Problem 14 - Calculus Early Transcendentals: Pearson New International Edition

Question

Sketch the graph of \(\bar{f}\). $$ f(x, y)=2-x-y^{2} $$

Step-by-Step Solution

Answer

The graph of \( f(x, y) = 2 - x - y^2 \) is a 3D surface with a linear decline in \( x \) and parabolic decline in \( y \).

Step 1: Rewrite the Function in Terms of z

Start by rewriting the given function in the form used to plot 3D surfaces: \[ z = f(x, y) = 2 - x - y^2 \]

Step 2: Understand the Shape of the Surface

The equation \( z = 2 - x - y^2 \) describes a parabolic cylinder. This is because when \( x \) is constant, the equation \( z = 2 - x - y^2 \) represents a parabola in \( y \) and \( z \). Similarly, when \( y \) is constant, the equation \( z = 2 - x \) represents a line in the \( x \) and \( z \) planes.

Step 3: Find and Plot Intersection Curves

Choose specific values for \( y \) and \( x \) to visualize the surface:- For \( y = 0 \), \( z = 2 - x \); this is a straight line that decreases as \( x \) increases.- For \( x = 0 \), \( z = 2 - y^2 \); this is an upside-down parabola opening downwards.

Step 4: Check Particular Values and Limits

Evaluate \( f(x, y) \) at a few specific points to confirm understanding:- \( f(0, 0) = 2 \)- \( f(2, 0) = 0 \)- \( f(0, 1) = 1 \)These examples help identify key points on the graph and the limits of the surface as \( x \) or \( y \) become very large.

Step 5: Sketch the Graph

Given the derived intersection curves and specific evaluated points, sketch the surface. The graph has linear decline in \( x \) and parabolic decline in \( y \). It starts at \( z = 2 \) when both \( x \) and \( y \) are zero.

Key Concepts

Parabolic Cylinder

A parabolic cylinder is a unique geometric shape that can be visualized as an extension of a simple parabola. Imagine taking the familiar U-shape of a 2-dimensional parabola and stretching it along an additional axis, creating a 3D object. In the context of the equation \( z = 2 - x - y^2 \), this shape is evident when either variable is held constant.

• Holding \( x \) constant makes the equation \( z = 2 - x - y^2 \) look like a parabola in the \( y \)-\( z \) plane.
• Conversely, holding \( y \) constant transforms the relationship into a straight line in the \( x \)-\( z \) plane.

This is what gives it the distinctive parabolic cylinder appearance. It's essential to understand this geometric transformation to correctly interpret the equation's implications in three-dimensional space. Recognizing how each change in variable alters the shape will also assist in sketching complex graphs.

Intersection Curves

Intersection curves are crucial for simplifying and understanding the surface's overall form. They are the lines and curves where our surface intersects with other planes. By examining the intersection of our parabolic cylinder with different planes (e.g., \( y = 0 \) or \( x = 0 \)), we can discern portions of the surface and how it behaves.

• For \( y = 0 \): The equation becomes \( z = 2 - x \), a straight downward sloping line in the \( x \)-\( z \) plane.
• For \( x = 0 \): The equation turns into \( z = 2 - y^2 \), an upside-down parabola in the \( y \)-\( z \) plane.

These intersection curves help us to not only visualize but also to verify the nature of the surface across different dimensions. They represent the "slices" of the surface, where evaluating these intersections clarifies the shifts in behavior from linear to parabolic.

Evaluating Points

Evaluating specific points on a 3D surface can make more abstract concepts tangible. In the parabolic cylinder equation \( f(x, y) = 2 - x - y^2 \), computing the function at different \((x, y)\) coordinates highlights crucial points on the graph and aids in understanding its shape.

• At \( (0, 0) \), the function value is \( f(0, 0) = 2 \).
• For \( (2, 0) \), \( f(2, 0) = 0 \), indicating that as \( x \) increases, the value of \( z \) declines.
• At \( (0, 1) \), \( f(0, 1) = 1 \), showing variation in \( z \) with changes in \( y \).

This evaluation strengthens our comprehension of the graph's key features and delineates the boundaries within which the surface varies. Checking particular points as \( x \) or \( y \) grow larger can also predict the behavior of the surface at these extents.

Plotting Surfaces

Plotting surfaces involves bringing together all the pieces - the equation, intersection curves, and evaluated points - to visually represent the surface. With \( f(x, y) = 2 - x - y^2 \), the plot will display the combination of linear and parabolic features.

• Start by sketching the key points derived in the evaluation process, forming a general outline.
• Overlay the intersection curves, emphasizing the line for \( y = 0 \) and the parabola for \( x = 0 \).
• Connect these insights to complete the graph, ensuring the curvature and slopes align with the algebraic expressions.

Plotting in this manner gives a clearer view of abstract mathematical relationships, enabling a deeper understanding of how a 3D object is shaped and behaves. This final visualization confirms your mathematical findings and provides a tangible model to further explore.