1.07b Gradient as rate of change: dy/dx notation

A is the point \((2, 1)\) on the curve \(y = \frac{4}{x^2}\). B is the point on the same curve with \(x\)-coordinate \(2.1\).

1. Calculate the gradient of the chord AB of the curve. Give your answer correct to 2 decimal places. [2]
2. Give the \(x\)-coordinate of a point C on the curve for which the gradient of chord AC is a better approximation to the gradient of the curve at A. [1]
3. Use calculus to find the gradient of the curve at A. [2]

In Fig. 5, A and B are the points on the curve \(y = 2^x\) with \(x\)-coordinates 3 and 3.1 respectively. \includegraphics{figure_5}

1. Find the gradient of the chord AB. Give your answer correct to 2 decimal places. [2]
2. Stating the points you use, find the gradient of another chord which will give a closer approximation to the gradient of the tangent to \(y = 2^x\) at A. [2]

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 temperature \(T°C\) of a liquid at time \(t\) minutes is given by the equation $$T = 30 + 20e^{-0.05t}, \quad \text{for } t \geq 0.$$ Write down the initial temperature of the liquid, and find the initial rate of change of temperature. Find the time at which the temperature is \(40°C\).

The gradient at any point \((x, y)\) on a curve is proportional to \(\sqrt{y}\). Given that the curve passes through the point with coordinates \((0, 4)\),

1. show that the equation of the curve can be written in the form $$2\sqrt{y} = kx + 4,$$ where \(k\) is a positive constant. [5]

Given also that the curve passes through the point with coordinates \((2, 9)\),

1. find the equation of the curve in the form \(y = \text{f}(x)\). [4]

A particle \(P\) moves in a straight line so that its velocity \(v\) ms\(^{-1}\) at time \(t\) seconds is given, for \(t > 1\), by the formula \(v = 2t + \frac{8}{t^2}\). Find the time when the acceleration of \(P\) is zero. [5 marks]

The velocity, \(v\) ms\(^{-1}\), of a particle at time \(t\) s is given by \(v = 4t^2 - 9\).

1. Find the acceleration of the particle when it is instantaneously at rest. [3 marks]
2. Find the distance travelled by the particle from time \(t = 0\) until it comes to rest. [4 marks]

A particle \(P\) moves in a plane such that its position vector \(\mathbf{r}\) metres at time \(t\) seconds, relative to a fixed origin \(O\), is \(\mathbf{r} = t^2\mathbf{i} - 2t\mathbf{j}\).

1. Find the velocity vector of \(P\) at time \(t\) seconds. [2 marks]
2. Show that the direction of the acceleration of \(P\) is constant. [2 marks]
3. Find the value of \(t\) when the acceleration of \(P\) has magnitude 12 ms\(^{-2}\). [3 marks]

Water is poured into an empty cone at a constant rate of 8 cm³/s After \(t\) seconds the depth of the water in the inverted cone is \(h\) cm, as shown in the diagram below. \includegraphics{figure_8} When the depth of the water in the inverted cone is \(h\) cm, the volume, \(V\) cm³, is given by $$V = \frac{\pi h^3}{12}$$

1. Show that when \(t = 3\) $$\frac{dV}{dh} = 6 \sqrt[3]{6\pi}$$ [4 marks]
2. Hence, find the rate at which the depth is increasing when \(t = 3\) Give your answer to three significant figures. [3 marks]

Water is being emptied out of a sink. The depth of water, \(y\)cm, at time \(t\) seconds, may be modelled by $$y = t^2 - 14t + 49 \quad\quad 0 \leqslant t \leqslant 7.$$

1. Find the value of \(t\) when the depth of water is 25cm. [3]
2. Find the rate of decrease of the depth of water when \(t = 3\). [3]

A car starts from the point \(A\). At time \(t\) s after leaving \(A\), the distance of the car from \(A\) is \(s\) m, where \(s = 30t - 0.4t^2\), \(0 \leq t \leq 25\). The car reaches the point \(B\) when \(t = 25\).

1. Find the distance \(AB\). [2]
2. Show that the car travels with a constant acceleration and state the value of this acceleration. [3]

A runner passes through \(B\) when \(t = 0\) with an initial velocity of \(2 \text{ m s}^{-1}\) running directly towards \(A\). The runner has a constant acceleration of \(0.1 \text{ m s}^{-2}\).

1. Find the distance from \(A\) at which the runner and the car pass one another. [8]

The number of members of a social networking site is modelled by \(m = 150e^{2t}\), where \(m\) is the number of members and \(t\) is time in weeks after the launch of the site.

1. State what this model implies about the relationship between \(m\) and the rate of change of \(m\). [2]
2. What is the significance of the integer 150 in the model? [1]
3. Find the week in which the model predicts that the number of members first exceeds 60 000. [3]
4. The social networking site only expects to attract 60 000 members. Suggest how the model could be refined to take account of this. [1]