Exercise 5 | Wiskunde op Tilburg University

The function

$z(x,y)$ is given by

$z(x,y)=5x^2y^4$ ,

$(x,y\geq 0)$ . Determine the point

$(x,y)$ where the line with slope

$-4$ is tangent to the level curve of the function

$z(x,y)$ with

$z$ -value 83886080.

Antwoord 1 correct

Correct

Antwoord 2 optie

$(x,y)=(1,64)$

Antwoord 2 correct

Fout

Antwoord 3 optie

$(x,y)=(2,16)$

Antwoord 3 correct

Fout

Antwoord 4 optie

$(x,y)=(4,1)$

Antwoord 4 correct

Fout

Antwoord 1 optie

$(x,y)=(4,32)$

Antwoord 1 feedback

Correct:

$\begin{align*} \dfrac{z'_x(x,y)}{z'_y(x,y)}&= \dfrac{10xy^4}{20x^2y^3}\\ & = \dfrac{y}{2x} \end{align*}$

$\begin{align*} -4&=- \dfrac{y}{2x}, \end{align*}$
and hence,

$y=8x$ .

Plugging in

$z(x,y)$ :

$\begin{align*} 5x^2(8x)^4 & = 83886080\\ 20480x^6&=83886080\\ x^6 & = 4096\\ x & = 4. \end{align*}$
Hence,

$y=8\cdot 4=32$ .

Go on.

Antwoord 2 feedback

Wrong:

$-4\neq - \dfrac{z'_x(1,64)}{z'_y(1,64)}$ .

See Tangent line to level curve.

Antwoord 3 feedback

Wrong:

$x^2\cdot x^4 \neq x^{12}$ .

See Properties power functions.

Antwoord 4 feedback

Wrong:

$\dfrac{z'_x(x,y)}{z'_y(x,y)}\neq -\dfrac{x}{y}$ .

See .