Problem 49 - Precalculus Mathematics for Calculus

Question

A great many laws of physics and chemistry are expressible as proportionalities. Give at least one example of a function that occurs in the sciences that is not a proportionality.

Step-by-Step Solution

Answer

Radioactive decay, which is exponential, is a non-proportional function in science.

Step 1: Understand Proportional Functions

Proportional functions describe relationships where one quantity varies directly at a constant rate with another. A key characteristic is that the graph of such functions passes through the origin, described as \( y = kx \).

Step 2: Find Examples of Proportional Functions in Science

Examples include Hooke's Law for springs, where force is directly proportional to displacement (\( F = kx \)), or Boyle’s Law, which states that pressure is inversely proportional to volume in gases (\( PV = k \)).

Step 3: Identify Non-proportional Functions

Non-proportional functions do not maintain a constant ratio. They include more complex functions where the relationship is not a simple direct or inverse proportion, like certain polynomial, logarithmic, or exponential relationships.

Step 4: Example from Science: Exponential Growth or Decay

An example of a non-proportional function in science is radioactive decay. This process is modeled by an exponential function \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the quantity remaining after time \( t \), \( N_0 \) is the initial quantity, and \( k \) is the decay constant.

Key Concepts

Proportional Functions

Proportional functions set a fascinating stage in both mathematics and science. They portray scenarios where two variables move in harmony, either increasing or decreasing together.

• The classic equation takes the form \( y = kx \), where \( k \) represents the constant of proportionality.
• This constant signifies how much one variable will change with a unit change in the other.
• A crucial feature of proportional functions is their linear graph that passes through the origin \((0,0)\).

For example, if you double the value of \( x \), \( y \) will also double, maintaining a steady march upwards or downwards along a straight line. This consistency makes them easy to graph and predict, explaining why they are common in foundational scientific contexts.

Exponential Functions

Exponential functions stand out due to their unique growth or decay patterns. Unlike proportional functions, they feature variable rates of change, becoming either extremely steep or gradually flattening over time.

• The general form is \( f(x) = a \cdot b^x \), where \( a \) is the initial value and \( b \) is the base or growth factor.
• In growth, \( b \) is greater than 1, causing \( f(x) \) to swiftly ascend as \( x \) increases.
• During decay, \( 0 < b < 1 \), leading \( f(x) \) to wane, shrinking towards zero.

These qualities make exponential functions well-suited for modeling phenomena like population increases, financial interest, and decay as in radioactive substances.

Radioactive Decay

Radioactive decay beautifully exemplifies a non-proportional function through its characteristic of exponential decrease. As particles in an unstable nucleus release energy, they drop in quantity at a non-linear rate.

• This progress is captured with the formula \( N(t) = N_0 e^{-kt} \), where \( N(t) \) is the remaining substance after time \( t \).
• Here, \( N_0 \) is the starting amount, and \( k \) represents the decay constant, unique to each radioactive material.
• The exponential nature indicates that the decay rate is proportional to the current quantity of substance, not to any fixed ratio.

Radioactive decay is pivotal in nuclear physics and various applications, including radiocarbon dating and medical diagnostics.

Hooke's Law

Hooke's Law introduces a neatly linear relationship between force and displacement in elastic springs. It's a quintessential proportional function observable in everyday mechanics.

• Described by \( F = kx \), this law shows that force \( F \) exerted on a spring is directly proportional to the displacement \( x \) it experiences.
• The constant \( k \) is known as the spring constant, representing the spring's stiffness.
• If you double the displacement, the force required will also double, and so on, maintaining a direct proportionality.

Hooke's Law is foundational in understanding dynamic systems involving elastic materials, ensuring safety and functionality in various engineering applications.

Boyle's Law

Boyle's Law presents an intriguing example of inverse proportionality in the behavior of gases. It balances pressure and volume in a closed system at constant temperature.

• The law can be written as \( PV = k \), where \( P \) signifies pressure and \( V \) the volume.
• This equation illustrates how, when pressure increases, volume decreases, and vice versa.
• The constant \( k \) represents the product of pressure and volume at a given state.

While it's different from direct proportionality, this inverse relationship is equally significant, highlighting core principles of thermodynamics and kinetic theory, essential for fields like chemistry and meteorology.