Exponential Functions Sample

Problem 1: Definition and Graph of f(x) = 2^x

An exponential function is defined as a mathematical function of the form f(x) = a · b^x, where a is a constant, b is a positive real number (base), and the exponent x is the variable. The concept traces its fundamental properties to mathematical developments such as Napier's Logarithms in 1614 and Euler's subsequent work (Stewart et al., 2015, p. 342).

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For the specific function f(x) = 2^x, the base is 2. The domain is the set of all real numbers, and the range is all positive real numbers (y > 0). The function intersects the y-axis at (0, 1), which is the y-intercept. The x-axis (y = 0) serves as the horizontal asymptote, restricting function values to strictly positive bounds (Larson, 2017, p. 285).

Table of Values

The graph exhibits continuous exponential growth, approaching the x-axis for negative values of x but never intersecting it (Stewart et al., 2015, p. 344).

Problem 2: Continuous Population Growth

The continuous exponential growth of a population can be modeled using the formula A = Pe^{rt}, a concept rooted in Jacob Bernoulli's discovery of the constant e in 1683. In this scenario, the initial population is given as P = 50,000. The continuous growth rate is r = 0.03 (3% per year), and the time period is t = 10 years.

Substituting the given variables into the formula yields:

A = 50,000 * e^(0.03 * 10)
A = 50,000 * e^(0.3)

Evaluating the exponential term, e^{0.3} ≈ 1.3498588. Multiplying this by the initial population gives:

A = 50,000 * 1.3498588
A ≈ 67,492.94

The projected population after 10 years is exactly 67,493 individuals.

Problem 3: Radioactive Decay and Half-life

Radioactive decay is modeled as an exponential decay function. The formula for the remaining amount of a substance based on half-life is N(t) = N_0(1/2)^{t/h}, where N_0 is the initial quantity, t is the elapsed time, and h is the half-life period. This continuous decay model relates closely to logarithmic scaling and half-life measurement applications (Larson, 2017, p. 289).

The given variables are: initial mass N_0 = 100 grams, half-life h = 15 days, and elapsed time t = 45 days. The number of half-lives elapsed is calculated as t/h = 45 / 15 = 3.

Substituting these values into the decay equation:

N(45) = 100 * (1/2)^3
N(45) = 100 * (1/8)
N(45) = 100 / 8
N(45) = 12.5

After 45 days, precisely 12.5 grams of the radioactive isotope will remain.

References

Larson, R. (2017). College Algebra. Cengage Learning.

Stewart, J., Redlin, L., & Watson, S. (2015). Precalculus: Mathematics for Calculus. Cengage Learning.