Exponential Growth and Decay

Population Growth Models

Exponential growth models are used to describe populations or investments that increase at a rate proportional to their current size. The general form of the exponential growth function is:

• Population Function: , where P is the initial population, r is the growth rate, and t is time in years.

• Example: If a city has a population of 45,000 in 2015 and grows at 3% per year, the population after t years is .

• Applications:

  • Predicting future population sizes

  • Determining when a population will reach a certain value

  • Calculating doubling time using

Uninhibited Population Growth (Differential Equation)

The differential equation for uninhibited population growth is:

• or , where k is the growth rate.

• The solution is , where c is the initial population.

• Interpretation: Population grows continuously at a rate proportional to its size.

Compound Interest and Investment Growth

Continuous compounding is modeled by exponential functions similar to population growth.

• Investment Value Function: , where P is the initial investment, r is the annual interest rate, and t is time in years.

• Example: Maria invests V(t) = 20000e^{0.054t}$.

• Applications:

  • Estimating future value of investments

  • Calculating present value needed for a future goal

  • Finding rate of change of balance:

Exponential Decay: Carbon Dating and Half-Life

Exponential decay models describe processes where quantities decrease at a rate proportional to their current value, such as radioactive decay.

• Decay Function: , where N_0 is the initial amount, k is the decay constant.

• Half-Life: The time required for half the substance to decay: .

• Example: Carbon-14 has a half-life of 5730 years. Used to date archaeological samples.

• Application: Determining age of artifacts by measuring remaining percentage of radioactive isotope.

Derivatives and Their Applications

Basic Derivative Rules

Derivatives measure the rate of change of a function. Key rules include:

• Power Rule:

• Exponential Rule:

• Logarithmic Rule:

• Example: ,

Critical Points, Extrema, and Concavity

Critical points occur where the derivative is zero or undefined. These points are candidates for local maxima, minima, or points of inflection.

• Second Derivative Test:

  • If , has a relative minimum at .

  • If , has a relative maximum at .

  • If , the test is inconclusive.

• Point of Inflection: Where changes sign.

• Concavity:

  • : function is concave up.

  • : function is concave down.

• Example: For , find critical values, points of inflection, and intervals of concavity.

Absolute Maximum and Minimum Values

To find absolute extrema on a closed interval:

1. Find and solve for critical points.

2. Evaluate at critical points and endpoints.

3. The largest value is the absolute maximum; the smallest is the absolute minimum.

Theorem 9 (Maximum–Minimum Principle 2): If , has an absolute maximum at ; if , has an absolute minimum at .

Asymptotes of Functions

Vertical Asymptotes

Vertical asymptotes occur where a function approaches infinity as approaches a certain value.

• Definition: For , vertical asymptotes occur where the denominator is zero.

• Find: Set and solve for .

Horizontal Asymptotes

Horizontal asymptotes describe the behavior of a function as approaches infinity.

• Definition: The line is a horizontal asymptote if or .

• Example: For , compare degrees of numerator and denominator to determine asymptotes.

Applications in Business and Science

Present Value and Investment Planning

Present value calculations determine the initial investment required to reach a future goal with continuous compounding.

• Formula: , where A is the future amount, r is the rate, and t is time.

• Example: To grow to in 20 years at 3.5% interest, solve for .

Radioactive Decay in Medicine

Half-life models are used to track decay of substances like iodine-125 in cancer treatment.

• Application: If iodine-125 decreases by 2% in storage, use the decay formula to find storage time.

Summary Table: Second Derivative Test for Relative Extrema

Key Definitions

• Critical Point: Where or does not exist.

• Point of Inflection: Where changes sign.

• Vertical Asymptote: Value of where approaches infinity.

• Horizontal Asymptote: Value of that approaches as goes to infinity.

Additional info: Some context and examples have been expanded for clarity and completeness, including standard formulas and definitions relevant to Business Calculus.