Differential and Integral Calculus (21-120) — Feedback on Homework 6

Homework 6 was due on Thursday 3rd October 2013 and consisted of:

• Section 3 Review Q 68, 74, 96, 106

I marked Q96 (out of 5) and Q106 (out of 4); a free mark was given for submitting the homework.

Section 3 Review Q96. This question was done very well on the whole. It followed Exercise 3 in Section 3.8 of the textbook. The error most people made in this question — and the textbook made this error too — is to round numbers too soon. For part (a) it was in fact possible to obtain an exact answer of $\frac{140}{3}$, which is approximately $46.667$, but people who rounded too early got fairly complicated numbers. What's worse is that if you didn't use a large number of decimal places then your final answer would have been out by many seconds, or even minutes! This matters when you're doing science and engeneering, when very high levels of precision are needed. The way around this problem is to round numbers to a certain number of decimal places only when you absolutely need to.

Section 3 Review Q106. The key to success in this question was to use the fact that $$f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$$ whenever $f$ is a function which is differentiable at a point $a$. Then we can just set $f(x)=x^{17}$ and $a=1$, and the limit in the question takes precisely this form.

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