Problem 18 - Advanced Engineering Mathematics

Question

Find an equation of the tangent plane to the graph of the given equation at the indicated point. $$ x y+y z+z x=7 ;(1,-3,-5) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \(8x + 4y + 2z = 30\).

Step 1: Write the Given Equation

The equation we are dealing with is \( x y + y z + z x = 7 \). This implicitly defines a surface in three-dimensional space.

Step 2: Identify the Point

We need to find the tangent plane to the surface at the point \((1, -3, -5)\). This point will be denoted as \((x_0, y_0, z_0) = (1, -3, -5)\).

Step 3: Compute Partial Derivatives

To find the tangent plane, we first compute the partial derivatives of the given equation with respect to \(x\), \(y\), and \(z\). \[ f_x = \frac{\partial}{\partial x}(xy + yz + zx) = y + z \] \[ f_y = \frac{\partial}{\partial y}(xy + yz + zx) = x + z \] \[ f_z = \frac{\partial}{\partial z}(xy + yz + zx) = y + x \]

Step 4: Evaluate Partial Derivatives at Given Point

Substitute \((x_0, y_0, z_0) = (1, -3, -5)\) into the partial derivatives calculated: \[ f_x(1, -3, -5) = -3 - 5 = -8 \] \[ f_y(1, -3, -5) = 1 - 5 = -4 \] \[ f_z(1, -3, -5) = -3 + 1 = -2 \]

Step 5: Write the Equation of the Tangent Plane

The equation of the tangent plane at the point \((x_0, y_0, z_0)\) is given by: \[ f_x(x_0, y_0, z_0)(x-x_0) + f_y(x_0, y_0, z_0)(y-y_0) + f_z(x_0, y_0, z_0)(z-z_0) = 0 \] Substitute the derivatives and the point: \[ -8(x-1) - 4(y+3) - 2(z+5) = 0 \] Simplify to: \[ -8x - 4y - 2z = -30 \] So, the equation of the tangent plane is: \[ 8x + 4y + 2z = 30 \]

Key Concepts

Partial Derivatives

Partial derivatives are like regular derivatives, but they help us understand how functions change with respect to individual variables. Imagine you have a function involving three variables, say \( f(x, y, z) \). Here, each variable plays a different role, so changing one variable might affect the output differently than changing another.
To find the partial derivative with respect to one variable, we imagine the other variables as constants, just like numbers. In our problem, we calculated the partial derivatives for the equation \( xy + yz + zx = 7 \):

• For the derivative with respect to \( x \): We calculated \( f_x = y + z \).


• For the derivative with respect to \( y \): We calculated \( f_y = x + z \).


• For the derivative with respect to \( z \): We found \( f_z = y + x \).

These derivatives show how the function changes as each variable changes, while the others stay fixed.

Surface in Three-Dimensional Space

Surfaces in three-dimensional space can be thought of as shapes that don't fit in our typical 2D view, like a sphere or a saddle. These surfaces are defined by equations that involve three variables. In our problem, the equation \( xy + yz + zx = 7 \) is such a surface. This means that the coordinates \( (x, y, z) \) on this surface satisfy the equation.
Imagine cutting through three layers of an onion to see the patterns. This is like exploring a surface in 3D space. Each slice can look different depending on how you cut it or from what angle you view it. Here, the challenge is to find a flat plane that just "touches" the surface at a specific point—this flat plane is our tangent plane.
Finding this tangent plane at the point \((1, -3, -5)\) provides valuable insights into the nature of the surface around that point, and these were observed by calculating where our surface equation balances out with respect to these coordinates.

Implicit Differentiation

Implicit differentiation is a technique used when dealing with equations where one variable is not isolated. It allows us to find derivatives even if we can't easily solve for one variable in terms of others. In cases like the equation \( xy + yz + zx = 7 \), it might be tricky to express one variable directly as a function of the others, hence implicit differentiation becomes handy.
Rather than struggle to rewrite the function, implicit differentiation lets us find the rates at which variables change by taking derivatives of both sides of the equation with respect to a variable. In our exercise, when we found partial derivatives like \( f_x = y + z \), it was done keeping one variable constant, effectively using implicit differentiation for each part of the equation.
This method is powerful when trying to understand how changes in one variable influence the system when the relations are tied closely together, just as in our surface equation. It helps define those derivatives needed to find the tangent plane without rearranging the entire equation for each variable.