1.07r Chain rule: dy/dx = dy/du * du/dx and connected rates

Find the equation of the tangent to the curve \(y = \sqrt{4x + 1}\) at the point \((2, 3)\). [5]

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

1. Show that \(x = 1 + \frac{\ln y}{\ln 5}\). [2]
2. Find an expression for \(\frac{dx}{dy}\) in terms of \(y\). [2]
3. Hence find the exact value of the gradient of the curve \(y = 5^{x-1}\) at the point \((3, 25)\). [2]

\includegraphics{figure_9} The diagram shows the curve with equation \(y = 2 \ln(x - 1)\). The point \(P\) has coordinates \((0, p)\). The region \(R\), shaded in the diagram, is bounded by the curve and the lines \(x = 0\), \(y = 0\) and \(y = p\). The units on the axes are centimetres. The region \(R\) is rotated completely about the \(y\)-axis to form a solid.

1. Show that the volume, \(V \text{ cm}^3\), of the solid is given by $$V = \pi(e^p + 4e^{\frac{p}{2}} + p - 5).$$ [8]
2. It is given that the point \(P\) is moving in the positive direction along the \(y\)-axis at a constant rate of \(0.2 \text{ cm min}^{-1}\). Find the rate at which the volume of the solid is increasing at the instant when \(p = 4\), giving your answer correct to 2 significant figures. [5]

\includegraphics{figure_8} The diagram shows the curve with equation \(y = x^8 e^{-x^2}\). The curve has maximum points at \(P\) and \(Q\). The shaded region \(A\) is bounded by the curve, the line \(y = 0\) and the line through \(Q\) parallel to the \(y\)-axis. The shaded region \(B\) is bounded by the curve, the line \(y = 0\) and the line \(PQ\).

1. Show by differentiation that the \(x\)-coordinate of \(Q\) is 2. [5]
2. Use Simpson's rule with 4 strips to find an approximation to the area of region \(A\). Give your answer correct to 3 decimal places. [4]
3. Deduce an approximation to the area of region \(B\). [2]

Differentiate each of the following with respect to \(x\).

1. \(x^3(x + 1)^5\) [2]
2. \(\sqrt{3x^4 + 1}\) [3]

Earth is being added to a pile so that, when the height of the pile is \(h\) metres, its volume is \(V\) cubic metres, where $$V = (h^6 + 16)^{\frac{1}{2}} - 4.$$

1. Find the value of \(\frac{dV}{dh}\) when \(h = 2\). [3]
2. The volume of the pile is increasing at a constant rate of 8 cubic metres per hour. Find the rate, in metres per hour, at which the height of the pile is increasing at the instant when \(h = 2\). Give your answer correct to 2 significant figures. [3]

The equation of a curve is \(y = (x^2 + 1)^8\).

1. Find an expression for \(\frac{dy}{dx}\) and hence show that the only stationary point on the curve is the point for which \(x = 0\). [4]
2. Find an expression for \(\frac{d^2y}{dx^2}\) and hence find the value of \(\frac{d^2y}{dx^2}\) at the stationary point. [5]

For each of the following curves, find the gradient at the point with \(x\)-coordinate 2.

1. \(y = \frac{3x}{2x + 1}\) [3]
2. \(y = \sqrt{4x^2 + 9}\) [3]

Find \(\frac{dy}{dx}\) in each of the following cases:

1. \(y = x^3 e^{2x}\), [2]
2. \(y = \ln(3 + 2x^2)\), [2]
3. \(y = \frac{x}{2x + 1}\). [2]

A curve has equation \(y = \frac{x}{2 + 3\ln x}\). Find \(\frac{dy}{dx}\). Hence find the exact coordinates of the stationary point of the curve.

Given that \(y = \sqrt[3]{1 + x^2}\), find \(\frac{dy}{dx}\). [4]

The area of a circular stain is growing at a rate of \(1 \text{ mm}^2\) per second. Find the rate of increase of its radius at an instant when its radius is \(2\) mm. [5]

Oil is leaking into the sea from a pipeline, creating a circular oil slick. The radius \(r\) metres of the oil slick \(t\) hours after the start of the leak is modelled by the equation $$r = 20(1 - e^{-0.2t}).$$

1. Find the radius of the slick when \(t = 2\), and the rate at which the radius is increasing at this time. [4]
2. Find the rate at which the area of the slick is increasing when \(t = 2\). [4]

Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x-2}}\), together with the lines \(y = x\) and \(x = 11\). The curve meets these lines at P and Q respectively. R is the point \((11, 11)\). \includegraphics{figure_8}

1. Verify that the \(x\)-coordinate of P is 3. [2]
2. Show that, for the curve, \(\frac{dy}{dx} = \frac{x-4}{2(x-2)^{\frac{3}{2}}}\). Hence find the gradient of the curve at P. Use the result to show that the curve is not symmetrical about \(y = x\). [7]
3. Using the substitution \(u = x - 2\), show that \(\int_3^{11} \frac{x}{\sqrt{x-2}} \, dx = 25\frac{1}{3}\). Hence find the area of the region PQR bounded by the curve and the lines \(y = x\) and \(x = 11\). [9]

The driving force \(F\) newtons and velocity \(v\) km s\(^{-1}\) of a car at time \(t\) seconds are related by the equation \(F = \frac{25}{v}\).

1. Find \(\frac{dF}{dv}\). [2]
2. Find \(\frac{dF}{dt}\) when \(v = 50\) and \(\frac{dv}{dt} = 1.5\). [3]

Find the exact gradient of the curve \(y = \ln(1 - \cos 2x)\) at the point with \(x\)-coordinate \(\frac{1}{4}\pi\). [5]

A spherical balloon of radius \(r\) cm has volume \(V\) cm\(^3\), where \(V = \frac{4}{3}\pi r^3\). The balloon is inflated at a constant rate of 10 cm\(^3\) s\(^{-1}\). Find the rate of increase of \(r\) when \(r = 8\). [5]

The volume \(V\) m³ of a pile of grain of height \(h\) metres is modelled by the equation $$V = 4\sqrt{h^3 + 1} - 4.$$

1. Find \(\frac{dV}{dh}\) when \(h = 2\). [4]

At a certain time, the height of the pile is 2 metres, and grain is being added so that the volume is increasing at a rate of 0.4 m³ per minute.

1. Find the rate at which the height is increasing at this time. [3]

Fig. 8 shows the curve \(y = \frac{x}{\sqrt{x+4}}\) and the line \(x = 5\). The curve has an asymptote \(l\). The tangent to the curve at the origin O crosses the line \(l\) at P and the line \(x = 5\) at Q. \includegraphics{figure_8}

1. Show that for this curve \(\frac{dy}{dx} = \frac{x + 8}{2(x + 4)^{\frac{3}{2}}}\). [5]
2. Find the coordinates of the point P. [4]
3. Using integration by substitution, find the exact area of the region enclosed by the curve, the tangent OQ and the line \(x = 5\). [9]

An oil slick is circular with radius \(r\) km and area \(A\) km\(^2\). The radius increases with time at a rate given by \(\frac{dr}{dt} = 0.5\), in kilometres per hour.

1. Show that \(\frac{dA}{dt} = \pi r\). [4]
2. Find the rate of increase of the area of the slick at a time when the radius is 6 km. [2]

Fig. 8 shows the graph of \(y = x\sqrt{1 + x}\). The point P on the curve is on the \(x\)-axis. \includegraphics{figure_8}

1. Write down the coordinates of P. [1]
2. Show that \(\frac{dy}{dx} = \frac{3x + 2}{2\sqrt{1 + x}}\). [4]
3. Hence find the coordinates of the turning point on the curve. What can you say about the gradient of the curve at P? [4]
4. By using a suitable substitution, show that \(\int_0^0 x\sqrt{1 + x} dx = \int_0^1 \left(u^{\frac{3}{2}} - u^{\frac{1}{2}}\right) du\). Evaluate this integral, giving your answer in an exact form. What does this value represent? [7]
5. Use your answer to part (ii) to differentiate \(y = x\sqrt{1 + x} \sin 2x\) with respect to \(x\). (You need not simplify your result.) [2]