Calculus for Economics,October 16, 2017
- I
-
We consider a Cobb-Douglas production function by
\begin{equation}
Q=F(K,L)=4K^{\frac 34}L^{\frac 14}
\end{equation}
Find an approximated value of $F(10^4+100,625+(-15))$
by using MPK,MPL $K=10^4,\ L=625$. Caculate the value of
$F(10^4+100,625+(-15))$ by an calculator etc.
- 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
A firm employs $x$ hours of unskilled labor and $y$ hours of skilled labor
to produce
\begin{equation*}
Q=F(x,y)=60x^{\frac 23}y^{\frac 13}
\end{equation*}
units of a product. Currently it uses $x=64$, $y=27$.
- (1) What is the current production?
- (2) In which direction should the firm change $(x,y)$
to increase output most?
- (3) The firm is planning to employ an addtional hour and a half hour of skilled labor. Estimate approximately the corresponding change of unskilled labor if the firm keeps its output.
- IV Use Clamer to solve the follwoing system of linear equations.
- (1)
$
\left\{
\begin{array}{lcl}
2x-3y&=&7\\
3x+5y&=&1
\end{array}
\right.
$
(2)
$
\left\{
\begin{array}{lcl}
2x-3y&=&-1\\
4x+7y&=&-1
\end{array}
\right.
$
(3)
$
\left\{
\begin{array}{lcl}
3x+5y&=&8\\
4x-2y&=&1
\end{array}
\right.
$
- V
- Use Clamer to express $x$ and $y$ by $z$ if
$(x,y,z)$ satisfies
$
\left\{
\begin{array}{lcl}
x+y-z&=&1\\
2x-y+z&=&-1
\end{array}
\right.
$