Show dy/dx equals given expression

The equation of a curve is \(2x^4 + xy^3 + y^4 = 10\).

1. Show that \(\frac{dy}{dx} = -\frac{8x^3 + y^3}{3xy^2 + 4y^3}\). [4]
2. Hence show that there are two points on the curve at which the tangent is parallel to the \(x\)-axis and find the coordinates of these points. [4]

\includegraphics{figure_4} Figure 4 shows a sketch of the closed curve with equation $$(x + y)^3 + 10y^2 = 108x$$

1. Show that $$\frac{dy}{dx} = \frac{108 - 3(x + y)^2}{20y + 3(x + y)^2}$$ [5]

The curve is used to model the shape of a cycle track with both \(x\) and \(y\) measured in km. The points \(P\) and \(Q\) represent points that are furthest north and furthest south of the origin \(O\), as shown in Figure 4. Using the result given in part (a),

1. find how far the point \(Q\) is south of \(O\). Give your answer to the nearest 100 m. [4]

A curve is defined by the equation \((x + y)^2 = 4x\). The point \((1, 1)\) lies on this curve. By differentiating implicitly, show that \(\frac{dy}{dx} = \frac{2}{x + y} - 1\). Hence verify that the curve has a stationary point at \((1, 1)\). [4]

You are given that \(y^2 = 4x + 7\).

1. Use implicit differentiation to find \(\frac{dy}{dx}\) in terms of \(y\). [2]
2. Make \(x\) the subject of the equation. Find \(\frac{dx}{dy}\) and hence show that in this case \(\frac{dx}{dy} = \frac{1}{\frac{dx}{dy}}\). [3]

A curve has the equation $$x^2 + 2xy^2 + y = 4.$$ Find an expression for \(\frac{dy}{dx}\) in terms of \(x\) and \(y\). [5]

A curve has equation $$x^2 + 4y^2 = k^2,$$ where \(k\) is a positive constant.

1. Verify that $$x = k\cos\theta, \quad y = \frac{k}{2}\sin\theta,$$ are parametric equations for the curve. [3]
2. Hence or otherwise show that \(\frac{dy}{dx} = -\frac{x}{4y}\). [3]
3. Fig. 8 illustrates the curve for a particular value of \(k\). Write down this value of \(k\). [1]

\includegraphics{figure_8}

1. Copy Fig. 8 and on the same axes sketch the curves for \(k = 1\), \(k = 3\) and \(k = 4\). [3]

On a map, the curves represent the contours of a mountain. A stream flows down the mountain. Its path on the map is always at right angles to the contour it is crossing.

1. Explain why the path of the stream is modelled by the differential equation $$\frac{dy}{dx} = \frac{4y}{x}.$$ [2]
2. Solve this differential equation. Given that the path of the stream passes through the point (2, 1), show that its equation is \(y = \frac{x^4}{16}\). [6]

In this question you must show detailed reasoning. \includegraphics{figure_6} The diagram shows the curve with equation \(4xy = 2(x^2 + 4y^2) - 9x\).

1. Show that \(\frac{dy}{dx} = \frac{4x - 4y - 9}{4x - 16y}\). [3] At the point \(P\) on the curve the tangent to the curve is parallel to the \(y\)-axis and at the point \(Q\) on the curve the tangent to the curve is parallel to the \(x\)-axis.
2. Show that the distance \(PQ\) is \(k\sqrt{5}\), where \(k\) is a rational number to be determined. [8]

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]

Given \(z = x\sin y + y\cos x\), show that \(\frac{\partial^2 z}{\partial x^2} + \frac{\partial^2 z}{\partial y^2} + z = 0\). [5]