Problem 1 - Calculus for the Life Sciences: A Modeling Approach Volume II

Question

Draw three dimensional graphs of a. $$ F(x, y)=2 \quad \text { b. } \quad F(x, y)=x $$ c. $$ F(x, y)=x^{2} $$ $$ \text { d. } \quad F(x, y)=(x+y) / 2 $$ e. \(F(x, y)=0.2 x+0.3 y \quad\) f. \(\quad F(x, y)=\left(x^{2}+y^{2}\right) / 4\) $$ F(x, y)=0.5 x e^{-y} $$ g. h. \(\quad F(x, y)=\sin y\) i. \(\quad F(x, y)=0.5 x+\sin y \quad\) j. \(\quad F(x, y)=x \sin y\) k. \(\quad F(x, y)=\sqrt{x^{2}+y^{2}} \quad\) 1. \(\quad F(x, y)=x y\) m. \(\quad F(x, y)=\frac{1}{0.4+x^{2}+y^{2}}\) n. \(\quad F(x, y)=e^{\left(-x^{2}-y^{2}\right)}\) o. \(F(x, y)=|x y| \quad\) p. \(\quad F(x, y)=\sin \left(x^{2}+y^{2}\right)\)

Step-by-Step Solution

Answer

Plot each function to visualize their corresponding 3D shapes based on the described characteristics.

Step 1: Analyzing the Constant Function

Function (a) is given by \( F(x, y) = 2 \). This is a constant function where for any value of \( x \) and \( y \), the output is always 2. In 3D space, this visualizes as a plane parallel to the \( xy \)-plane at \( z = 2 \).

Step 2: Understanding the Linear Function in One Variable

Function (b) is \( F(x, y) = x \). This function represents a plane in 3D space sloping along the \( x \)-axis, where \( z \) takes the values of \( x \) for any level of \( y \).

Step 3: Quadratic Function Over the X-Domain

Function (c) is \( F(x, y) = x^2 \). Here, \( z \) is the square of \( x \), resulting in a parabolic cylinder stretching infinitely along the \( y \)-axis.

Step 4: Averaging Function

Function (d) \( F(x, y) = \frac{x+y}{2} \) results in a plane linearly increasing equally as both \( x \) and \( y \) increase. It's sloped at an angle depending on both \( x \) and \( y \).

Step 5: Linear Combination Function

Function (e) \( F(x, y) = 0.2x + 0.3y \) relates to a plane rising with slopes of 0.2 and 0.3 respectively along the \( x \) and \( y \) axes.

Step 6: Quadratic Average Over Two Axes

Function (f) \( F(x, y) = \frac{x^2 + y^2}{4} \) forms a parabolic surface, symmetrically rising from the origin along both axes.

Step 7: Exponential Decay Function

Function (g) \( F(x, y) = 0.5x e^{-y} \) describes an exponential decay along the \( y \)-axis. As \( y \) increases, \( z \) diminishes rapidly, while being directly proportional to \( x \).

Step 8: Sine Function Over Y

Function (h) \( F(x, y) = \sin y \) generates a wave, parallel to the \( xz \) plane, oscillating only along the \( y \)-axis.

Step 9: Combined Linear and Sine Function

Function (i) \( F(x, y) = 0.5x + \sin y \) merges linear changes in the \( x \) direction with oscillations along the \( y \)-axis, creating a wave along \( y \) superimposed on a plane.

Step 10: Product of Sine and Linear Function

Function (j) \( F(x, y) = x \sin y \) yields an oscillatory pattern modulated by \( x \). As \( x \) increases, the amplitude of the sine wave along \( y \) also increases.

Step 11: Euclidean Distance Function

Function (k) \( F(x, y) = \sqrt{x^2 + y^2} \) interprets as a conical surface, expanding outward symmetrically from the origin.

Step 12: Bilinear Form

Function (l) \( F(x, y) = xy \) gives a saddle-shaped surface, increasing in one diagonal and decreasing in the other.

Step 13: Reciprocal Quadric Function

Function (m) \( F(x, y) = \frac{1}{0.4+x^2+y^2} \) visualizes as a circular bump peaking at the origin, diminishing as \( x \) and \( y \) move away from zero.

Step 14: Gaussian Decay

Function (n) \( F(x, y) = e^{-\left(x^2+y^2\right)} \) presents a Gaussian curve or bell shape, high at the origin and exponentially flattening outwards.

Step 15: Absolute Product

Function (o) \( F(x, y) = |xy| \) results in a surface that reflects about \( x \) and \( y \) axes due to the absolute value, creating a saddle but without negative values.

Step 16: Sine of Sum of Squares

Function (p) \( F(x, y) = \sin(x^2 + y^2) \) represents a complex surface, oscillating in both axes, leading to ripple-like shapes emanating from the origin.

Key Concepts

Constant Function

Constant functions are a fascinating aspect of mathematics because no matter the input, the output remains the same. In three-dimensional space, imagine this as a flat sheet of paper floating parallel above the xy-plane. For the exercise, the constant function is represented by \( F(x, y) = 2 \), meaning that for all possible values of \( x \) and \( y \), the output value is always 2.
This concept is simple because regardless of how much you vary the \( x \) or \( y \), the height (or z-value) doesn't change. This leads to a perfectly horizontal plane situated at the height of 2 along the z-axis. A constant function teaches us about consistency and invariability in mathematical scenarios, often serving as a foundational case for more complex functions.

Linear Function

Linear functions extend the concept of a straight line into three dimensions by allowing one variable to determine the outcome, creating a visible slope or incline. For example, the function \( F(x, y) = x \) involves only changes along the x-axis. As you move along the y-direction, the height changes directly with x values, while y remains constant.
This results in a plane that looks like a tilted sheet of paper, tilting in the direction of the x-axis but lying flat and straight across the y-axis. The essence of linear functions in 3D helps us understand direct relationships where changes in one dimension affect the outcome proportionally. This clarity provides an introduction to more complex potential variations seen in other mathematical functions.

Quadratic Function

Quadratic functions offer an intriguing glimpse into curves and how variables can dramatically affect outcomes. Specifically, \( F(x, y) = x^2 \) results in a parabolic cylinder. In simple terms, while moving along the x-axis, the curve opens upwards like the arms of a U-shaped parabola.
The y-axis doesn't affect the curve's shape, making it stretch infinitely along it. This means any slice parallel to the yz-plane remains the same curve. Quadratic functions are vital for understanding how squaring a variable can lead to significant changes, helping to illustrate acceleration and curvature in space.

Exponential Decay

Exponential decay describes how values diminish strongly over time or distance, providing essential insights into natural processes like population decrease or radioactive decay. The function \( F(x, y) = 0.5x e^{-y} \) showcases this behavior, where moving along the y-axis causes a rapid decline in the function's value.
Here, x contributes a linear factor, but as y increases, the exponential term \( e^{-y} \) quickly brings the function value close to zero. This means that at large y-values, even significant x-values won't dramatically influence the result. Understanding exponential decay in 3D provides a model for processes that reduce rapidly over time or with distance, making it essential in many scientific and engineering contexts.