Exercise 3 | Wiskunde op Tilburg University

The function

$z(x,y)$ is given by

$z(x,y)=4x^2y^{\frac{1}{3}}$ . Determine the tangent line to the level curve through the point

$(5,1)$ .

Antwoord 1 correct

Correct

Antwoord 2 optie

$t(x)=-\frac{6}{5}x+6\frac{1}{5}$

Antwoord 2 correct

Fout

Antwoord 3 optie

$t(x)=\frac{11}{25}x-\frac{6}{5}$

Antwoord 3 correct

Fout

Antwoord 4 optie

$t(x)=6\frac{1}{5}x-\frac{6}{5}$

Antwoord 4 correct

Fout

Antwoord 1 optie

$t(x)=-\frac{6}{5}x+7$

Antwoord 1 feedback

Correct:

$\begin{align*} \dfrac{z'_x(x,y)}{z'_y(x,y)}&= \dfrac{8xy^{-\frac{1}{3}}}{\frac{4}{3}x^2y^{-\frac{2}{3}}}\\ & = \dfrac{6y}{x} \end{align*}$

$\begin{align*} \textrm{slope} &=- \dfrac{6\cdot 1}{5}\\ &=-\dfrac{6}{5}. \end{align*}$

Hence,

$t(x)=-\frac{6}{5}x+b$ . Moreover,

$t(5)=-\frac{6}{5} \cdot 5+b=1$ gives

$b=7$ .

Consequently,

$t(x)=-\frac{6}{5}x+7$ .

Go on.

Antwoord 2 feedback

Wrong:

$x=5$ and

$y=1$ , not the other way around.

Try again.

Antwoord 3 feedback

Wrong: The slope of the tangent line is the number preceeding

$x$ .

Try again.

Antwoord 4 feedback

Wrong: The slope of the tangent line is the number preceeding

$x$ .

Try again.