Quiz 9

$There~is~a~3-dimension~graph,~z~=~2x~+~3y~+~5.~If~it~is~projected~to~2-dimensional~(x,~z)~space~while~the~value~of~y~is~fixed~to~10,~what~is~$ $~the~equation~of~the~line~in~the~x,z~space?$

z = 2x – 35

z = 2x + 3y + 5

z = 2x + 35

z = 2x + 5

$There~is~a~3-dimension~graph,~z~=~2x~+~3y~+~5.~If~it~is~projected~to~2-dimensional~(y,z)~space~while~the~value~of~x~is~fixed~to~15,~what~is~$ $~the~equation~of~the~line~in~the~y,z~space?$

z = 3y – 35

z = 3y + 35

z = 3y + 5

z = 3y + 2x + 35

$Given~z~=~2x~+~3y~+~5,~find~the~partial~derivative~~\frac{\partial~z}{\partial~x}.$

2 + 3y

2x + 5

$~Given~z~=~2x~+~3y~+~5,~find~the~partial~derivative~\frac{\partial~z}{\partial~y}.$

3 + 2x

$Find~the~partial~derivative,~\frac{\partial~z}{\partial~x},~of~the~function~z=3x^{2}+2x+3xy+5y^{2}+10.$

$\frac{\partial~z}{\partial~x}=6x+3y+2$

$\frac{\partial~z}{\partial~x}=6x+3xy+2x$

$\frac{\partial~z}{\partial~x}=6x+2$

$\frac{\partial~z}{\partial~x}=6x-3y-2$

$Find~the~partial~derivative,~\frac{\partial~z}{\partial~y},~of~the~function~z=3x^{2}+2x+3xy+5y^{2}+10.$

$\frac{\partial~z}{\partial~y}=10y+3xy$

$\frac{\partial~z}{\partial~y}=10y+2x+3x$

$\frac{\partial~z}{\partial~y}=10y+3x$

$\frac{\partial~z}{\partial~y}=10y+3xy+10$

$Find~the~second~order~partial~derivative,~\frac{\partial^{2}z}{\partial~x^{2}},~of~the~function~z=3x^{2}+2x+3xy+5y^{2}+10.$

$\frac{\partial^{2}z}{\partial~x^{2}}=6$

$\frac{\partial^{2}z}{\partial~x^{2}}=6x$

$\frac{\partial^{2}z}{\partial~x^{2}}=3y$

$\frac{\partial^{2}z}{\partial~x^{2}}=3y+10$

$Find~the~second~order~partial~derivative,~\frac{\partial^{2}z}{\partial~y^{2}},~of~the~function~z=3x^{2}+2x+3xy+5y^{2}+10.$

$\frac{\partial^{2}z}{\partial~y^{2}}=10y$

$\frac{\partial^{2}z}{\partial~y^{2}}=3x$

$\frac{\partial^{2}z}{\partial~y^{2}}=10$

$\frac{\partial^{2}z}{\partial~y^{2}}=10x$

$Find~the~cross~partial~derivative,~\frac{\partial^{2}z}{\partial~y\partial~x},~of~the~function~z=3x^{2}+2x+3xy+5y^{2}+10.$

$\frac{\partial^{2}z}{\partial~y\partial~x}=2$

$\frac{\partial^{2}z}{\partial~y\partial~x}=6x$

$\frac{\partial^{2}z}{\partial~y\partial~x}=3y$

$\frac{\partial^{2}z}{\partial~y\partial~x}=3$

$Find~the~marginal~productivity~of~labour,~\frac{\partial~Y}{\partial~L},~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~productivity.$

$\frac{3}{4}AL^{\frac{3}{4}}K^{\frac{1}{4}}$

$\frac{3}{4}AL^{-\frac{1}{4}}K^{\frac{1}{4}}$

$AL^{-\frac{1}{4}}K^{\frac{1}{4}}$

$AL^{-\frac{3}{4}}K^{\frac{1}{4}}$

$Find~the~marginal~productivity~of~capital,~\frac{\partial~Y}{\partial~K},~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~productivity.$

$\frac{1}{4}AL^{\frac{3}{4}}K^{-\frac{3}{4}}$

$\frac{3}{4}AL^{\frac{3}{4}}K^{-\frac{3}{4}}$

$AL^{\frac{3}{4}}K^{-\frac{3}{4}}$

$\frac{1}{4}AL^{\frac{1}{4}}K^{-\frac{1}{4}}$

$Find~the~second~order~partial~derivative~w.r.t.~labour,~\frac{\partial^{2}Y}{\partial~L^{2}},~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~productivity.$

$\frac{3}{16}AL^{-\frac{1}{4}}K^{\frac{1}{4}}$

$AL^{-\frac{5}{4}}K^{\frac{1}{16}}$

$-\frac{1}{16}AL^{-\frac{1}{4}}K^{\frac{3}{4}}$

$-\frac{3}{16}AL^{-\frac{5}{4}}K^{\frac{1}{4}}$

$Find~the~second~order~partial~derivative~w.r.t.~capital,~\frac{\partial^{2}Y}{\partial~K^{2}},~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~productivity.$

$-\frac{3}{16}AL^{\frac{3}{4}}K^{-\frac{7}{4}}$

$\frac{1}{16}AL^{\frac{3}{4}}K^{-\frac{7}{4}}$

$AL^{\frac{3}{4}}K^{-\frac{7}{4}}$

$-\frac{3}{2}AL^{\frac{1}{4}}K^{-\frac{3}{4}}$

$Find~the~cross~partial~derivative,~\frac{\partial^{2}Y}{\partial~K\partial~L},~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~productivity.$

$\frac{1}{16}AL^{-\frac{1}{4}}K^{-\frac{3}{2}}$

$-\frac{3}{16}AL^{-\frac{1}{4}}K^{-\frac{3}{4}}$

$\frac{3}{16}AL^{-\frac{1}{4}}K^{-\frac{3}{4}}$

$\frac{3}{8}AL^{\frac{1}{4}}K^{\frac{3}{4}}$

$Find~all~the~correct~arguments~about~MPL(the~marginal~labor~productivity)~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~$ $productivity.$

MPL is declining as L rises.

MPL is rising as L rises

MPL is rising as K rises.

MPL is rising as A rises.

MPL is declining as K rises.

$Find~all~the~correct~arguments~about~MPK(the~marginal~productivity~of~Kapital)~of~the~production~function~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~A~is~a~constant~representing~$ $productivity.$

MPK is declining as K rises.

MPK is rising as K rises

MPK is declining as L rises.

MPK is rising as L rises.

MPK is rising as A rises.

$The~production~function~is~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~Find~the~gradient~of~the~isoquant~of~the~production~function,~\frac{\partial~K}{\partial~L},~given~the~value~of~Y~fixed.$

$(\frac{Y}{AL})^{4}$

$-3(\frac{Y}{AL})^{4}$

$3(\frac{Y}{AL})^{3}$

$-3(\frac{Y}{L})^{2}$

$The~production~function~is~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~Find~the~equation~of~the~isoquant~of~the~production~function,~given~the~value~of~Y~fixed.$

$K=\left(\frac{Y}{A}\right)^{4}L^{-3}$

$K=\left(\frac{Y}{A}\right)^{-4}L^{3}$

$K=\left(\frac{Y}{A}\right)^{2}L^{-4}$

$K=\left(AY\right)^{4}L^{-3}$

$The~production~function~is~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~Find~the~gradient~of~the~isoquant~of~the~production~function,\frac{dK}{dL}.~$

$3\left(\frac{Y}{A}\right)^{4}L^{-3}$

$-3\left(\frac{Y}{A}\right)^{4}L^{-4}$

$\left(\frac{Y}{A}\right)^{3}L^{-5}$

$\left(\frac{Y}{A}\right)^{-4}L^{-4}$

$The~production~function~is~Y=AL^{\frac{3}{4}}K^{\frac{1}{4}}.~Find~all~the~correct~arguments~about~the~isoquant~of~the~production~function.~The~y~axis~is~the~$ $~level~of~capital~-~K,~and~the~x~axis~is~L.$

The slope is always negative (downward).

The slope is positive (upward).

The slope is decreasing as L increases.

The slope is increasing as L increases.

It becomes easier to substitute labour with capital as L increases.

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Department of Economics

Quiz 9

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