordinate of each of the stationary points of the graph of \(y = g(x)\), giving your answers in exact form. [1 mark]
2. Use Simpson's rule with 3 ordinates to estimate $$\int_0^\pi g(x) \, \mathrm{d}x$$ giving your answer to two decimal places. [3 marks]
3. Explain how Simpson's rule could be used to find a more accurate estimate of the integral in part (b). [1 mark]

A particle moves in a circular path so that at time \(t\) seconds its position vector, \(\mathbf{r}\) metres, is given by $$\mathbf{r} = 4\sin(2t)\mathbf{i} + 4\cos(2t)\mathbf{j}$$ Find the velocity of the particle, in m s\(^{-1}\), when \(t = 0\) Circle your answer. [1 mark] \(8\mathbf{i}\) \quad \(-8\mathbf{j}\) \quad \(8\mathbf{j}\) \quad \(8\mathbf{i} - 8\mathbf{j}\)

A particle \(P\) of mass \(0.5\) kg moves on a horizontal plane such that its velocity vector \(\mathbf{v}\) ms\(^{-1}\) at time \(t\) seconds is given by $$\mathbf{v} = 12\cos(3t)\mathbf{i} - 5\sin(2t)\mathbf{j}.$$

1. Find an expression for the force acting on \(P\) at time \(t\) s. [3]
2. Given that when \(t = 0\), \(P\) has position vector \((\mathbf{4i} + \mathbf{7j})\) m relative to the origin \(O\), find an expression for the position vector of \(P\) at time \(t\) s. [4]
3. Hence determine the distance of \(P\) from \(O\) at time \(t = \frac{\pi}{2}\). [2]

\includegraphics{figure_5} Figure 5 shows a sketch of the curve with equation \(y = f(x)\), where $$f(x) = \frac{4\sin 2x}{e^{\sqrt{2}x-1}}, \quad 0 \leq x \leq \pi$$ The curve has a maximum turning point at \(P\) and a minimum turning point at \(Q\) as shown in Figure 5.

1. Show that the \(x\) coordinates of point \(P\) and point \(Q\) are solutions of the equation $$\tan 2x = \sqrt{2}$$ [4]
2. Using your answer to part (a), find the \(x\)-coordinate of the minimum turning point on the curve with equation $$y = 3 - 2f(x)$$ [2]

Find \(\frac{dy}{dx}\) for the following functions, simplifying your answers as far as possible.

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

In all questions you must show all stages of your working, justifying solutions and not relying solely on calculator technology.

1. Differentiate with respect to \(x\)
   1. \(x^2 e^{3x + 2}\), [4]
   2. \(\frac{\cos(2x^4)}{3x}\). [4]

The curve \(C\) has equation $$y = \frac{3 + \sin 2x}{2 + \cos 2x}$$

1. Show that $$\frac{dy}{dx} = \frac{6\sin 2x + 4\cos 2x + 2}{(2 + \cos 2x)^2}$$ [4]
2. Find an equation of the tangent to \(C\) at the point on \(C\) where \(x = \frac{\pi}{2}\). Write your answer in the form \(y = ax + b\), where \(a\) and \(b\) are exact constants. [4]

In this question you must show detailed reasoning. The diagram below shows the curve \(r = \sqrt{\sin \theta} e^{\frac{1}{2}\cos \theta}\) for \(0 \leqslant \theta \leqslant \pi\). \includegraphics{figure_13}

1. Find the exact area enclosed by the curve. [4]
2. Show that the greatest value of \(r\) on the curve is \(\sqrt{\frac{3}{2}} e^{\frac{1}{6}}\). [7]

A curve has equation \(y = xe^{\frac{x}{2}}\) Show that the curve has a single point of inflection and state the exact coordinates of this point of inflection. [8 marks]

The function f is defined by \(f : t \mapsto 2 \sin t + \cos 2t\) for \(0 \leqslant t < 2\pi\).

1. Show that \(\frac{df}{dt} = 2 \cos t(1 - 2 \sin t)\). [2]
2. Determine the range of f. [5]

A curve \(C\) is given parametrically by \(x = 2 \cos t + \sin 2t\), \(y = f(t)\) for \(0 \leqslant t < 2\pi\).

1. Show that \(x^2 + y^2 = 5 + 4 \sin 3t\). [3]
2. Deduce that \(C\) lies between two circles centred at the origin, and touches both. [2]
3. Find the gradient of the tangent to \(C\) at the point at which \(t = 0\). [3]