Problem 181 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(z=x^{2}-2 x y+y^{2}, P(1,2,1)\)

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = -2x + 2y - 3 \).

Step 1: Formula for the Tangent Plane

The equation for the tangent plane to a surface given by \( z = f(x, y) \) at a point \( P(x_0, y_0, z_0) \) is: \[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \]where \( f_x \) and \( f_y \) are partial derivatives of \( f \) with respect to \( x \) and \( y \), respectively.

Step 2: Calculate Partial Derivatives

Find the partial derivatives \( f_x \) and \( f_y \) of \( z = x^2 - 2xy + y^2 \).- \( f_x = \frac{\partial}{\partial x}(x^2 - 2xy + y^2) = 2x - 2y \).- \( f_y = \frac{\partial}{\partial y}(x^2 - 2xy + y^2) = -2x + 2y \).

Step 3: Evaluate Partial Derivatives at P(1,2)

Substitute \( x = 1 \) and \( y = 2 \).- \( f_x(1, 2) = 2(1) - 2(2) = 2 - 4 = -2 \).- \( f_y(1, 2) = -2(1) + 2(2) = -2 + 4 = 2 \).

Step 4: Plug Values into Tangent Plane Equation

Substitute \( (x_0, y_0, z_0) = (1, 2, 1) \), \( f_x(1, 2) = -2 \), and \( f_y(1, 2) = 2 \) into the tangent plane equation:\[ z - 1 = -2(x - 1) + 2(y - 2)\]Simplify to obtain the equation of the tangent plane.

Step 5: Simplify the Equation

Expand and simplify the equation:\[ z - 1 = -2x + 2 + 2y - 4\]Combine like terms:\[ z = -2x + 2y - 3 + 1 \]Thus the equation of the tangent plane is:\[ z = -2x + 2y - 3\]

Key Concepts

Partial Derivatives

Understanding partial derivatives is key in finding the tangent plane to a surface. When you have a function of several variables, such as \( z = f(x, y) \), the partial derivative with respect to one variable helps to measure the rate of change of \( z \) along that particular direction, while keeping other variables constant.

In our case, for the surface represented by \( z = x^2 - 2xy + y^2 \), you need to differentiate with respect to \( x \) and \( y \) individually.

• Partial derivative with respect to \( x \): This captures how \( z \) changes as \( x \) changes, while \( y \) is constant. It is calculated as \( f_x = \frac{\partial}{\partial x}(x^2 - 2xy + y^2) = 2x - 2y \).
• Partial derivative with respect to \( y \): This shows how \( z \) changes as \( y \) changes, keeping \( x \) constant. It is calculated as \( f_y = \frac{\partial}{\partial y}(x^2 - 2xy + y^2) = -2x + 2y \).

These partial derivatives are plugged into the tangent plane equation to analyze the behavior of the surface at a given point.

Surface Calculus

Surface calculus provides a framework for understanding the geometry of surfaces like the one given by \( z = x^2 - 2xy + y^2 \). Calculus allows us to zoom in on a surface and study its properties at a microscopic level through local linear approximations.

For a surface given by \( z = f(x, y) \), the tangent plane at a point is the best linear approximation near that point. The formula for a tangent plane involves partial derivatives, making it one of the fundamental tools in surface calculus.

Consider the point \( P(1, 2, 1) \) on our surface. By using the partial derivatives \( f_x \) and \( f_y \) at \( P \), we draw a plane that just "kisses" the surface at this point, showing us the direction in which the surface is heading along each axis. The tangent plane thus becomes \( z = -2x + 2y - 3 \), capturing the surface's behavior at \( P \).

Equation Simplification

Simplifying equations is a crucial step in mathematical problem-solving, as it helps to express findings in the most straightforward form. Once you substitute the calculated partial derivatives into the tangent plane equation, you must simplify it.

Initially, the equation may look complex. You begin with something like \( z - 1 = -2(x - 1) + 2(y - 2) \).
Through proper distribution and combining like terms, you gradually transform this expression into the simplified form \( z = -2x + 2y - 3 \).

• Distribute terms: Spread out the coefficients to each term inside the parentheses.
• Combine like terms: Gather similar variables together to streamline the equation.

The result clearly depicts the tangent plane's structure, making it easier to analyze and interpret the plane's role in relation to the surface.