Error Analysis in Approximations

1 You are given that \(\left( x _ { 1 } , y _ { 1 } \right) = ( 0.9,2.3 )\) and \(\left( x _ { 2 } , y _ { 2 } \right) = ( 1.1,2.7 )\).
The values of \(x _ { 1 }\) and \(x _ { 2 }\) have been rounded to \(\mathbf { 1 }\) decimal place.

1. Determine the range of possible values of \(x _ { 2 } - x _ { 1 }\). The values of \(y _ { 1 }\) and \(y _ { 2 }\) have been chopped to \(\mathbf { 1 }\) decimal place.
2. Determine the range of possible values of \(y _ { 2 } - y _ { 1 }\). You are given that \(m = \frac { y _ { 2 } - y _ { 1 } } { x _ { 2 } - x _ { 1 } }\).
3. Determine the range of possible values of \(m\).
4. Explain why your answer to part (c) is much larger than your answer to part (a) and your answer to part (b).

6

1. 1. Calculate the relative error when \(\pi\) is chopped to \(\mathbf { 2 }\) decimal places in approximating $$\pi ^ { 2 } + 2 .$$
   2. Without doing any calculation, explain whether the relative error would be the same when \(\pi\) is chopped to 2 decimal places when approximating \(( \pi + 2 ) ^ { 2 }\). The table shows some spreadsheet output. The values of \(x\) in column A are exact.

      	A	B	C
      1	\(x\)	\(10 ^ { x }\)	\(\log _ { 10 } 10 ^ { x }\)
      2	\(1 \mathrm { E } - 12\)	1	\(1.00001 \mathrm { E } - 12\)
      3	\(1 \mathrm { E } - 11\)	1	\(9.99998 \mathrm { E } - 12\)

      The formula in cell B2 is \(= 10 ^ { \wedge } \mathrm { A } 2\).
      This has been copied down to cell B3.
      The formula in cell C2 is \(\quad =\) LOG(B2) .
      This formula has been copied down to cell C3.
2. 1. Write the value displayed in cell C 2 in standard mathematical notation.
   2. Explain why the values in cells C 2 and C 3 are neither zero nor the same as the values in cells A2 and A3 respectively.

2 You are given that \(a = \tanh ( 1 )\) and \(b = \tanh ( 2 )\).
\(A\) is the approximation to \(a\) formed by rounding \(\tanh ( 1 )\) to 1 decimal place.
\(B\) is the approximation to \(b\) formed by rounding \(\tanh ( 2 )\) to 1 decimal place.

1. Calculate the following.
   • The relative error \(\mathrm { R } _ { \mathrm { A } }\) when \(A\) is used to approximate \(a\).
   • The relative error \(\mathrm { R } _ { \mathrm { B } }\) when \(B\) is used to approximate \(b\).
   • Calculate the relative error \(\mathrm { R } _ { \mathrm { C } }\) when \(\mathrm { C } = \frac { \mathrm { A } } { \mathrm { B } }\) is used to approximate \(\mathrm { c } = \frac { \mathrm { a } } { \mathrm { b } }\).
   • Comment on the relationship between \(R _ { A } , R _ { B }\) and \(R _ { C }\).

1 Fig. 1 shows some spreadsheet output. \begin{table}[h] \end{table}

1. Write the value displayed in cell A3 in standard mathematical notation. The formula in cell A3 is
   \(= \mathrm { A } 2 - \mathrm { A } 1\)
2. Explain why the value displayed in cell A3 is non zero.
3. Write down the value of the number stored in cell A2 to the highest precision possible.
4. Explain why your answer to part (c) may be different to the actual value stored in cell A2.

3 At Heathwick airport each passenger’s luggage is weighed before being loaded into the hold of the aeroplane. Each weight is displayed digitally in kg to 1 decimal place. Some examples are given in Fig. 3. \begin{table}[h] \end{table} On each flight, the total weight of luggage is calculated to ensure compliance with health and safety regulations. Winston models this situation by assuming that the displayed weights are rounded to 1 decimal place, and that the total weight of luggage is calculated using the displayed values. On a flight to Athens, there are 154 items of passengers’ luggage.

1. Determine the maximum possible error, according to Winston's model, when the total weight of luggage is calculated for the flight to Athens. Piotre models this situation by assuming that the displayed weights are chopped to 1 decimal place, and that the total weight of luggage is calculated using the displayed values.
2. Determine the maximum possible error, according to Piotre's model, when the total weight of luggage is calculated for the flight to Athens. A health and safety inspector notes that the total of the displayed weights is 3080.2 kg . However, when the luggage is all weighed together in the loading bay, the total weight is found to be 3089.44 kg .
3. Determine whether Winston's model or Piotre's model is a better fit for the data.

1

1. 1. Determine the relative error when
      • 1.414214 is used to approximate \(\sqrt { 2 }\),
2. \(1.414214 ^ { 2 }\) is used to approximate 2.
   (ii) Write down the relationship between your answers to part (a)(i).
3. Fig. 1 shows some spreadsheet output.
\begin{table}[h]

	A	B	C
1	2	1.414214	2

\captionsetup{labelformat=empty} \caption{Fig. 1}
\end{table} The formula in cell B1 is = SQRT (A1)
and the formula in cell C 1 is \(\quad = \mathrm { B } 1 \wedge 2\).
Ben evaluates \(1.414214 ^ { 2 }\) on his calculator and obtains 2.000001238 . He states that this shows that the value displayed in cell C1 is wrong. Explain whether Ben is correct.

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.

1

1. Solve the following simultaneous equations. $$\begin{aligned} & x + \quad y = 1
   & x + 0.99 y = 2 \end{aligned}$$
2. The coefficient 0.99 is correct to two decimal places. All other coefficients in the equations are exact. With the aid of suitable calculations, explain why your answer to part (i) is unreliable.