Implicit functions

Verify that the equation

$$x^3+y^3+xy=3$$

implicitly defines a function $y=\phi (x)$ near the point $(1,1)$ and compute ${\phi }^{\prime }(1)$.

Solution available to those who have unlocked the subject.

Verify that the equation

$$x{e}^y+y=1$$

implicitly defines a function $y=\phi (x)$ near the point $(1,0)$ and compute ${\phi }^{\prime }(1)$.

Solution available to those who have unlocked the subject.

Write the equation of the tangent line to the curve

$$x^{2}y+y^3=2$$

at the point $(1,1)$.

Solution available to those who have unlocked the subject.

Write the equation of the tangent line to the curve

$$\ln (x+y)+xy=0$$

at the point $(0,1)$.

Solution available to those who have unlocked the subject.

Find the points of the curve

$$x^2+xy+y^2=3$$

at which the tangent line is horizontal.

Solution available to those who have unlocked the subject.

Verify that the equation

$$y^3+y-2x^2=0$$

implicitly defines a function $y=\phi (x)$ near the point $(1,1)$ and compute ${\phi }^{\prime }(1)$ and ${\phi }^{\prime \prime }(1)$.

Solution available to those who have unlocked the subject.

Verify that the equation

$$y^3+3y-x^2+2x-5=0$$

implicitly defines a function $y=\phi (x)$ near the point $(1,1)$, and determine whether $x_0=1$ is a local maximum or minimum point for $\phi$.

Solution available to those who have unlocked the subject.

Verify that the equation

$$z^3+2z+x^2-y^2=3$$

implicitly defines a function $z=g(x,y)$ near the point $(1,1,1)$ and compute $g_x(1,1)$ and $g_y(1,1)$.

Solution available to those who have unlocked the subject.

Write the equation of the tangent plane to the surface

$$xz+y{z}^2+xy=3$$

at the point $(1,1,1)$.

Solution available to those who have unlocked the subject.

Find the points of the curve

$$x^3+y^2-3x=0$$

at which Dini’s theorem does not apply with respect to the variable $y$, and determine what happens at those points with respect to the variable $x$.

Solution available to those who have unlocked the subject.

Verify that the equation

$$e^{x+y}-x^2-y^2=1$$

implicitly defines a function $y=\phi (x)$ near the point $(0,0)$, compute ${\phi }^{\prime }(0)$ and write the tangent line to the curve at the origin.

Solution available to those who have unlocked the subject.

Verify that the system of equations

$$\{\begin{array}{l}{x}^2+y^2+z^2=6\\ x+2y-z=1\end{array}$$

implicitly defines two functions $y=y(x)$ and $z=z(x)$ near the point $(1,1,2)$ and compute $y^{\prime }(1)$ and $z^{\prime }(1)$.

Solution available to those who have unlocked the subject.