Math Insight

Elementary partial derivative problems

Problem 1

Let $h(y,z)=ye^{-z}$. Calculate $\pdiff{h}{y}$ and $\pdiff{h}{z}$.

Problem 2

Let $z(s,u)=s^2+su+u^2$. Calculate $\pdiff{z}{s}$ and $\pdiff{z}{u}$.

Problem 3

Let $q(y,z) = (y+3)(10-z)$. Calculate $\pdiff{q}{y}$ and $\pdiff{q}{z}$.

Problem 4

Let $r(a,b)=ab\ln(ca)$ where $c$ is a positive parameter. Calculate $\pdiff{r}{a}$ and $\pdiff{r}{b}$.

Problem 5

Let $r(a,b,c)=ab\ln(ca)$. Calculate $\pdiff{r}{a}$, $\pdiff{r}{b}$, and $\pdiff{r}{c}$.

Problem 6

Let $t(q,k)$ be the amount a person tips a server as a function of the food quality $q$ and the kindness $k$ of the server.

1. What quantity indicates how much the tip will increases as the quality of the food increases while the kindness of the server stays constant?
2. Let $t(q,k) = e^{2q+5k}$. Calculate $\pdiff{t}{k}$ and $\pdiff{t}{q}$.

Problem 7

Let $h(c, v, s, g)$ be your risk of heart disease as function of the amount of cholesterol $c$, vegetables $v$, sodium $s$, and whole grains $g$ you eat.

1. Describe what $\pdiff{h}{c}$, $\pdiff{h}{v}$, $\pdiff{h}{s}$, and $\pdiff{h}{g}$ mean.
2. If eating more vegetables and whole grains decrease your risk of heart disease, but eating more cholesterol and sodium increase your risk of heart disease, describe what must be true about these partial derivatives.

One you have worked on a few problems, you can compare your solutions to the ones we came up with.