### Calculus in Three Dimensions (21-259) — Feedback on Homework 3

Homework 3 was due on Tuesday 30th September 2014. Questions 2, 4, 5, 8 and 10 were graded.

Question 2. This was mostly done very well. A couple of common errors were:

• Not labelling contour lines or axes. Please make sure you do this on the test, if required to do so!
• In part (a) many people inexplicably only drew the curves for $x>0$, even though negative values of $x$ were perfectly permissible.
• Also in part (a) there was some asymmetry in a few people's drawings. Since $\ln(\frac{1}{k}) = -\ln(k)$, the curve corresponding to $k=\frac{1}{3}$ should have been the reflection in the $x$-axis of the curve corresponding to $k=3$, and likewise for $\frac{1}{2}$,$2$.

Question 4. Arithmetic errors notwithstanding, this question was done well.

Question 5. This was done surprisingly well: I expected far more people to slip on the application of the rule $\frac{d}{dx}(a^x)=(\ln a)a^x$, but this only happened a couple of times! Yay! The biggest error that I saw was misapplication of the quotient rule.

Question 8. Another question done well by most people, so long as they differentiated correctly and plugged in the correct values. One thing to be mindful of is that you may be asked to put the equation of a plane in a particular form. If this is the case, make sure you do the appropriate rearrangements.

Question 10. Most errors on this question were arithmetical. Very few people did a 'sanity check' at the end: the value of the linear approximation at $(3.01, 3.98)$ should be close to the value of the function at $(3,4)$, namely it should be close to $\sqrt{3^2+4^2}=5$. If your linear approximation wasn't close to $5$, you could be instantly sure that it was incorrect!

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