### Differential and Integral Calculus (21-120) — Feedback on Homework 10

Homework 10 was due on Thursday 7th November 2013 and consisted of:

• Section 5.1 Q 22, 24
• Section 5.2 Q 22, 30

I marked 5.1/24 (out of 6) and 5.2/30 (out of 4).

Section 5.1 Q24. Most people who dropped marks on this question did so because they only answered half of the question. Another common error relates to the general comment below.

Section 5.2 Q30. By far the most common error here was writing $x_i=\frac{9i}{n}$. When you have $\displaystyle \int_a^b f(x)\, dx$, for a given $n$, the interval is split up into segments of length $\Delta x = \frac{b-a}{n}$, and the right-hand endpoint of each segment is $x_i = a + i\Delta x$. In this case $b-a=10=1=9$, so $\Delta x = \frac{9}{n}$ and $x_i = 1 + \frac{9i}{n}$.

Both questions. As you saw in recitation, there are three summation identities that you need to know: $$\sum_{i=1}^n i = \frac{n(n+1)}{2}, \quad \sum_{i=1}^n i^2 = \frac{n(n+1)(2n+1)}{6}, \quad \sum_{i=1}^n i^3 = \left( \frac{n(n+1)}{2} \right)^2$$ It does not follow from this that, for example, $\displaystyle \sum_{i=1}^n \ln(i) = \ln \left( \frac{n(n+1)}{2} \right)$. (After all, why should it? The logarithm of a sum isn't the sum of the logarithms!) Be very careful when applying these identities!

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