Problem 10 - Calculus Early Transcendentals

Question

\(\begin{array}{l}{9-10 \text { Draw the graph of } f \text { and its tangent plane at the given }} \\ {\text { point. (Use your computer algebra system both to compute the }} \\ {\text { partial derivatives and to graph the surface and its tangent plane. }} \\ {\text { Then zoom in until the surface and the tangent plane become }} \\ {\text { indistinguishable. }}\end{array}\) $$ f(x, y)=e^{-x y / 10}(\sqrt{x}+\sqrt{y}+\sqrt{x y}), \quad\left(1,1,3 e^{-01}\right) $$

Step-by-Step Solution

Answer

Graph the surface and tangent plane at \((1,1,3e^{-0.1})\); use CAS for derivatives and graphing.

Step 1: Evaluate the Function at the Given Point

Begin by plugging the values of \(x = 1\) and \(y = 1\) into the function \(f(x, y) = e^{-xy/10}(\sqrt{x} + \sqrt{y} + \sqrt{xy})\).For \(x = 1, y = 1:\) \[ f(1, 1) = e^{-1 imes 1/10}(\sqrt{1} + \sqrt{1} + \sqrt{1 imes 1}) = e^{-0.1}(1 + 1 + 1) = 3e^{-0.1}. \]The function value at the point \((1,1)\) is \(3e^{-0.1}\), which matches the point given.

Step 2: Calculate Partial Derivatives

Find the partial derivatives \(f_x\) and \(f_y\) at the point \(x=1, y=1\). First, compute the partial derivative with respect to \(x\): \[ f_x(x, y) = \frac{\partial}{\partial x}\left(e^{-xy/10}(\sqrt{x} + \sqrt{y} + \sqrt{xy})\right). \]Evaluate \(f_x(1, 1)\) using a computer algebra system (CAS).Second, compute the partial derivative with respect to \(y\):\[ f_y(x, y) = \frac{\partial}{\partial y}\left(e^{-xy/10}(\sqrt{x} + \sqrt{y} + \sqrt{xy})\right). \]Evaluate \(f_y(1, 1)\) using CAS.

Step 3: Equation of the Tangent Plane

The equation for the tangent plane at a point \((a, b)\) is:\[ z = f(a, b) + f_x(a, b)(x-a) + f_y(a, b)(y-b). \]Using the values from Step 1 and Step 2, plug in \(f(a,b) = 3e^{-0.1}\), \(f_x(1, 1)\), and \(f_y(1, 1)\) to write the tangent plane equation.

Step 4: Graph Using a Computer Algebra System

Utilize a computer algebra system to generate a 3D graph of the function \(f(x, y)\) and its tangent plane at the point \((1, 1, 3e^{-0.1})\). Display both the surface and its tangent plane on the same graph.

Step 5: Zoom In on the Graph

Gradually zoom in on the graph around the point \((1, 1, 3e^{-0.1})\) until the surface \(f(x, y)\) and the tangent plane appear indistinguishable. This visualization highlights the accuracy of the tangent plane in approximating the surface locally.

Key Concepts

Partial Derivatives

A partial derivative represents the rate of change of a function with respect to one of its variables, while keeping the other variables constant. When dealing with a multivariable function like \( f(x, y) \), it's essential to understand how changes in one variable impact the function's behavior.

• For the function \( f(x, y) = e^{-xy/10}(\sqrt{x} + \sqrt{y} + \sqrt{xy}) \), we calculate the partial derivative with respect to \( x \) denoted as \( f_x \), to observe how the function changes as \( x \) varies, while \( y \) remains fixed.
• Similarly, the partial derivative with respect to \( y \), denoted as \( f_y \), helps us understand changes in the function as \( y \) varies, with \( x \) held constant.

At the point \((1, 1)\), these derivatives help us determine the slope of the tangent plane in the \( x \) and \( y \) directions respectively. They are crucial for finding the tangent plane equation because they provide the necessary slopes in each direction.

3D Graph

Graphs of functions of two variables, such as \( f(x, y) \), are best visualized in three dimensions. A 3D graph showcases the relationship between the variables \( x \) and \( y \) and the resulting function value \( f(x, y) \). This visualization helps in better understanding the topography of the surface.

• The main challenge of interpreting a 3D graph is the ability to see how the surface behaves at different points. It showcases peaks, valleys, and flat regions, each representing different function values.
• By graphing both the surface of \( f(x, y) \) and its tangent plane, one can see how well the tangent plane approximates the curve at a specific point.

The goal is to zoom into the graph at points like \( (1, 1, 3e^{-0.1}) \), until the surface and its tangent plane look the same. This close-up view demonstrates how accurate the tangent plane approximation is at that point.

Computer Algebra System

A Computer Algebra System (CAS) is a powerful tool used to perform symbolic mathematical calculations. It simplifies the process of solving complex mathematical problems, like finding derivatives and plotting functions, which can be cumbersome to do manually.

• Using a CAS to compute partial derivatives allows for precise analytical results without manual calculation errors.
• In graphing, a CAS takes care of rendering the 3D visuals of both the function and its tangent plane. It's advantageous because it can scale the graph seamlessly as you zoom into or out of specific areas.

Overall, a CAS streamlines the process of mathematical analysis, making it an essential component for anyone studying multivariable calculus. Its role is to aid conceptual understanding by providing quick computations and facilitating the visualization of complex mathematical ideas.