Function Operations Worksheets Grade 12

1. Given the functions f(x) = 2x^3 - 5x and g(x) = 4e^(3x), find the exact value of the derivative h'(0) where h(x) = f(x) * g(x).
2. f(x) = 3x³ - 5x + 7, g(x) = 2x³ + x - 9. Find (f - g)(x) = ?
3. A marine biologist is studying the population growth of an endangered coral species. The population function is P(t) = 2000e^(0.03t) / (1 + 0.5e^(0.03t)), where t is time in years since monitoring began. The conservation effectiveness is modeled by E(p) = ln(p/500), where p is the population size. What is the rate of change of conservation effectiveness with respect to time when t = 20 years?

…and 6 more problems

1. Emma is a water quality analyst studying two pollutants in a river. The concentration of pollutant A (in parts per million) over time t (in years) is modeled by the function A(t) = 5t^2 + 20t, and the concentration of pollutant B is modeled by B(t) = 15t + 10. Emma needs to determine the function that represents the total concentration of both pollutants combined, T(t) = A(t) + B(t). She then plans to use the quotient function R(t) = T(t) / B(t) to analyze the ratio of total pollutants to pollutant B. Find both T(t) and R(t), and simplify R(t) as much as possible.
2. f(x) = 2x² - 7x + 3, g(x) = x² + 2x - 5. Find (f - g)(x) = ?
3. f(x) = 7x³ - 11x² + 4x - 8, g(x) = 3x³ + 9x - 15. Find (f - g)(x)

…and 3 more problems

1. Tane is a wildlife biologist studying the population of two competing bird species on an island. The population of the Kākā (in hundreds) is modeled by the function K(t) = 3t² + 5, and the population of the Tūī (in hundreds) is modeled by T(t) = 7t - 9, where t is the number of years since the study began. Tane needs to determine the combined population function, C(t) = (K + T)(t), to analyze the total bird population. He also needs to find the difference function, D(t) = (K - T)(t), to understand the population advantage of one species over the other. Determine both C(t) and D(t), and calculate the combined population (in hundreds) when t = 5 years.
2. f(x) = 7x³ - 13x, g(x) = 5x² + 9. Find (f + g)(x)
3. A pharmaceutical company is modeling the concentration of a new drug in a patient's bloodstream over time. The concentration function is C(t) = 50te^(-0.2t) mg/L, where t is hours after administration. The therapeutic window for this drug is between 15 mg/L and 35 mg/L. Determine the time interval during which the drug concentration remains within the therapeutic window.

…and 4 more problems