Problem 7 - Calculus for Biology and Medicine

Question

The tangent plane at the indicated poini \(\left(x_{0}, y_{0}, z_{0}\right)\) exists. Find its equation. \(f(x, y)=e^{2 x^{2}+y^{2}} ;\left(1,0, e^{2}\right)\)

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = 4e^2x - 3e^2 \).

Step 1: Find Partial Derivatives

To find the equation of the tangent plane, we first need to compute the partial derivatives of the function. The partial derivatives are:1. With respect to \(x\): \[ f_x(x, y) = \frac{d}{dx} \left( e^{2x^2 + y^2} \right) = e^{2x^2 + y^2} \cdot \frac{d}{dx}(2x^2) = 4xe^{2x^2 + y^2} \]2. With respect to \(y\): \[ f_y(x, y) = \frac{d}{dy} \left( e^{2x^2 + y^2} \right) = e^{2x^2 + y^2} \cdot \frac{d}{dy}(y^2) = 2ye^{2x^2 + y^2} \]

Step 2: Evaluate the Partial Derivatives at the Given Point

Next, we evaluate the partial derivatives at the point \((x_0, y_0) = (1, 0)\):1. \( f_x(1, 0) = 4 \cdot 1 \cdot e^{2 \cdot 1^2 + 0^2} = 4e^{2} \)2. \( f_y(1, 0) = 2 \cdot 0 \cdot e^{2 \cdot 1^2 + 0^2} = 0 \)

Step 3: Use the Tangent Plane Formula

The tangent plane to the surface \(z = f(x, y)\) at the point \((x_0, y_0, z_0)\) is given by:\[ z - z_0 = f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \].Substituting the computed values and the coordinates of the given point \((1, 0, e^2)\), we get:\[ z - e^2 = 4e^2(x - 1) + 0(y - 0) \].

Step 4: Simplify the Tangent Plane Equation

Simplifying the equation, we find:\[ z - e^2 = 4e^2(x - 1) \]Expanding this, we obtain:\[ z = 4e^2x - 4e^2 + e^2 \]\[ z = 4e^2x - 3e^2 \]

Key Concepts

Partial Derivatives

Partial derivatives are an extension of the derivative concept into functions with more than one variable. They allow us to see how the function changes, just like regular derivatives, but they focus on one variable at a time while holding others constant.
For a function like \( f(x, y) = e^{2x^2 + y^2} \), we compute the partial derivatives with respect to \( x \) and \( y \).
Here's how we do it:

• To find the partial derivative with respect to \( x \), treat \( y \) as a constant and differentiate \( f(x, y) \) as though it's a one-variable function of \( x \).
• To find the partial derivative with respect to \( y \), treat \( x \) as a constant and differentiate accordingly.

These partial derivatives give us the slopes or rates of change of the function in the direction of each variable, which are critical in establishing the tangent plane equations in 3D spaces.

Equation of a Plane

In mathematics, specifically in three-dimensional analytic geometry, the equation of a plane is a formula that lets us describe a flat surface in 3D space.

• The general form of the equation of a plane is \( Ax + By + Cz + D = 0 \).
• In the context of tangent planes, the equation often has a specific form \( z - z_0 = A(x - x_0) + B(y - y_0) \).

This equation can be derived using the partial derivatives as shown in the problem solution. The terms \( A \) and \( B \) correspond to partial derivatives \( f_x(x_0, y_0) \) and \( f_y(x_0, y_0) \), while \( (x_0, y_0, z_0) \) is the point of tangency.
Understanding how to construct this equation helps describe tangency and slopes, allowing for better insights into surface behavior at specific points.

Surface in 3D

Visualizing and understanding surfaces in three dimensions is vital in multivariable calculus. A surface is essentially a two-dimensional shape residing in a three-dimensional space.
For a given function such as \( f(x, y) = e^{2x^2 + y^2} \), the surface represents all points \((x, y, z)\) where \( z = f(x, y) \).

• These surfaces can be complicated and curved, unlike flat 2D planes.
• Finding tangent planes to these surfaces at certain points provides local linear approximations that aid in understanding how the surface behaves around that point.

In our example, by calculating the tangent plane at a specific point, we can "flatten" the surface locally, simplifying complex geometry into more manageable linear equations. This process is similar to how a map simplifies the complexity of a three-dimensional Earth onto a flat surface.