Business Calculus Study Guide: Exponential Growth, Derivatives, and Extrema

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: $A(t) = Pe^{rt}$, 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 $A(t) = 45000e^{0.03t}$.

• Applications:

  • Predicting future population sizes

  • Determining when a population will reach a certain value

  • Calculating doubling time using $t = \frac{\ln 2}{r}$

Uninhibited Population Growth (Differential Equation)

The differential equation for uninhibited population growth is:

• $\frac{dP}{dt} = kP$ or $P'(t) = kP(t)$, where k is the growth rate.

• The solution is $P(t) = ce^{kt}$, 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: $V(t) = Pe^{rt}$, where P is the initial investment, r is the annual interest rate, and t is time in years.

• Example: Maria invests $20,000 at 5.4% annual rate:$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: $\frac{dV}{dt} = rV(t)$

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: $N(t) = N_0e^{-kt}$, where N_0 is the initial amount, k is the decay constant.

• Half-Life: The time required for half the substance to decay: $t_{1/2} = \frac{\ln 2}{k}$.

• 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: $\frac{d}{dx}x^n = nx^{n-1}$

• Exponential Rule: $\frac{d}{dx}a^x = a^x \ln a$

• Logarithmic Rule: $\frac{d}{dx}\ln x = \frac{1}{x}$

• Example: $f(x) = \log_2(3x-1)$, $y = 4^x \cdot \log_5 x$

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 $f''(c) > 0$, $f$ has a relative minimum at $c$.

  • If $f''(c) < 0$, $f$ has a relative maximum at $c$.

  • If $f''(c) = 0$, the test is inconclusive.

• Point of Inflection: Where $f''(x)$ changes sign.

• Concavity:

  • $f''(x) > 0$: function is concave up.

  • $f''(x) < 0$: function is concave down.

• Example: For $h(x) = x^3 - 6x^2 - 15x + 1$, 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 $f'(x)$ and solve $f'(x) = 0$ for critical points.

2. Evaluate $f(x)$ 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 $f''(c) < 0$, $f$ has an absolute maximum at $c$; if $f''(c) > 0$, $f$ has an absolute minimum at $c$.

Asymptotes of Functions

Vertical Asymptotes

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

• Definition: For $f(x) = \frac{x+2}{x^2+6x+8}$, vertical asymptotes occur where the denominator is zero.

• Find: Set $x^2+6x+8=0$ and solve for $x$.

Horizontal Asymptotes

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

• Definition: The line $y = b$ is a horizontal asymptote if $\lim_{x \to \infty} f(x) = b$ or $\lim_{x \to -\infty} f(x) = b$.

• Example: For $f(x) = \frac{5x^4 + 2x^3 + x}{x^3 + x^4 + 8}$, 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: $P = \frac{A}{e^{rt}}$, where A is the future amount, r is the rate, and t is time.

• Example: To grow $P$ to $40,000$ in 20 years at 3.5% interest, solve for $P$.

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

Case	f'(c)	f''(c)	Conclusion
Relative Maximum	0	< 0	f has a relative maximum at c
Relative Minimum	0	> 0	f has a relative minimum at c
No Relative Max/Min	0	0	Test is inconclusive

Key Definitions

• Critical Point: Where $f'(x) = 0$ or $f'(x)$ does not exist.

• Point of Inflection: Where $f''(x)$ changes sign.

• Vertical Asymptote: Value of $x$ where $f(x)$ approaches infinity.

• Horizontal Asymptote: Value of $y$ that $f(x)$ approaches as $x$ 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.