We are asked to find the equation of the tangent plane to the given surfaces at the indicated points for problems 1 to 5. Problem 1: $x^2 + y^2 + z^2 = 16$ at $(2, 3, \sqrt{3})$. Problem 2: $8x^2 + y^2 + 8z^2 = 16$ at $(1, 2, \sqrt{2}/2)$. Problem 3: $x^2 - y^2 + z^2 + 1 = 0$ at $(1, 3, \sqrt{7})$. Problem 4: $x^2 + y^2 - z^2 = 4$ at $(2, 1, 1)$. Problem 5: $z = \frac{x^2}{4} + \frac{y^2}{4}$ at $(2, 2, 2)$.

1. Problem Description

We are asked to find the equation of the tangent plane to the given surfaces at the indicated points for problems 1 to

5.

Problem 1: $x^2 + y^2 + z^2 = 16$ at $(2, 3, \sqrt{3})$.

Problem 2: $8x^2 + y^2 + 8z^2 = 16$ at $(1, 2, \sqrt{2}/2)$.

Problem 3: $x^2 - y^2 + z^2 + 1 = 0$ at $(1, 3, \sqrt{7})$.

Problem 4: $x^2 + y^2 - z^2 = 4$ at $(2, 1, 1)$.

Problem 5: $z = \frac{x^2}{4} + \frac{y^2}{4}$ at $(2, 2, 2)$.

2. Solution Steps

The equation of the tangent plane to the surface $F(x, y, z) = 0$ at the point $(x_0, y_0, z_0)$ is given by

$$F_x(x_0, y_0, z_0)(x - x_0) + F_y(x_0, y_0, z_0)(y - y_0) + F_z(x_0, y_0, z_0)(z - z_0) = 0$$

Problem 1: $F(x, y, z) = x^2 + y^2 + z^2 - 16 = 0$.

$F_x = 2x$, $F_y = 2y$, $F_z = 2z$.

At $(2, 3, \sqrt{3})$, $F_x = 4$, $F_y = 6$, $F_z = 2\sqrt{3}$.

The equation of the tangent plane is $4(x - 2) + 6(y - 3) + 2\sqrt{3}(z - \sqrt{3}) = 0$, which simplifies to $4x - 8 + 6y - 18 + 2\sqrt{3}z - 6 = 0$.

$4x + 6y + 2\sqrt{3}z = 32$ or $2x + 3y + \sqrt{3}z = 16$.

Problem 2: $F(x, y, z) = 8x^2 + y^2 + 8z^2 - 16 = 0$.

$F_x = 16x$, $F_y = 2y$, $F_z = 16z$.

At $(1, 2, \sqrt{2}/2)$, $F_x = 16$, $F_y = 4$, $F_z = 16(\sqrt{2}/2) = 8\sqrt{2}$.

The equation of the tangent plane is $16(x - 1) + 4(y - 2) + 8\sqrt{2}(z - \sqrt{2}/2) = 0$, which simplifies to $16x - 16 + 4y - 8 + 8\sqrt{2}z - 8 = 0$.

$16x + 4y + 8\sqrt{2}z = 32$ or $4x + y + 2\sqrt{2}z = 8$.

Problem 3: $F(x, y, z) = x^2 - y^2 + z^2 + 1 = 0$.

$F_x = 2x$, $F_y = -2y$, $F_z = 2z$.

At $(1, 3, \sqrt{7})$, $F_x = 2$, $F_y = -6$, $F_z = 2\sqrt{7}$.

The equation of the tangent plane is $2(x - 1) - 6(y - 3) + 2\sqrt{7}(z - \sqrt{7}) = 0$, which simplifies to $2x - 2 - 6y + 18 + 2\sqrt{7}z - 14 = 0$.

$2x - 6y + 2\sqrt{7}z = -2$ or $x - 3y + \sqrt{7}z = -1$.

Problem 4: $F(x, y, z) = x^2 + y^2 - z^2 - 4 = 0$.

$F_x = 2x$, $F_y = 2y$, $F_z = -2z$.

At $(2, 1, 1)$, $F_x = 4$, $F_y = 2$, $F_z = -2$.

The equation of the tangent plane is $4(x - 2) + 2(y - 1) - 2(z - 1) = 0$, which simplifies to $4x - 8 + 2y - 2 - 2z + 2 = 0$.

$4x + 2y - 2z = 8$ or $2x + y - z = 4$.

Problem 5: $F(x, y, z) = \frac{x^2}{4} + \frac{y^2}{4} - z = 0$.

$F_x = \frac{x}{2}$, $F_y = \frac{y}{2}$, $F_z = -1$.

At $(2, 2, 2)$, $F_x = 1$, $F_y = 1$, $F_z = -1$.

The equation of the tangent plane is $(x - 2) + (y - 2) - (z - 2) = 0$, which simplifies to $x - 2 + y - 2 - z + 2 = 0$.

$x + y - z = 2$.

3. Final Answer

Problem 1: $2x + 3y + \sqrt{3}z = 16$

Problem 2: $4x + y + 2\sqrt{2}z = 8$

Problem 3: $x - 3y + \sqrt{7}z = -1$

Problem 4: $2x + y - z = 4$

Problem 5: $x + y - z = 2$