### Differential and Integral Calculus (21-120) — Feedback on Homework 11

Homework 11 was due on Thursday 14th November 2013 and consisted of:

- Section 5.3 Q 14, 42, 70
- Section 5.5 Q 44, 62

I marked 5.3/70, 5.5/44 and 5.5/62 (each out of 3); a free mark was given for submitting the homework.

**Section 5.3 Q70.** A lot of people correctly identified $f(x)$, $\Delta x$ and $x_i$ but then didn't use them to write the limit as an integral. Unlike in Sections 5.1 and 5.2, in this section you're not prohibited from doing integration! (It's a section about the Fundamental Theorem of Calculus after all.) The way to go was to notice that $$\lim_{n \to \infty} \sum_{i=1}^n \frac{1}{n} \sqrt{\frac{i}{n}} = \int_0^1 \sqrt{x}\, dx$$ and then use FTC part 2 to evaluate it. Every attempt to evaluate the limit without writing it as an integral contained some kind of error.

**Section 5.5 Q44.** This question was done very well on the whole, provided the substitution made was $u=x^2$. The most common error after this was thinking the antiderivative of $\frac{1}{1+u^2}$ was a logarithm rather than an inverse tangent.

**Section 5.5 Q62.** Most people made the substitution $u=\sin x$ correctly. A few people forgot to change the limits (grr!) and a few seemed to think that $\cos(0)=0$ (when in fact $\cos(0)=1$). A large proportion of people got out their calculators to write $\cos(1)$ as a decimal: this was unnecessary, and in some cases led to the wrong answer because the calculator was in degrees mode when it should have been in radians mode. Careful!

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