Paper Homework 5

Problem 1: Some students made mistakes taking the partial derivatives of z in part (b). A good thing to do is to check whether the tangent planes you got in (a) and (b) are the same. If not, one of your result must be wrong.

Problem 5: Subtracting $dz$ (the linear approximation) from $\Delta z$ gives us $\epsilon_1\Delta x + \epsilon_2\Delta y$. The rest is to specify $\epsilon_1$ and $\epsilon_2$ and show they go to 0 as $(\Delta x, \Delta y)\to (0,0)$.

Problem 7: Some students used an alternative way of implicit differentiation instead of using the formula given on the textbook. The idea is to directly take the partial differentiation on the equation. This idea is good but needs precise execution. For instance, the partial derivative of $e^y\sin x$ with respect to $x$ is $e^y\frac{\partial y}{\partial x}\sin x+e^y\cos x$. Here we have used the chain rule and the product rule.

Problem 9: Almost all have done well on this problem.