Problem 10 - Applied Calculus

Question

Sketch a contour diagram for the function with at least four labeled contours. Describe in words the contours and how they are spaced. $$f(x, y)=y-x^{2}$$

Step-by-Step Solution

Answer

The contours are upward-opening parabolas shifted vertically by whole units along the y-axis.

Step 1: Understand the Function

The given function is \( f(x, y) = y - x^2 \). This function represents a surface in a 3D space where the height at any point \((x, y)\) is given by \( y - x^2 \). The goal is to sketch its contour diagram, which is a 2D representation of points where the function has constant values.

Step 2: Determine Contour Levels

To sketch a contour diagram, choose several contour levels, which are constant values of \( f(x, y) \). For example, let the contour levels be \( c = -1, 0, 1, 2 \). This means you will set \( f(x, y) = -1 \), \( f(x, y) = 0 \), \( f(x, y) = 1 \), and \( f(x, y) = 2 \), and solve for \( y \).

Step 3: Solve for y in Terms of x for Each Contour Level

For each contour level \( c \):1. \( c = -1: y - x^2 = -1 \Rightarrow y = x^2 - 1 \)2. \( c = 0: y - x^2 = 0 \Rightarrow y = x^2 \)3. \( c = 1: y - x^2 = 1 \Rightarrow y = x^2 + 1 \)4. \( c = 2: y - x^2 = 2 \Rightarrow y = x^2 + 2 \)These are equations of parabolas that open upwards.

Step 4: Sketch the Contours

Plot each equation on a coordinate plane. These are parabolas with vertex at \( (0, -1) \), \( (0, 0) \), \( (0, 1) \), and \( (0, 2) \). The contours will be evenly spaced apart and rise in horizontal direction as the contour value \( c \) increases. Each parabola represents a line of constant \( f(x, y) \).

Step 5: Describe the Contours

The contours are parabolas opening upwards. They are centered along the \( y \)-axis, with each successive parabola shifted vertically upwards by 1 unit compared to the previous one. More specifically:- For \( c = -1 \), the parabola is down by 1 unit.- For \( c = 0 \), the parabola is at the origin.- For \( c = 1 \), the parabola is up by 1 unit.- For \( c = 2 \), the parabola is up by 2 units. This indicates an evenly spaced increase in \( y \) coordinates as \( f(x,y) \) increases.

Key Concepts

3D Surfaces

In mathematics, a 3D surface is a graphical representation of a function that exists in three-dimensional space. For a function like \( f(x, y) = y - x^2 \), this surface maps every point \((x, y)\) to a corresponding height or value \( f(x, y) \). This concept is akin to how a geographical terrain map would represent landscapes, peaks, and valleys. The function \( y - x^2 \) creates a 3D surface which could be imagined as rolling hills.

Understanding a 3D surface is critical as it allows us the freedom to visualize the effect of changing \( x \) and \( y \) on the surface height. This visualization is not just theoretical; it's crucial for applications such as physics and engineering, where function-derived surfaces can inform design and analysis, such as aerodynamics on a car hood or the stresses on a bridge.

The value \( y - x^2 \) signifies that for any additional step in \( y \), the effect of \( -x^2 \) pulls the height downward, creating a unique surface that is quadratic in form along the \( x \)-axis direction.

Parabolas

A parabola is a specific type of curve described by a quadratic equation, often in the form \( y = ax^2 + bx + c \). In our contour diagram equation solutions, we observe equations of the form \( y = x^2 + k \), where \( k \) signifies a vertical shift in the parabola's position.

Each parabola has a vertex, the point where it changes direction, and it opens either upward or downward. Our given contours show parabolas that open upwards, a typical outcome when the coefficient of \( x^2 \) is positive.

• For \( c = -1 \), the vertex is at \( (0, -1) \).
• For \( c = 0 \), the vertex moves to \( (0, 0) \).
• For \( c = 1 \), it shifts upwards to \( (0, 1) \).
• For \( c = 2 \), it further ascends to \( (0, 2) \).

The spacing among these parabolas makes it clear that the vertical shift \( k \) corresponds directly to our contour levels, showing increments of 1 as specified by the constant \( c \).

Contour Levels

Contour levels are a fundamental part of a contour diagram; they represent specific constant values within a function's range. For our function \( f(x, y) = y - x^2 \), contour levels like \( c = -1, 0, 1, 2 \) indicate fixed values of \( f \) which translates graphically into lines or curves that pass through points all sharing this same height.

The contour level \( c \) sets a benchmark:

• If \( c = 0 \), all points on that contour have height zero from the perspective of \( y = x^2 \).
• With \( c = 1 \), those points share a height of 1, courtesy of shift 1 upwards, shown as \( y = x^2 + 1 \).
• Similarly, \( c = 2 \) draws another level uplifting the parabola by 2.

By choosing these levels, one can readily visualize how the function varies and appreciate the uniform spacing of contours, guiding intuition in understanding variations across the surface.

Constant Values

In the context of a contour diagram, constant values allow a 3D surface to be dialogued on a 2D plane. These values, symbolized as contour levels, define where the function \( f(x, y) \) holds a fixed value. Such constants simplify complex surface interactions into manageable flat structures.

Consider \( c = -1, 0, 1, 2 \), each a showcase of constancy in the continuum of change.

• Each chosen constant leads to an equation \( y - x^2 = c \), resulting in unique parabolas for each \( c \).
• These curves clearly depict what stays uniform while emphasizing variances elsewhere, a foundational step for concise mathematical representation.

This makes constant values powerful. They provide solid reference points that can be related back to overarching patterns and behaviors within the function's surface.