Question $1$: Calculate the partial derivatives $\frac{\text{d}\text{e}\text{l}\text{f}(x_1,x_2)}{del{x}_1}$ and $\frac{\text{d}\text{e}\text{l}\text{f}(x_1,x_2)}{del{x}_2}$ for each of the following functions:

$(1)f(x_1,x_2)=3x_1+4x_2^2$

$(2)f(x_1,x_2)=x_1{x}_2^4$

$(3)f(x_1,x_2)=3x_1^{\frac{3}{4}}{x}_2^{\frac{1}{4}}$

$(4)f(x_1,x_2)=x_1^{\frac{1}{3}}{x}_2^{\frac{1}{3}}$

Question $2$: Consider the market for flights from the US to the Caribbean. Fill in the blanks to describe what happens after the following events. The first blank in each response should be supply or demand and the second blank should be left or right.

$(1)$ After the cost of airplane fuel increases, the curve shifts to the

$(2)$ After it is determined that a virus plaguing the Caribbean is no longer a threat, the curve shifts to the

$(3)$ After airplanes are reconfigured to fit $10\%$ more seats on each plane $($without consumers finding out$),$ the curve shifts to the

$(4)$ After the weather in the US gets much warmer, the curve shifts to the

Now fill in the blanks to describe what happens to the market equilibrium after the following events. Both blanks should be increases, decreases, or changes ambiguously. Draw graphs to support your answers.

$(1)$ After airplanes are reconfigured to fit $10\%$ more seats on each plane $($without consumers finding out$),$ the equilibrium price quantity and the equilibrium

$1$