Problem 7 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find the equation of the tangent plane to the given surface at the indicated point. $$ z=2 e^{3 y} \cos 2 x ;(\pi / 3,0,-1) $$

Step-by-Step Solution

Answer

The equation of the tangent plane is \( z = 2\sqrt{3}x - 3y - \frac{2\sqrt{3}\pi}{3} - 1 \).

Step 1: Identify the Given Surface and Point

The given surface is described by the equation \( z = 2 e^{3y} \cos 2x \). We need to find the equation of the tangent plane at the point \( (x_0, y_0, z_0) = (\pi/3, 0, -1) \).

Step 2: Find the Partial Derivatives

To find the equation of the tangent plane, we first need the partial derivatives of the function with respect to both \( x \) and \( y \).\[ \frac{\partial z}{\partial x} = -4 e^{3y} \sin 2x \] and \[ \frac{\partial z}{\partial y} = 6 e^{3y} \cos 2x \].

Step 3: Evaluate Partial Derivatives at the Point

Substitute the point \( (x_0, y_0) = (\pi/3, 0) \) into the partial derivatives calculated in Step 2.\[ \frac{\partial z}{\partial x} \bigg|_{(x_0, y_0)} = -4 e^{0} \sin\left(2\cdot\frac{\pi}{3}\right) = -4 (-\frac{\sqrt{3}}{2}) = 2\sqrt{3} \] and \[ \frac{\partial z}{\partial y} \bigg|_{(x_0, y_0)} = 6 e^{0} \cos\left(2\cdot\frac{\pi}{3}\right) = 6 (-\frac{1}{2}) = -3. \]

Step 4: Write the Equation of the Tangent Plane

The general formula for the equation of the tangent plane to a surface \( z = f(x, y) \) at the point \((x_0, y_0, z_0)\) is given by: \[ z - z_0 = \frac{\partial z}{\partial x}(x - x_0) + \frac{\partial z}{\partial y}(y - y_0) \].Substitute \( x_0 = \pi/3 \), \( y_0 = 0 \), \( z_0 = -1 \), \( \frac{\partial z}{\partial x} \bigg|_{(x_0, y_0)} = 2\sqrt{3} \), and \( \frac{\partial z}{\partial y} \bigg|_{(x_0, y_0)} = -3 \).\[ z + 1 = 2\sqrt{3}(x - \frac{\pi}{3}) - 3(y - 0) \].

Step 5: Simplify the Tangent Plane Equation

Simplify \( z + 1 = 2\sqrt{3}(x - \frac{\pi}{3}) - 3y \) to obtain the final form of the tangent plane equation.\[ z = 2\sqrt{3}x - \frac{2\sqrt{3}\pi}{3} - 3y - 1 \].Thus, the equation of the tangent plane is: \[ z = 2\sqrt{3}x - 3y - \frac{2\sqrt{3}\pi}{3} - 1 \].

Key Concepts

Partial Derivatives

Partial derivatives are a fundamental concept in calculus, especially when dealing with functions of multiple variables. When you have a function like \( z = f(x, y) \), which depends on two variables \( x \) and \( y \), a partial derivative measures how \( z \) changes as one of these variables changes while the other is kept constant. This helps us understand how the function behaves in different directions.

To find the partial derivative of \( z \) with respect to \( x \), noted as \( \frac{\partial z}{\partial x} \), you treat \( y \) as a constant and differentiate the function with respect to \( x \). Similarly, the partial derivative with respect to \( y \), \( \frac{\partial z}{\partial y} \), is found by treating \( x \) as a constant and differentiating with respect to \( y \).

• The partial derivative with respect to \( x \) gives the slope of the tangent line to the curve showing how \( z \) increases or decreases as \( x \) changes.
• The partial derivative with respect to \( y \) does the same for changes in \( y \).

These derivatives are crucial for finding the tangent plane to a surface at a given point.

Surface Equation

A surface equation is a mathematical representation that describes a 3D surface in space. For example, \( z = 2 e^{3y} \cos 2x \) defines a surface where the height \( z \) above each \( (x, y) \) point is calculated using this formula. It combines both exponential and trigonometric functions, depicting a surface that varies in complex ways over different values of \( x \) and \( y \).

Studying such equations helps in understanding the topography of surfaces, i.e., their peaks, valleys, slopes, and other features. Each part of this equation affects the surface's shape:

• \( 2 e^{3y} \) suggests a rapid increase in one direction, influenced strongly by \( y \).
• \( \cos 2x \) introduces oscillation in the \( x \) direction, creating waves or peaks and troughs.

This diverse behavior means the surface will have different gradients, which is useful for more complex calculations like finding a tangent plane.

Tangent Plane Equation

A tangent plane is like a flat sheet that just touches a surface at a particular point and gives us a way to approximate the surface near that point. Think of it as a snapshot of the surface's slope and tilt in all directions at that single point. The equation for a tangent plane can be derived using partial derivatives, as they give us the rates of change in different directions.

The general formula for a tangent plane to a surface \( z = f(x, y) \) at a point \((x_0, y_0, z_0)\) is:\[ z - z_0 = \frac{\partial z}{\partial x}(x - x_0) + \frac{\partial z}{\partial y}(y - y_0).\]This equation allows us to determine how changes in \( x \) and \( y \) affect \( z \), giving us an approximation of the surface near our point of interest.

• \( \frac{\partial z}{\partial x} \) shows the surface's slope along the \( x \) direction.
• \( \frac{\partial z}{\partial y} \) shows the slope along the \( y \) direction.

By plugging in specific values (like \( x_0 = \pi/3 \), \( y_0 = 0 \), \( z_0 = -1 \)), we get a linear equation representing our tangent plane.

Calculus

Calculus is a branch of mathematics focused on change. It includes finding how things vary, optimally represented through derivatives and integrals. When dealing with surfaces, calculus allows us to understand how variables interact to shape the surface.

In our scenario, calculus revealed the tangent plane equation which helps approximate a complex surface using straight lines. This simplification is powerful in many real-world applications like physics, engineering, economics, and more.

• Derivatives help us calculate slopes and curves efficiently.
• Partial derivatives extend this to functions with multiple inputs, critical for understanding surfaces.

With these tools, calculus allows us to break down complicated, multidimensional shapes into manageable pieces, making problem-solving significantly easier.