Problem 40 - Calculus for Biology and Medicine

Question

Determine the equation of the level curves \(f(x, y)=c\) and sketch the level curves for the specified values of \(c\). \(f(x, y)=y-x^{2} ; c=0,1,2\)

Step-by-Step Solution

Answer

The level curves are parabolas: \(y = x^2, y = x^2 + 1,\) and \(y = x^2 + 2\).

Step 1: Set up the level curve equation

To find the level curves of the function \(f(x, y) = y - x^2\), we set \(f(x, y) = c\), so the equation becomes \(y - x^2 = c\). Rearrange this equation to get \(y = x^2 + c\). This is the equation of the level curve for a specific value of \(c\).

Step 2: Analyze the equation for different values of c

For different values of \(c\), the equation of the level curve changes. For \(c = 0\), the equation is \(y = x^2\), a parabola opening upwards. For \(c = 1\), the equation is \(y = x^2 + 1\), which is a parabola shifted 1 unit upwards. For \(c = 2\), the equation is \(y = x^2 + 2\), which is a parabola shifted 2 units upwards.

Step 3: Sketch the level curves

To sketch the level curves, plot each parabola on the same graph for the values \(c = 0, 1, 2\). All curves will be parabolas opening upwards, with their vertices at the points (0, 0), (0, 1), and (0, 2) for \(c = 0, 1,\) and \(2\) respectively.

Key Concepts

Parabolas

Parabolas are U-shaped curves that are key features of quadratic functions. In our context, a parabola is represented by the equation \(y = x^2 + c\), where the graph opens upwards. The parameter "c" in the equation affects the vertical position of the parabola's vertex. This means:

• For \(c = 0\), the parabola \(y = x^2\) has its vertex at \( (0, 0) \).
• For \(c = 1\), the parabola shifts up by 1 unit, resulting in \(y = x^2 + 1\) with a vertex at \( (0, 1) \).
• For \(c = 2\), the parabola shifts up by 2 units, giving the equation \(y = x^2 + 2\) with the vertex at \( (0, 2) \).

Parabolas are symmetrical around their vertical axis. The standard form \(y = x^2 + c\) helps in easily identifying this symmetry and accurately sketching these curves.

Equation Rearrangement

Equation rearrangement is a crucial skill in solving mathematical problems and drawing insights from them. In our exercise, we started with the level curve function \(f(x, y) = y - x^2\) and set it equal to a constant \(c\), leading to the equation \(y - x^2 = c\). To isolate \(y\), we rearranged the equation to get \(y = x^2 + c\). This rearrangement allows us to see that the equation represents the set of all points \( (x, y) \) where the function \(f(x, y)\) remains constant, i.e., a level curve. Such transformations are essential:

• They help simplify equations, making them easier to understand and analyze.
• They allow us to express variables in terms of others, providing insights into relationships between variables.

By rearranging equations, we can better visualize the geometry and relationships contained within a problem.

Graph Sketching

Graph sketching is an invaluable tool for visualizing functions and equations. When sketching the level curves \(y = x^2 + c\) for different values of \(c\), you need to understand the basic shape of a parabola, its symmetry, and its apex position.To sketch correctly:

• First, identify the vertex of each parabola. For \(y = x^2 + c\), the vertex is located at \( (0, c) \).
• Next, draw the parabolas opening upwards. All the level curves are vertically translated forms of \(y = x^2\).
• Finally, ensure each graph maintains its symmetry around the y-axis. This symmetry helps ensure the accuracy of your sketch.

Graph sketching brings the math to life, turning abstract equations into tangible visual representations. It's a practical way to solidify your understanding of functions and their characteristics.