Problem 174 - Calculus 3

Question

For the following exercises, find the equation for the tangent plane to the surface at the indicated point. (Hint: Solve for \(z\) in terms of \(x\) and \(y . )\) $$z=e^{7 x^{2}+4 y^{2}}, \quad P(0,0,1)$$

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = 1 \).

Step 1: Partial Derivatives

First, we need to find the partial derivatives of the function defining the surface with respect to both variables, x and y. The function given is \( z = e^{7x^2 + 4y^2} \).The partial derivative with respect to \( x \) is:\[ \frac{\partial z}{\partial x} = \frac{\partial}{\partial x} e^{7x^2 + 4y^2} = e^{7x^2 + 4y^2} \cdot 14x \]The partial derivative with respect to \( y \) is:\[ \frac{\partial z}{\partial y} = \frac{\partial}{\partial y} e^{7x^2 + 4y^2} = e^{7x^2 + 4y^2} \cdot 8y \]

Step 2: Evaluate Partial Derivatives at the Point P(0,0,1)

Next, evaluate the partial derivatives at the given point \( P(0, 0, 1) \).Substitute \( x = 0 \) and \( y = 0 \) into the partial derivatives:\[ \frac{\partial z}{\partial x} \bigg|_{(0,0)} = e^{7(0)^2 + 4(0)^2} \cdot 14(0) = 0 \]\[ \frac{\partial z}{\partial y} \bigg|_{(0,0)} = e^{7(0)^2 + 4(0)^2} \cdot 8(0) = 0 \]

Step 3: Form the Equation of the Tangent Plane

Using the point \( P(0, 0, 1) \) and the values from Step 2, the tangent plane equation can be formed.The formula for the tangent plane at point \((x_0, y_0, z_0)\) is:\[ z - z_0 = \frac{\partial z}{\partial x} \bigg|_{(x_0, y_0)} (x - x_0) + \frac{\partial z}{\partial y} \bigg|_{(x_0, y_0)} (y - y_0) \]Substitute \( x_0 = 0, y_0 = 0, z_0 = 1 \):\[ z - 1 = 0 \cdot (x - 0) + 0 \cdot (y - 0) \]Thus, the equation simplifies to \( z = 1 \).

Key Concepts

Partial Derivatives

When you're dealing with multivariable calculus, partial derivatives are a powerful tool. They allow us to see how a function changes as we adjust one variable at a time, while keeping the others constant. Think of it as focusing on the effect of just one component in a more complex system.
For the surface given by the equation \( z = e^{7x^2 + 4y^2} \), we want to calculate the partial derivatives with respect to \( x \) and \( y \).
Let's break this down:

• For \( x \), the partial derivative, \( \frac{\partial z}{\partial x} \), involves the chain rule, as the exponent of \( e \) depends on \( x \). Calculating this derivative, we get:\[ \frac{\partial z}{\partial x} = e^{7x^2 + 4y^2} \cdot 14x\]
• Similarly, for \( y \), the partial derivative, \( \frac{\partial z}{\partial y} \), is:\[ \frac{\partial z}{\partial y} = e^{7x^2 + 4y^2} \cdot 8y\]

These expressions tell us the slope of the surface in the direction of the \( x \)-axis and \( y \)-axis, respectively.

Surface Equation

The surface equation is a mathematical expression representing the shape or form that we’re investigating. In this problem, the surface is given by \( z = e^{7x^2 + 4y^2} \).
This is an exponential function that depends on both \( x \) and \( y \). It forms a curved shape, which can often be visualized as a hill or a valley.
When we talk about the surface in this context, we're interested in how it changes from one point to another.

• The exponential nature of this surface implies that values of \( z \) can change rapidly with small changes in \( x \) and \( y \).
• This surface isn't flat - its degree of incline and curvature vary depending on the location in the \( xy \)-plane.

Understanding the surface equation helps us determine the behavior of \( z \) as \( x \) and \( y \) are manipulated. This forms the basis for computing derivatives and eventually finding the tangent plane.

Evaluate at a Point

Evaluating at a given point means we're calculating the numerical value of expressions like derivatives at specific coordinates. This gives us a "snapshot" of what's happening at that exact location.
In our context, the point \( P(0, 0, 1) \) provides the base of our tangent plane.
The process involves:

• Substituting \( x = 0 \) and \( y = 0 \) into our partial derivatives. Here, both result in zero:\[ \frac{\partial z}{\partial x} \bigg|_{(0,0)} = 0, \quad \frac{\partial z}{\partial y} \bigg|_{(0,0)} = 0\]
• These zero values tell us that the slope of the tangent plane at point \( P \) is zero in both the \( x \) and \( y \) directions, which means the surface is flat there.
• We can then use these values and the given point to write the tangent plane equation as:\[z = 1\]

This equation tells us the tangent plane is a horizontal plane resting at \( z = 1 \). Understanding how to evaluate at a point is crucial for graphically and numerically analyzing 3D surfaces.