First-order integration

6 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3 x ^ { \frac { 1 } { 2 } } - 6\) and the point \(( 9,2 )\) lies on the curve.

1. Find the equation of the curve.
2. Find the \(x\)-coordinate of the stationary point on the curve and determine the nature of the stationary point.

8 The equation of a curve is such that \(\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 x - 1\). Given that the curve has a minimum point at \(( 3 , - 10 )\), find the coordinates of the maximum point.

7 A curve is such that \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 5 - \frac { 8 } { x ^ { 2 } }\). The line \(3 y + x = 17\) is the normal to the curve at the point \(P\) on the curve. Given that the \(x\)-coordinate of \(P\) is positive, find

1. the coordinates of \(P\),
2. the equation of the curve.

9 The curve \(y = \mathrm { f } ( x )\) has a stationary point at \(( 2,10 )\) and it is given that \(\mathrm { f } ^ { \prime \prime } ( x ) = \frac { 12 } { x ^ { 3 } }\).

1. Find \(\mathrm { f } ( x )\).
2. Find the coordinates of the other stationary point.
3. Find the nature of each of the stationary points.

6. The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac { \mathrm { d } n } { \mathrm {~d} t } = \mathrm { e } ^ { 0.5 t } - 5 , \quad t \geq 0$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue.
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation.
3. Explain why this model would not be appropriate for large values of \(t\).

7. \begin{figure}[h] \end{figure} Figure 2 Figure 2 shows a cylindrical tank of height 1.5 m .
Initially the tank is full of water.
The water starts to leak from a small hole, at a point \(L\), in the side of the tank.
While the tank is leaking, the depth, \(H\) metres, of the water in the tank is modelled by the differential equation $$\frac { \mathrm { d } H } { \mathrm {~d} t } = - 0.12 \mathrm { e } ^ { - 0.2 t }$$ where \(t\) hours is the time after the leak starts.
Using the model,

1. show that $$H = A \mathrm { e } ^ { - 0.2 t } + B$$ where \(A\) and \(B\) are constants to be found,
2. find the time taken for the depth of the water to decrease to 1.2 m . Give your answer in hours and minutes, to the nearest minute. In the long term, the water level in the tank falls to the same height as the hole.
3. Find, according to the model, the height of the hole from the bottom of the tank.

11 In this question you must show detailed reasoning.
The variables \(x\) and \(y\) are such that \(\frac { \mathrm { dy } } { \mathrm { dx } }\) is directly proportional to the square root of \(x\).
When \(x = 4 , \frac { d y } { d x } = 3\).

1. Find \(\frac { \mathrm { dy } } { \mathrm { dx } }\) in terms of \(x\). When \(\mathrm { x } = 4 , \mathrm { y } = 10\).
2. Find \(y\) in terms of \(x\).

5. At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac { \mathrm { d } y } { \mathrm {~d} t } = - k \mathrm { e } ^ { - 0.2 t }$$ where \(k\) is a positive constant,

1. Find an expression for \(y\) in terms of \(k\) and \(t\). Given that two hours after being filled the depth of water in the tank is 1.6 metres,
2. find the value of \(k\) to 4 significant figures. Given also that the hole in the tank is \(h \mathrm {~cm}\) above the base of the tank,
3. show that \(h = 79\) to 2 significant figures.
   5. continued

1. The variables \(x\) and \(y\) satisfy the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } = 2 y ^ { 2 } - x - 1$$ where \(\frac { \mathrm { d } y } { \mathrm {~d} x } = 3\) and \(y = 0\) at \(x = 0\) Use the approximations $$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$ with \(h = 0.1\) to find an estimate for the value of \(y\) at \(x = 0.2\)

1. The variables \(x\) and \(y\) satisfy the differential equation

$$\frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } + 15 \frac { \mathrm {~d} y } { \mathrm {~d} x } - 3 y ^ { 2 } = 2 x$$ where \(y = 1\) at \(x = 0\) and where \(y = 2\) at \(x = 0.1\) Use the approximations $$\left( \frac { \mathrm { d } ^ { 2 } y } { \mathrm {~d} x ^ { 2 } } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - 2 y _ { n } + y _ { n - 1 } \right) } { h ^ { 2 } } \text { and } \left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { \left( y _ { n + 1 } - y _ { n - 1 } \right) } { 2 h }$$ with \(h = 0.1\) to find an estimate for the value of \(y\) when \(x = 0.3\)

The number of people, \(n\), in a queue at a Post Office \(t\) minutes after it opens is modelled by the differential equation $$\frac{dn}{dt} = e^{0.5t} - 5, \quad t \geq 0.$$

1. Find, to the nearest second, the time when the model predicts that there will be the least number of people in the queue. [3]
2. Given that there are 20 people in the queue when the Post Office opens, solve the differential equation. [4]
3. Explain why this model would not be appropriate for large values of \(t\). [1]

At time \(t = 0\), a tank of height 2 metres is completely filled with water. Water then leaks from a hole in the side of the tank such that the depth of water in the tank, \(y\) metres, after \(t\) hours satisfies the differential equation $$\frac{dy}{dt} = -ke^{-0.2t},$$ where \(k\) is a positive constant.

1. Find an expression for \(y\) in terms of \(k\) and \(t\). [4]

Given that two hours after being filled the depth of water in the tank is 1.6 metres,

1. find the value of \(k\) to 4 significant figures. [2]

Given also that the hole in the tank is \(h\) cm above the base of the tank,

1. show that \(h = 79\) to 2 significant figures. [3]

A particle is moving in a straight line through the origin \(O\) The displacement of the particle, \(r\) metres, from \(O\), at time \(t\) seconds is given by $$r = p + 2t - qe^{-0.2t}$$ where \(p\) and \(q\) are constants. When \(t = 3\), the acceleration of the particle is \(-1.8\) m s\(^{-2}\)

1. Show that \(q \approx 82\) [5 marks]
2. The particle has an initial displacement of 5 metres. Find the value of \(p\) Give your answer to two significant figures. [2 marks]