Problem 8 - Calculus for Biology and Medicine

Question

Evaluate each function at the given point. (a) \(f_{1}(x, y)=\frac{x}{y}\) at \((3,2)\) (b) \(f_{2}(y, x)=\frac{x}{y}\) at \((3,2)\) (c) \(f_{3}(y, x)=\frac{y}{x}\) at \((3,2)\)

Step-by-Step Solution

Answer

f1 at (3, 2) is 3/2; f2 at (3, 2) is 2/3; f3 at (3, 2) is 3/2.

Step 1: Evaluate function f1

The function given is \( f_1(x, y) = \frac{x}{y} \). We need to evaluate it at the point \((3, 2)\), where \(x = 3\) and \(y = 2\). Substituting these values into the function gives us: \[ f_1(3, 2) = \frac{3}{2} \] Thus, the value of the function \( f_1 \) at \((3, 2)\) is \( \frac{3}{2} \).

Step 2: Evaluate function f2

The function given is \( f_2(y, x) = \frac{x}{y} \). This function expects \( y \) as the first variable and \( x \) as the second. At the point \((3, 2)\), we interpret it as \( y = 3 \) and \( x = 2 \), so we substitute these values into the function: \[ f_2(3, 2) = \frac{2}{3} \] Thus, the value of the function \( f_2 \) at \((3, 2)\) is \( \frac{2}{3} \).

Step 3: Evaluate function f3

The function given is \( f_3(y, x) = \frac{y}{x} \). We need to evaluate it at the point \((3, 2)\), where \( y = 3 \) and \( x = 2 \). Substituting these values gives us: \[ f_3(3, 2) = \frac{3}{2} \] Therefore, the value of the function \( f_3 \) at \((3, 2)\) is \( \frac{3}{2} \).

Key Concepts

Function Evaluation

Function evaluation is the process of computing the output of a function given specific input values. This involves substituting values into the function's formula and simplifying the result. Let's consider our first function, where we have a function of two variables, \(f_1(x, y) = \frac{x}{y}\). To find the value of this function at a given point, say \((3, 2)\), we simply plug in \(x = 3\) and \(y = 2\) into the function. This gives us:

• For \(f_1(3, 2)\): Substitute \(x = 3\) and \(y = 2\) into the formula.
• Compute \(\frac{3}{2}\).

The result of this computation is \(\frac{3}{2}\), which is the value of the function at the given point. Function evaluation is crucial in various fields, helping us understand how a function behaves at specific locations.

Variables Substitution

Variable substitution is a method used in mathematics to replace variables with specific values in order to simplify or solve an expression. In the context of our exercise, substitution plays a vital role. Each function we evaluated required this action. We have varied setups with functions like \(f_2(y, x) = \frac{x}{y}\), where the order of variables is different.It's important to identify which number corresponds to which variable based on the function's parameter order:

• In \(f_2(y, x)\) at point \((3, 2)\), \(y = 3\) and \(x = 2\).
• Swap the typical conventions to meet the function's expectations.

As substitution brings exact numerical values into abstract expressions, it transforms general expressions to their specific instance, enabling precise evaluations.

Ratio of Variables

The ratio of variables refers to the division of two variables, often used in functions to depict relationships or proportions. In our functions such as \(f_3(y, x) = \frac{y}{x}\), this concept is fundamental in understanding the output.When we take the function \(f_3(y, x) = \frac{y}{x}\) and evaluate it at \((3, 2)\):

• Recognize that \(3\) is the value of \(y\) and \(2\) is the value of \(x\)
• The ratio becomes \(\frac{3}{2}\)

Ratios can express how one quantity compares to another, providing a tangible outcome for calculations based on variable divisions. By practicing these computations, the power of understanding relationships through ratios becomes clear, fostering deeper insight into functions made of variable interactions.