Which of the following graphs depicts the level curves of $z(x,y)=x^2 + y^2$ with values $k=1$ and $k=4$?

Antwoord 1 correct

Correct

Antwoord 2 optie

Antwoord 2 correct

Fout

Antwoord 3 optie

Antwoord 3 correct

Fout

Antwoord 4 optie

It is not possible to draw the level curves of this function, because $z(x,y)=k$ cannot be rewritten.

Antwoord 4 correct

Fout

Antwoord 1 optie

Antwoord 1 feedback

Correct: Rewriting of $x^2 + y^2 = k$ for $k=1$ gives:

$$ x^2 + y^2 = 1 \iff y^2 = 1-x^2 \iff y = \pm\sqrt{1-x^2}.$$

The graph of this is the central circle; you might recognize $x^2+y^2=1$ as the circle equation with center $(0,0)$ and radius $\sqrt{1}=1$. Rewriting of $x^2 + y^2 = k$ for $k=4$ gives:

$$ x^2 + y^2 = 4 \iff y^2 = 4-x^2 \iff y = \pm\sqrt{4-x^2}.$$

The graph of this is the outside circle; you might recognize $x^2+y^2=4$ as the circle equation with center $(0,0)$ and radius $\sqrt{4}=2$.

Go on.

$$ x^2 + y^2 = 1 \iff y^2 = 1-x^2 \iff y = \pm\sqrt{1-x^2}.$$

The graph of this is the central circle; you might recognize $x^2+y^2=1$ as the circle equation with center $(0,0)$ and radius $\sqrt{1}=1$. Rewriting of $x^2 + y^2 = k$ for $k=4$ gives:

$$ x^2 + y^2 = 4 \iff y^2 = 4-x^2 \iff y = \pm\sqrt{4-x^2}.$$

The graph of this is the outside circle; you might recognize $x^2+y^2=4$ as the circle equation with center $(0,0)$ and radius $\sqrt{4}=2$.

Go on.

Antwoord 2 feedback

Antwoord 3 feedback