Problem 174 - Calculus Volume 3

Question

Find the equation for the tangent plane to the surface at the indicated point. \(z=e^{7 x^{2}+4 y^{2}}, \quad P(0,0,1)\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We need to find the equation of the tangent plane to the given surface at the point \(P(0,0,1)\). The surface is defined by \(z = e^{7x^2 + 4y^2}\).

Step 2: Find the Partial Derivatives

Compute the partial derivatives \(\frac{\partial z}{\partial x}\) and \(\frac{\partial z}{\partial y}\). For \(z = e^{7x^2 + 4y^2}\), use the chain rule:\[ \frac{\partial z}{\partial x} = \frac{d}{dx}(e^{7x^2 + 4y^2}) = 14xe^{7x^2 + 4y^2} \]\[ \frac{\partial z}{\partial y} = \frac{d}{dy}(e^{7x^2 + 4y^2}) = 8ye^{7x^2 + 4y^2} \]

Step 3: Evaluate Partial Derivatives at P(0,0,1)

Substitute \((x, y) = (0, 0)\) into the partial derivatives:\[ \frac{\partial z}{\partial x}(0, 0) = 14 \cdot 0 \cdot e^{7 \cdot 0^2 + 4 \cdot 0^2} = 0 \]\[ \frac{\partial z}{\partial y}(0, 0) = 8 \cdot 0 \cdot e^{7 \cdot 0^2 + 4 \cdot 0^2} = 0 \]

Step 4: Write the Equation of the Tangent Plane

The equation of the tangent plane at \(P(x_0, y_0, z_0)\) is:\[ z = z_0 + \frac{\partial z}{\partial x}(x_0, y_0)(x - x_0) + \frac{\partial z}{\partial y}(x_0, y_0)(y - y_0) \]Substitute \((x_0, y_0, z_0) = (0, 0, 1)\) and the values of the partial derivatives:\[ z = 1 + 0 \cdot (x - 0) + 0 \cdot (y - 0) \]Thus, the equation is \(z = 1\).

Key Concepts

Partial Derivatives

Partial derivatives are essential in calculus, especially when dealing with functions of multiple variables. In our exercise, we had a surface function, which is a kind of multivariable function. When we take a partial derivative, we focus on how the function changes as we vary one variable while holding the others constant.
To find the partial derivatives of the function defining the given surface, we do:

• Compute the partial derivative of the function with respect to x, treating y as a constant.
• Compute the partial derivative with respect to y, treating x as a constant.

Understanding partial derivatives helps us determine the slope of the surface in the directions parallel to the x and y-axes.

Chain Rule

The chain rule is an invaluable tool in calculus for finding derivatives of composite functions. In the context of our surface equation, the function is composite because it involves the exponential function and a quadratic expression inside the exponent.
Using the chain rule involves:

• Identifying the "outer" and "inner" functions. Here, the outer function is the exponential function, and the inner function is the quadratic expression in the exponent.
• Differentiating the outer function, keeping the inner function unchanged, and then multiplying by the derivative of the inner function.

The chain rule simplifies our task by making the differentiation of complex expressions more manageable.

Surface Equation

The equation of a surface can be thought of as a formula that gives us the z-coordinate of points on the surface based on their x and y-coordinates. In the current problem, the surface equation is given as:
\[ z = e^{7x^2 + 4y^2} \]
This equation describes a surface where z changes exponentially with changes in x and y. Each point (x, y, z) on this surface satisfies this equation. The nature of this surface is such that it rises quickly as x or y move away from the origin.

Calculus Problem Solving

Calculus problem solving often involves a systematic approach. For example, when dealing with problems involving tangent planes:

• Understand and rewrite the problem if necessary. Know the formulas and their roles.
• Use appropriate calculus tools, such as partial derivatives and the chain rule, to tackle varying parts of the problem.
• Evaluate essential derivatives at given points to simplify the solution.
• Apply these evaluated results to calculate the desired outcomes like equations of tangent planes.

Combining these steps leads to a precise solution, as we saw when finding the tangent plane equation at a specific point on a surface.