A-Level Edexcel Maths Pure Modelling with Functions: The table below shows correspond

The table below shows corresponding values of $x$ and $y$ for $y = ext{log}_2 x$ The values of $y$ are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Question 5

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The table below shows corresponding values of $x$ and $y$ for $y = ext{log}_2 x$ The values of $y$ are given to 2 decimal places as appropriate. | $x$ | 3 | 4.5... show full transcript

Worked Solution & Example Answer:The table below shows corresponding values of $x$ and $y$ for $y = ext{log}_2 x$ The values of $y$ are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Step 1

Using the trapezium rule with all the values of $y$ in the table, find an estimate for $$\int_3^8 \text{log}_2 2x \, dx$$

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Answer

To estimate the integral using the trapezium rule, we first identify the width of each segment, denoted as $h$ . For the given data, we have:

$h = 8 - 3 = 5$ and there are 5 segments in total, thus: $h = \frac{5}{4} = 1.5$ .

Using the trapezium rule: $\int_3^8 \text{log}_2 2x \, dx \approx \frac{h}{2} [y_0 + 2(y_1 + y_2 + y_3) + y_4]$ Substituting the values: $= \frac{1.5}{2} [1.63 + 2(2.26 + 2.46 + 2.63)]$ Calculating the terms: $= 0.75 [1.63 + 2(7.35)]$ $= 0.75 [1.63 + 14.70]$ $= 0.75 \times 16.33 = 12.2475$ . Hence, the estimated value is approximately $12.25$ .

Step 2

Estimate $$\int_3^8 \text{log}_2 (2x)^{10} \, dx$$

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Answer

Using the result from part (a), we have: $\int_3^8 \text{log}_2 (2x)^{10} \, dx = 10 \int_3^8 \text{log}_2 2x \, dx$ Thus: $\approx 10 \times 12.25 = 122.5$ .

Step 3

Estimate $$\int_3^8 \text{log}_2 18x \, dx$$

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Answer

We can break this integral into parts: $\int_3^8 \text{log}_2 18x \, dx = \int_3^8 \text{log}_2 18 \, dx + \int_3^8 \text{log}_2 x \, dx.$

The first part simplifies as follows: $\int_3^8 \text{log}_2 18 \, dx = \text{log}_2 18 * (8 - 3) = \text{log}_2 18 * 5 = 5 \times \text{log}_2 18.$

For the second part, we use the result from part (a): $= 5 \times \text{log}_2 x |^8_3 + 5(12.25).$

So: $\int_3^8 \text{log}_2 18x \, dx \approx 5 \times 4.169 \approx 20.845 + 61.25 = 82.095.$

The table below shows corresponding values of $x$ and $y$ for $y = ext{log}_2 x$ The values of $y$ are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Worked Solution & Example Answer:The table below shows corresponding values of $x$ and $y$ for $y = ext{log}_2 x$ The values of $y$ are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 5 - 2022 - Paper 2

Using the trapezium rule with all the values of $y$ in the table, find an estimate for $$\int_3^8 \text{log}_2 2x \, dx$$

Estimate $$\int_3^8 \text{log}_2 (2x)^{10} \, dx$$

Estimate $$\int_3^8 \text{log}_2 18x \, dx$$

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