### 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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