A-Level Edexcel Maths Pure Working with Data: 4. (a) Express $$ \lim

4. (a) Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

Question 4

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4. (a) Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral. (b) Hence show that $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x = ... show full transcript

Worked Solution & Example Answer:4. (a) Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

Step 1

Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral.

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Answer

To express the given limit as an integral, we begin by interpreting the summation. The expression can be related to the concept of definite integrals:

$\lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x \approx \int_{2.1}^{6.3} 2 \, dx$

In this case, we can see that as $\alpha$ approaches 0, the width of the partitions approaches 0, thus turning the Riemann sum into the integral above.

Step 2

Hence show that $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x = \ln k $$ where $k$ is a constant to be found.

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Answer

From the previous step, we have:

$\int_{2.1}^{6.3} 2 \, dx$

Calculating this integral:

$\int 2 \, dx = 2x$

Evaluating from 2.1 to 6.3:

$2[6.3 - 2.1] = 2(6.3 - 2.1) = 2(4.2) = 8.4$

Now, according to the marking scheme, we then need to equate this to $\ln k$ :

$8.4 = \ln k$

To find $k$ , we exponentiate both sides:

$k = e^{8.4}$

Hence, we have shown that:

$\lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x = \ln(e^{8.4}) = 8.4$ , where $k = e^{8.4}$ .

4. (a) Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

Worked Solution & Example Answer:4. (a) Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral - Edexcel - A-Level Maths Pure - Question 4 - 2022 - Paper 1

Express $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x $$ as an integral.

Hence show that $$ \lim_{\alpha \to 0} \sum_{x=2.1}^{6.3} 2 \delta x = \ln k $$ where $k$ is a constant to be found.

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