1.07s Parametric and implicit differentiation

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$

1. Find the gradient of the curve at the point where \(t = -2\) [4 marks]
2. Find a Cartesian equation of the curve. [2 marks]

A curve has equation $$x^2y^2 + xy^4 = 12$$

1. Prove that the curve does not intersect the coordinate axes. [2 marks]
2. 1. Show that \(\frac{dy}{dx} = -\frac{2xy + y^3}{2x^2 + 4xy^2}\) [5 marks]
   2. Prove that the curve has no stationary points. [4 marks]
   3. In the case when \(x > 0\), find the equation of the tangent to the curve when \(y = 1\) [4 marks]

A design for a surfboard is shown in Figure 1. Figure 1 \includegraphics{figure_6_1} The curve of the top half of the surfboard can be modelled by the parametric equations $$x = -2t^2$$ $$y = 9t - 0.7t^2$$ for \(0 \leq t \leq 9.5\) as shown in Figure 2, where \(x\) and \(y\) are measured in centimetres. Figure 2 \includegraphics{figure_6_2}

1. Find the length of the surfboard. [2 marks]
2. 1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\). [3 marks]
   2. Hence, show that the width of the surfboard is approximately one third of its length. [4 marks]

A water slide is the shape of a curve \(PQ\) as shown in Figure 1 below. \includegraphics{figure_9} The curve can be modelled by the parametric equations $$x = t - \frac{1}{t} + 4.8$$ $$y = t + \frac{2}{t}$$ where \(0.2 \leq t \leq 3\) The horizontal distance from O is \(x\) metres. The vertical distance above the point O at ground level is \(y\) metres. P is the point where \(t = 0.2\) and Q is the point where \(t = 3\)

1. To make sure speeds are safe at Q, the difference in height between P and Q must be less than 7 metres. Show that the slide meets this safety requirement. [3 marks]
2. 1. Find an expression for \(\frac{dy}{dx}\) in terms of \(t\) [3 marks]
   2. A vertical support, RS, is to be added between the ground and the lowest point on the slide as shown in Figure 2 below. \includegraphics{figure_9b} Find the length of RS [4 marks]
   3. Find the acute angle the slide makes with the horizontal at Q Give your answer to the nearest degree. [2 marks]

The parametric equations of a curve are \(x = 2 + 5\cos\theta\) and \(y = 1 + 5\sin\theta\), where \(0 \leq \theta < 2\pi\).

1. Determine the cartesian equation of the curve. [3]
2. Hence or otherwise, find the equation of the tangent to the curve at the point \((5, -3)\), giving your answer in the form \(ax + by + c = 0\), where \(a\), \(b\) and \(c\) are integers to be determined. [4]

Fig. 12 shows the curve \(2x^3 + y^3 = 5y\). \includegraphics{figure_12}

1. Find the gradient of the curve \(2x^3 + y^3 = 5y\) at the point \((1,2)\), giving your answer in exact form. [4]
2. Show that all the stationary points of the curve lie on the \(y\)-axis. [2]

The diagram shows part of a spiral curve. The point \(P\) has polar coordinates \((r, \theta)\) where \(0 \leq \theta \leq \frac{\pi}{2}\) The points \(T\) and \(S\) lie on the initial line and \(O\) is the pole. \(TPQ\) is the tangent to the curve at \(P\). \includegraphics{figure_15}

1. Show that the gradient of \(TPQ\) is equal to $$\frac{\frac{dr}{d\theta} \sin \theta + r \cos \theta}{\frac{dr}{d\theta} \cos \theta - r \sin \theta}$$ [4 marks]
2. The curve has polar equation $$r = e^{(\cot b)\theta}$$ where \(b\) is a constant such that \(0 < b < \frac{\pi}{2}\) Use the result of part (a) to show that the angle between the line \(OP\) and the tangent \(TPQ\) does not depend on \(\theta\). [7 marks]

The equation of a curve \(C\) is given by the parametric equations $$x = \cos 2\theta, \quad y = \cos\theta.$$

1. Find the Cartesian equation of \(C\). [2]
2. Show that the line \(x - y + 1 = 0\) meets \(C\) at the point \(P\), where \(\theta = \frac{\pi}{3}\), and at the point \(Q\), where \(\theta = \frac{\pi}{2}\). Write down the coordinates of \(P\) and \(Q\). [5]
3. Determine the equations of the tangents to \(C\) at \(P\) and \(Q\). Write down the coordinates of the point of intersection of the two tangents. [7]

A curve is defined parametrically by $$x = 2\theta + \sin 2\theta, \quad y = 1 + \cos 2\theta.$$

1. Show that the gradient of the curve at the point with parameter \(\theta\) is \(-\tan\theta\). [6]
2. Find the equation of the tangent to the curve at the point where \(\theta = \frac{\pi}{4}\). [4]

A curve is defined by the parametric equations $$x = t^3 + 2, \quad y = t^2 - 1$$ Find the gradient of the curve at the point where \(t = -2\) [2]

A curve \(C\) is given by the equation $$\sin x + \cos y = 0.5 \quad -\frac{\pi}{2} \leq x < \frac{3\pi}{2}, -\pi < y < \pi$$ A point \(P\) lies on \(C\). The tangent to \(C\) at the point \(P\) is parallel to the \(x\)-axis. Find the exact coordinates of all possible points \(P\), justifying your answer. (Solutions based entirely on graphical or numerical methods are not acceptable.) [5]

A curve has parametric equations $$x = 2\sin t, \quad y = \cos 2t + 2\sin t$$ for \(-\frac{1}{2}\pi \leqslant t \leqslant \frac{1}{2}\pi\).

1. Show that \(\frac{dy}{dx} = 1 - 2\sin t\) and hence find the coordinates of the stationary point. [5]
2. Find the cartesian equation of the curve. [3]
3. State the set of values that \(x\) can take and hence sketch the curve. [3]

A curve has equation \(x^3 - 3x^2y + y^2 + 1 = 0\).

1. Show that \(\frac{dy}{dx} = \frac{6xy - 3x^2}{2y - 3x^2}\). [4]
2. Find the equation of the normal to the curve at the point \((1, 2)\). [4]

The curve defined by the parametric equations $$x = 2\cos\theta, \quad y = 3\sin(2\theta) \quad \text{and} \quad \theta \in [0, 2\pi]$$ is shown below. The point \(P\left(\sqrt{3}, \frac{3\sqrt{3}}{2}\right)\) is marked on the curve. \includegraphics{figure_curve}

1. Show that the equation of the normal to the curve at \(P\) can be written as \(3y - x = \frac{7\sqrt{3}}{2}\) [5]
2. Show that the Cartesian equation of the curve may be written as \(ay^2 + bx^4 + cx^2 = 0\) where \(a\), \(b\) and \(c\) are integers to be found. [3]

The curve \(C\) has parametric equations $$x = \sin 2\theta \quad y = \cos\text{ec}^3 \theta \quad 0 < \theta < \frac{\pi}{2}$$

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(\theta\) [3]
2. Hence find the exact value of the gradient of the tangent to \(C\) at the point where \(y = 8\) [3]

In this question you must show detailed reasoning. A curve has parametric equations $$x = \cos t - 3t \text{ and } y = 3t - 4\cos t - \sin 2t, \text{ for } 0 \leqslant t \leqslant \pi.$$ Show that the gradient of the curve is always negative. [7]

The curve \(C\) has parametric equations $$x = 2\cos t, \quad y = \sqrt{3}\cos 2t, \quad 0 \leq t \leq \pi$$

1. Find an expression for \(\frac{\mathrm{d}y}{\mathrm{d}x}\) in terms of \(t\). [2]

The point \(P\) lies on \(C\) where \(t = \frac{2\pi}{3}\) The line \(l\) is the normal to \(C\) at \(P\).

1. Show that an equation for \(l\) is $$2x - 2\sqrt{3}y - 1 = 0$$ [5]

The line \(l\) intersects the curve \(C\) again at the point \(Q\).

1. Find the exact coordinates of \(Q\). You must show clearly how you obtained your answers. [6]

Fig. 10 shows the graph of \(x^3 + y^3 = xy\). \includegraphics{figure_10}

1. Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [4]
2. P is the maximum point on the curve. The parabola \(y = kx^2\) intersects the curve at P. Find the value of the constant \(k\). [2]

In this question you must show detailed reasoning. \includegraphics{figure_12} The curve \(C\) has parametric equations $$x = \frac{1}{\sqrt{2 + t}}, \quad y = \ln(1 + t), \quad 2 \leq t < \infty$$ The point \(P\) on curve \(C\) has \(x\)-coordinate \(\frac{1}{2}\).

1. Find the exact \(y\)-coordinate of \(P\). [1]

The tangent to \(C\) at \(P\) meets the \(y\)-axis at point \(Y\).

1. Determine the exact coordinates of \(Y\). [4]

The curve \(C\) and the line segment \(PY\) are rotated \(2\pi\) radians about the \(y\)-axis.

1. Determine the exact volume of the solid generated. Give your answer in the form \(\pi(\ln p + q)\), where \(p\) and \(q\) are rational numbers. [8]

[You are given that the volume of a cone with radius \(r\) and height \(h\) is \(\frac{1}{3}\pi r^2 h\)]