1.09g Numerical methods in context

5 When Nina does the weekly grocery shopping she models the total cost by adding up the cost of each item in her head as she goes along. To simplify matters she rounds the cost of each item to the nearest pound. One week Nina buys 48 items.

1. Calculate the maximum possible error in Nina's model in this case. Nina estimated the total cost of her shopping to be \(\pounds 92\). The actual cost is \(\pounds 90.23\).
2. Explain whether this is consistent with Nina's model. The next week her husband, Kareem, does the weekly shopping. He models the total cost by chopping the cost of each item to the nearest pound as he goes along. On this occasion Kareem buys 52 items.
3. Calculate the expected error in Kareem's model in this case. Using his model Kareem estimates the total cost as \(\pounds 76\). The total cost of the shopping is \(\pounds 103.24\).
4. Explain how such a large error could arise. The next week Kareem buys \(n\) items.
5. Write down a formula for the maximum possible error when Kareem uses his model to estimate the total cost of his shopping.
6. Explain how Kareem's model could be adapted so that his formula gives the same expected error as Nina's model when they are both used to estimate the total cost of the shopping.

4 The table below gives values of a function \(y = \mathrm { f } ( x )\).

1. Calculate three estimates of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) using the central difference method.
2. State the value of \(\frac { \mathrm { d } y } { \mathrm {~d} x }\) at \(x = 0.4\) to an appropriate degree of accuracy. Justify your answer.

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 \(x = 0.3\)

1. A population of deer was introduced onto an island.

The number of deer, \(P\), on the island at time \(t\) years following their introduction is modelled by the differential equation $$\frac { \mathrm { d } P } { \mathrm {~d} t } = \frac { P } { 5000 } \left( 1000 - \frac { P ( t + 1 ) } { 6 t + 5 } \right) \quad t > 0$$ It was estimated that there were 540 deer on the island six months after they were introduced.
Use two applications of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the number of deer on the island 10 months after they were introduced.

1. A teacher made a cup of coffee. The temperature \(\theta ^ { \circ } \mathrm { C }\) of the coffee, \(t\) minutes after it was made, is modelled by the differential equation

$$\frac { \mathrm { d } \theta } { \mathrm {~d} t } + 0.05 ( \theta - 20 ) = 0$$ Given that

• the initial temperature of the coffee was \(95 ^ { \circ } \mathrm { C }\)
• the coffee can only be safely drunk when its temperature is below \(70 ^ { \circ } \mathrm { C }\)
• the teacher made the cup of coffee at 1.15 pm
• the teacher needs to be able to start drinking the coffee by 1.20 pm
  use two iterations of the approximation formula

$$\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }$$ to estimate whether the teacher will be able to start drinking the coffee at 1.20 pm .

1. An area of woodland contains a mixture of blue and yellow flowers.

A study found that the proportion, \(x\), of blue flowers in the woodland area satisfies the differential equation $$\frac { \mathrm { d } x } { \mathrm {~d} t } = \frac { x t ( 0.8 - x ) } { x ^ { 2 } + 5 t } \quad t > 0$$ where \(t\) is the number of years since the start of the study.
Given that exactly 3 years after the start of the study half of the flowers in the woodland area were blue,

1. use one application of the approximation formula \(\left( \frac { \mathrm { d } y } { \mathrm {~d} x } \right) _ { n } \approx \frac { y _ { n + 1 } - y _ { n } } { h }\) to estimate the proportion of blue flowers in the woodland area half a year later.
2. Deduce from the differential equation the proportion of flowers that will be blue in the long term.

The temperature \(\theta\) °C of a room \(t\) hours after a heating system has been turned on is given by $$\theta = t + 26 - 20e^{-0.5t}, \quad t \geq 0.$$ The heating system switches off when \(\theta = 20\). The time \(t = \alpha\), when the heating system switches off, is the solution of the equation \(\theta - 20 = 0\), where \(\alpha\) lies in the interval \([1.8, 2]\).

1. Using the end points of the interval \([1.8, 2]\), find, by linear interpolation, an approximation to \(\alpha\). Give your answer to 2 decimal places. [4]
2. Use your answer to part (a) to estimate, giving your answer to the nearest minute, the time for which the heating system was on. [1]

$$f(x) = 3x^2 + x - \tan \left( \frac{x}{2} \right) - 2, \quad -\pi < x < \pi.$$ The equation \(f(x) = 0\) has a root \(\alpha\) in the interval \([0.7, 0.8]\). Use linear interpolation, on the values at the end points of this interval, to obtain an approximation to \(\alpha\). Give your answer to 3 decimal places. [4]

A curve passes through the point \((9, 6)\) and satisfies the differential equation $$\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{1}{2 + \sqrt{x}}$$ Use a step-by-step method with a step length of \(0.25\) to estimate the value of \(y\) at \(x = 9.5\). Give your answer to four decimal places. [5 marks]