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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