Problem 214 - Calculus 3

Question

Find the equation of the tangent plane to the surface \(z=f(x, y)=\sin \left(x+y^{2}\right)\) at point \(\left(\frac{\pi}{4}, 0,0\right),\) and graph the surface and the tangent plane.

Step-by-Step Solution

Answer

The tangent plane at \( \left(\frac{\pi}{4}, 0, 0\right) \) is \( z = \frac{\sqrt{2}}{2}x - \frac{\sqrt{2}\pi}{8} \).

Step 1: Identify the function and point

The given surface is described by the function \( z = f(x, y) = \sin(x + y^2) \), and we are required to find the equation of the tangent plane at the point \( \left( \frac{\pi}{4}, 0, 0 \right) \). We need to evaluate the function and its partial derivatives at this point.

Step 2: Calculate partial derivatives

Find the partial derivatives of the function with respect to \( x \) and \( y \). The partial derivative with respect to \( x \) is \( f_x = \cos(x + y^2) \). The partial derivative with respect to \( y \) is \( f_y = 2y \cos(x + y^2) \).

Step 3: Evaluate partial derivatives at the given point

Evaluate \( f_x \) and \( f_y \) at \( \left( \frac{\pi}{4}, 0 \right) \). For \( f_x \left( \frac{\pi}{4}, 0 \right) = \cos\left(\frac{\pi}{4} + 0^2\right) = \cos\left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \). For \( f_y \left( \frac{\pi}{4}, 0 \right) = 2(0) \cos\left( \frac{\pi}{4} + 0^2 \right) = 0 \).

Step 4: Construct the tangent plane equation

The general formula for the tangent plane to a surface \( z = f(x, y) \) at a point \( (x_0, y_0, z_0) \) is \( z = z_0 + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0) \). Substituting the values calculated, we get \( z = 0 + \frac{\sqrt{2}}{2}\left(x - \frac{\pi}{4}\right) \), thus the tangent plane equation is \( z = \frac{\sqrt{2}}{2}x - \frac{\sqrt{2}\pi}{8} \).

Step 5: Graph the surface and tangent plane

Plot the function \( z = \sin(x + y^2) \) and the tangent plane \( z = \frac{\sqrt{2}}{2}x - \frac{\sqrt{2}\pi}{8} \) using graphing software. Ensure that the surface is plotted for a range of \( x \) and \( y \) values around the given point, and the plane is depicted so that it intersects the surface at \( \left(\frac{\pi}{4}, 0, 0\right) \).

Key Concepts

Partial Derivatives

When analyzing surfaces or functions of two variables, partial derivatives help in understanding how the function changes with respect to each variable independently. In our exercise, we have a function defined as \( z = f(x, y) = \sin(x + y^2) \). To find the equation of the tangent plane, we first need to determine the partial derivatives of the function with respect to its variables, \( x \) and \( y \).

• Partial derivative with respect to \( x \): This measures how \( z \) changes as \( x \) changes, keeping \( y \) constant. For our function, \( f_x = \cos(x + y^2) \).
• Partial derivative with respect to \( y \): This measures the change in \( z \) as \( y \) changes, keeping \( x \) constant. Here, \( f_y = 2y \cos(x + y^2) \).

Once you have these derivatives, evaluate them at the specific point given in the exercise to proceed with constructing the tangent plane.

Function Evaluation

After finding the partial derivatives, the next step is to evaluate these derivatives at the specific point provided, \( \left( \frac{\pi}{4}, 0 \right) \). This process involves substituting the given values of \( x \) and \( y \) into the partial derivatives.

• Evaluate \( f_x \): At \( \left( \frac{\pi}{4}, 0 \right) \), \( f_x = \cos( \frac{\pi}{4} + 0^2 ) = \cos( \frac{\pi}{4} ) = \frac{\sqrt{2}}{2} \).
• Evaluate \( f_y \): At \( \left( \frac{\pi}{4}, 0 \right) \), \( f_y = 2(0) \cos( \frac{\pi}{4} + 0^2 ) = 0 \).

These evaluated derivatives, \( f_x \) and \( f_y \), are crucial for determining the slope of the tangent plane at the specified point.

Graphing Surfaces

To visualize mathematical concepts like the tangent plane on a surface, we employ graphing. In this case, our surface is characterized by \( z = \sin(x + y^2) \), and we also need to graph the tangent plane equation. Using graphing software, you can plot both the surface and the tangent plane to see how they interact.

A few tips for graphing:

• Choose a suitable range for \( x \) and \( y \) to capture the behavior of the surface around the point \( \left( \frac{\pi}{4}, 0, 0 \right) \).
• Ensure the graphing software depicts the surface and the tangent plane clearly, showing how the plane touches the surface at the given point.
• Look for intersections to understand where the tangent is applied — it simplifies understanding of the mathematical relationship.

Graphing not only aids in solving the problem but also enhances comprehension by providing a visual representation of the mathematical concepts.

Surface Equation

The equation of the tangent plane to a surface serves to approximate the surface at a specific point. For a function \( z = f(x, y) \), the tangent plane at the point \( (x_0, y_0, z_0) \) is given by:

\[z = z_0 + f_x(x_0, y_0)(x - x_0) + f_y(x_0, y_0)(y - y_0)\]

For our exercise, substituting in the evaluated derivatives and the given point, we find:

• \( z_0 = 0 \)
• \( f_x(x_0, y_0) = \frac{\sqrt{2}}{2} \)
• \( f_y(x_0, y_0) = 0 \)

Thus, the tangent plane equation simplifies to:
\[z = \frac{\sqrt{2}}{2}(x - \frac{\pi}{4}) = \frac{\sqrt{2}}{2}x - \frac{\sqrt{2}\pi}{8} \]

This plane is a linear approximation of the surface at the point specified, providing an insightful way to understand the surface's local behavior.