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The table below shows corresponding values of x and y for $y = \frac{x}{\sqrt{1+x}}$ - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 2

Question 3

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The table below shows corresponding values of x and y for $y = \frac{x}{\sqrt{1+x}}$. The values of y are given to 4 significant figures. | x | 0.5 | 1 | ... show full transcript

Worked Solution & Example Answer:The table below shows corresponding values of x and y for $y = \frac{x}{\sqrt{1+x}}$ - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 2

Step 1

Use the trapezium rule, with all the values of y in the table, to find an estimate for $$\int_{0.5}^{2.5} \frac{x}{1+x} \, dx$$ giving your answer to 3 significant figures.

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Answer

To apply the trapezium rule, we first establish the values from the table:

$x$ : 0.5, 1.0, 1.5, 2.0, 2.5
$y$ : 0.5774, 0.7071, 0.7746, 0.8165, 0.8452

The formula for the trapezium rule is: $A = \frac{h}{2} \left( y_0 + 2\sum_{i=1}^{n-1} y_i + y_n \right)$

Where:

$h = \frac{{b-a}}{n} = \frac{2.5-0.5}{4} = 0.5$ (the width of each interval)
$y_0 = 0.5774$ , $y_1 = 0.7071$ , $y_2 = 0.7746$ , $y_3 = 0.8165$ , $y_4 = 0.8452$

Plugging these values into the trapezium formula: $A = \frac{0.5}{2} \left( 0.5774 + 2(0.7071 + 0.7746 + 0.8165) + 0.8452 \right)$

Calculating step-by-step:

Calculate the sum inside the parentheses:
- $0.5774 + 2(0.7071 + 0.7746 + 0.8165) + 0.8452 = 0.5774 + 2(2.2982) + 0.8452 = 0.5774 + 4.5964 + 0.8452 = 6.0190$
Next, calculate:
- $A = 0.25 \times 6.0190 = 1.50475$

Rounding to 3 significant figures gives: 1.50.

Step 2

Using your answer to part (a), deduce an estimate for $$\int_{0.5}^{2.5} \frac{9x}{1+x} \, dx$$

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Answer

From part (a), we have estimated: $\int_{0.5}^{2.5} \frac{x}{1+x} \, dx \approx 1.50$

Now, note that: $\int_{0.5}^{2.5} \frac{9x}{1+x} \, dx = 9 \int_{0.5}^{2.5} \frac{x}{1+x} \, dx$

Therefore: $\int_{0.5}^{2.5} \frac{9x}{1+x} \, dx \approx 9 \times 1.50 = 13.5$

Step 3

comment on the accuracy of your answer to part (b).

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Answer

Given that the actual value is: $\int_{0.5}^{2.5} \frac{9x}{1+x} \, dx = 4.535$ to 4 significant figures, our estimate of 13.5 is significantly higher than the actual value.

This discrepancy indicates that there may have been a miscalculation in applying the trapezium rule or a misinterpretation of the function.

Since 13.5 is an order of magnitude larger than 4.535, further investigation on potential errors in calculation methods or approximations used during the trapezium application should be performed.

The table below shows corresponding values of x and y for $y = \frac{x}{\sqrt{1+x}}$ - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 2

Worked Solution & Example Answer:The table below shows corresponding values of x and y for $y = \frac{x}{\sqrt{1+x}}$ - Edexcel - A-Level Maths Pure - Question 3 - 2020 - Paper 2

Use the trapezium rule, with all the values of y in the table, to find an estimate for $$\int_{0.5}^{2.5} \frac{x}{1+x} \, dx$$ giving your answer to 3 significant figures.

Using your answer to part (a), deduce an estimate for $$\int_{0.5}^{2.5} \frac{9x}{1+x} \, dx$$

comment on the accuracy of your answer to part (b).

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