Calculus for Economics, October 02, 2017
- I Find the tangent plane of the graph
$z=f(x,y)$ at $\mathrm{P}_0(a,b)$
- (1)
$z=xy-2x+2y-1$ at $P_0(0,0,-1)$
- (2)
$z=\frac x{x+y}$ at $P_0(1,-2,-1)$
- (3)
$z=x^2-xy+2y^2$ at $P_0(2,1,4)$
- (4)
$z=\frac y{1+x^2}$ at $P_0(0,0,0)$
- II Find the tangent line of
the curve $g(x,y)=0$ at $\mathrm{P}_0$.
- (1)
$g(x,y)=x^2+4y^2-1=0$ at $P_0(\frac 1{\sqrt 2}, \frac 1{2\sqrt 2})$
- (2)
$g(x,y)=x^{\frac 13}y^{\frac 13}-1=0$ at
$P_0(1,1)$
- (3)
$g(x,y)=x^2-xy+y^2-1=0$ at $P_0(0,1)$
III We consider the production function
\begin{equation*}
Q=F(K,L)=9K^{\frac 13}L^{\frac 23}
\end{equation*}
of Capital input $K$ and Labor input $L$.
- (1)
Find the output $Q$ when $K=216$ and $L=10^3$.
- (2) Find MPK and MPL at $(K,L)=(216,10^3)$ and
get the approximate value for $F(216,998)$ and $F(217.5,10^3)$ by
using MPK and MPL obtained above.