Problem 18 - Calculus Early Transcendentals

Question

Find an equation of the plane tangent to the following surfaces at the given points. $$z=2+2 x^{2}+\frac{y^{2}}{2} ;\left(-\frac{1}{2}, 1,3\right) \text { and }(3,-2,22)$$

Step-by-Step Solution

Answer

Based on the given step by step solution, the question could be: "Find the equations of the tangent planes to the surface \(z = 2 + 2x^2 + \frac{y^2}{2}\) at the points \((-1/2, 1, 3)\) and \((3, -2, 22)\). Use the equation of a tangent plane and find the partial derivatives of the surface function with respect to x and y to solve the problem."

Step 1: Compute the partial derivatives with respect to x and y

First we will find the partial derivatives of the given function \(z = 2 + 2x^2 + \frac{y^2}{2}\) with respect to x and y. The partial derivative with respect to x: $$ \frac{\partial z}{\partial x} = 4x $$ The partial derivative with respect to y: $$ \frac{\partial z}{\partial y} = y $$

Step 2: Use the given points and the partial derivatives to find the tangent planes

The equation of a tangent plane to a surface \(z = f(x, y)\) at a point \((x_0, y_0, z_0)\) is given by: $$ z - z_0 = \frac{\partial f}{\partial x}(x_0, y_0)(x-x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0) $$ We will now use this equation to find the tangent planes at the given points.

Step 3: Find the tangent plane at point \((-\frac{1}{2}, 1, 3)\)

Plug in the point \((-\frac{1}{2}, 1, 3)\), the partial derivatives \(\frac{\partial z}{\partial x} = 4x\) and \(\frac{\partial z}{\partial y} = y\) into the equation: $$ z - 3 = 4(-\frac{1}{2})(x + \frac{1}{2}) + 1(y - 1) $$ Simplifying the equation, we obtain: $$ z = -2x + y + 5 $$ This is the equation of the tangent plane at the point \((-\frac{1}{2}, 1, 3)\).

Step 4: Find the tangent plane at point \((3, -2, 22)\)

Plug in the point \((3, -2, 22)\), the partial derivatives \(\frac{\partial z}{\partial x} = 4x\) and \(\frac{\partial z}{\partial y} = y\) into the equation: $$ z - 22 = 4(3)(x - 3) + (-2)(y + 2) $$ Simplifying the equation, we obtain: $$ z = 12x - 2y - 14 $$ This is the equation of the tangent plane at the point \((3, -2, 22)\). Hence, the tangent plane equations are: 1. At point \((-\frac{1}{2}, 1, 3)\): \(z = -2x + y + 5\) 2. At point \((3, -2, 22)\): \(z = 12x - 2y - 14\)

Key Concepts

Partial Derivatives

When we deal with surfaces and need to find tangent planes, partial derivatives are key. They help us understand how a function changes in each direction. For a given function like \( z = 2 + 2x^2 + \frac{y^2}{2} \), we want to compute derivatives with respect to both \( x \) and \( y \). This gives us a slope showing how much \( z \) changes if we move in the \( x \)-direction or \( y \)-direction separately.

• To find the partial derivative with respect to \( x \), treat \( y \) as constant. Thus, we focus on the \( x^2 \) term: \( \frac{\partial z}{\partial x} = 4x \).
• Similarly, for \( y \), we focus on the \( \frac{y^2}{2} \) term: \( \frac{\partial z}{\partial y} = y \).

These derivatives are crucial when forming the equation of a plane as they form part of the slope parameters.

Equation of a Plane

The equation of a tangent plane is a mathematical way of representing a flat surface that just touches a curved one at a specific point.

For a surface \( z = f(x, y) \), the tangent plane at any point \( (x_0, y_0, z_0) \) is given by the formula: \[ z - z_0 = \frac{\partial f}{\partial x}(x_0, y_0)(x-x_0) + \frac{\partial f}{\partial y}(x_0, y_0)(y - y_0) \]

• In this formula, \( \frac{\partial f}{\partial x} \) and \( \frac{\partial f}{\partial y} \) are the partial derivatives found earlier, providing the slope of the plane.
• The point \( (x_0, y_0, z_0) \) denotes where the plane touches the surface, ensuring it is tangent.

This formula essentially gives a linear approximation of the surface at the point, using the slopes in the \( x \) and \( y \) directions.

Surface Equations

Surface equations like \( z = 2 + 2x^2 + \frac{y^2}{2} \) define complex shapes in three-dimensional space. These equations portray surfaces that can curve and twist, unlike simple planes.

• Such an equation represents all points \( (x, y, z) \) making up the surface.
  
• For every \( x \) and \( y \), the equation gives a unique \( z \)-value, helping to visualize the surface.
• When looking for how a surface behaves locally, or around a specific point, we focus on tangent planes because they provide a simpler surface that approximates the original.

In mathematics and various applications like physics and engineering, understanding surface equations and their tangents allows for deeper insights into how objects behave.