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y = \frac{x}{\sqrt{(1 + x)}} (a) Complete the table below with the value of y corresponding to x = 1.3, giving your answer to 4 decimal places - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 5
Question 5
y = \frac{x}{\sqrt{(1 + x)}} (a) Complete the table below with the value of y corresponding to x = 1.3, giving your answer to 4 decimal places. | x | y | |-... show full transcript
Worked Solution & Example Answer:y = \frac{x}{\sqrt{(1 + x)}} (a) Complete the table below with the value of y corresponding to x = 1.3, giving your answer to 4 decimal places - Edexcel - A-Level Maths Pure - Question 5 - 2013 - Paper 5
Step 1
Complete the table with the value of y corresponding to x = 1.3, giving your answer to 4 decimal places.
Answer
To find the value of y for x = 1.3, we substitute x into the given function:
[ y = \frac{1.3}{\sqrt{(1 + 1.3)}} = \frac{1.3}{\sqrt{2.3}} \approx 0.8572 ]
Hence, the completed value for y when x = 1.3 is 0.8572.
Step 2
Use the trapezium rule, with all the values of y in the completed table, to obtain an approximate value for \( \int_1^{1.5} \frac{x}{\sqrt{(1 + x)}} \, dx \)
Answer
To apply the trapezium rule, we have:
- The width of each segment, ( h = \frac{b - a}{n} = \frac{1.5 - 1}{5 - 1} = 0.125 )
- The values of y corresponding to the x values:
- At x = 1: y = 0.7071
- At x = 1.1: y = 0.7591
- At x = 1.2: y = 0.8090
- At x = 1.3: y = 0.8572
- At x = 1.4: y = 0.9037
- At x = 1.5: y = 0.9487
Using the trapezium rule formula: [ ext{Area} = \frac{h}{2} \left( y_0 + 2\sum_{i=1}^{n-1} y_i + y_n \right) ] Substituting in: [ ext{Area} = \frac{0.125}{2} \left( 0.7071 + 2(0.7591 + 0.8090 + 0.8572 + 0.9037) + 0.9487 \right) ] Calculating the sum: [ = \frac{0.125}{2} \left( 0.7071 + 2(3.3280) + 0.9487 \right) ] [ = \frac{0.125}{2} \left( 0.7071 + 6.6560 + 0.9487 \right) ] [ = \frac{0.125}{2} \left( 7.3118 \right) ] [ = 0.125 \times 3.6559 = 0.4570 ] Rounding this to three decimal places gives: [ \approx 0.457 ]