$1$

$\frac{v(P+xP,C+yC)-v(P,C)}{v(P,C)}$

Expression $1$ gives the fractional change in speed that results from a change $x$ in power and a change $y$ in drag. Show that this reduces to the function

$f(x,y)=(\frac{1+x}{1+y}{)}^{\frac{1}{3}}-1$

Given the context, what is the domain of $f?$

$2.$ Suppose that the possible changes in power $x$ and drag $y$ are small. Find the linear approximation to the function $f(x,y).$ What does this approximation tell you about the effect of a small increase in power versus a small decrease in drag?

$3.$ Calculate $f_{}(x,y)$ and $f_{yy}(x,y).$ Based on the signs of these derivatives, does the linear approximation in Problem $2$ result in an overestimate or an underestimate for an increase in power? What about for a decrease in drag? Use your answer to explain why, for changes in power or drag that are not very small, a decrease in drag is more effective.

$4.$ Graph the level curves of $f(x,y).$ Explain how the shapes of these curves relate to your answers to Problems $2$ and $3.$