Photo AI

The table below shows corresponding values of x and y for y = log₂x The values of y are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Question 6

The-table-below-shows-corresponding-values-of-x-and-y-for-y-=-log₂x-The-values-of-y-are-given-to-2-decimal-places-as-appropriate-Edexcel-A-Level Maths Pure-Question 6-2022-Paper 2.png

The table below shows corresponding values of x and y for y = log₂x The values of y are given to 2 decimal places as appropriate. x | 3 | 4.5 | 6 | 7.5 | 8 ---|---|... show full transcript

Worked Solution & Example Answer:The table below shows corresponding values of x and y for y = log₂x The values of y are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Step 1

Using the trapezium rule with all the values of y in the table, find an estimate for $$\int_{3}^{8} log_{2} x \, dx$$

96%

114 rated

Answer

To apply the trapezium rule, we first identify the intervals and the widths between the points on the x-axis:

The intervals: [3, 4.5], [4.5, 6], [6, 7.5], and [7.5, 8]
The widths of these intervals (h) are: h = 1.5, 1.5, 1.5, and 0.5 respectively.

Using the trapezium rule: $ext{Area} = \frac{h}{2} \left( y_1 + 2y_2 + 2y_3 + 2y_4 + y_5 \right)$ Where:

$y_1 = 1.63$ , $y_2 = 2.26$ , $y_3 = 2.46$ , $y_4 = 2.63$ , and $y_5 = 2.63$ .

Calculating: $\text{Area} \approx \frac{1.5}{2} \left( 1.63 + 2(2.26) + 2(2.46) + 2.63 \right)$

= $\frac{1.5}{2} \left( 1.63 + 4.52 + 4.92 + 2.63 \right) = \frac{1.5}{2} \left( 13.8 \right) \approx 10.35$ .

Thus, the estimate for $\int_{3}^{8} log_{2} x \, dx$ is approximately 10.35.

Step 2

Estimate $$\int_{3}^{8} log_{2} (2x)^{10} \ dx$$

99%

104 rated

Answer

From part (a), we know: $\int_{3}^{8} log_{2} 2x \ dx = \int_{3}^{8} log_{2} 2 + \int_{3}^{8} log_{2} x \ dx$

Using properties of logarithms: $log_{2} (2x)^{10} = 10 \, log_{2} (2x) = 10 (log_{2} 2 + log_{2} x) = 10 (1 + log_{2} x).$

Thus, the integral becomes: $\int_{3}^{8} log_{2} (2x)^{10} \ dx = 10 \int_{3}^{8} log_{2} x \, dx + 10 \times 5 = 10(10.35) + 50 = 153.5.$

Step 3

Estimate $$\int_{3}^{8} log_{2} 18x \ dx$$

96%

101 rated

Answer

Using linearity of integrals: $\int_{3}^{8} log_{2} 18x \ dx = \int_{3}^{8} log_{2} 18 + \int_{3}^{8} log_{2} x \ dx$

We know: $\int_{3}^{8} log_{2} 18 = log_{2} 18 \times (8-3) = log_{2} 18 \times 5$

The value of $log_{2} 18$ is approximately: $log_{2} 18 \approx log_{2} 2^{4.17} ext{ (using a calculator)}$

Thus, we can compute: $= \approx 4.17 \times 5 + 10.35 = 20.85 + 10.35 = 31.2.$

Therefore, the estimate for $\int_{3}^{8} log_{2} 18x \ dx$ is approximately 31.2.

The table below shows corresponding values of x and y for y = log₂x The values of y are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Worked Solution & Example Answer:The table below shows corresponding values of x and y for y = log₂x The values of y are given to 2 decimal places as appropriate - Edexcel - A-Level Maths Pure - Question 6 - 2022 - Paper 2

Using the trapezium rule with all the values of y in the table, find an estimate for $$\int_{3}^{8} log_{2} x \, dx$$

Estimate $$\int_{3}^{8} log_{2} (2x)^{10} \ dx$$

Estimate $$\int_{3}^{8} log_{2} 18x \ dx$$

Join the A-Level students using SimpleStudy...