The population of a town can be modelled by the function P(t)= 20 (4t+3)/(2t+5), where P is the population, in thousands, and t is the time, in years, after the year 2000 (t > 0). Clearly label all cases (a, b, c,…). Answers must be clear to read and show all steps. Use appropriate units for your answers. Determine x-intercept(s) if it exists. Determine y-intercept(s) if it exists. Determine the equations of the horizontal asymptote if it exists.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
The population of a town can be modelled by the function
P(t)= 20 (4t+3)/(2t+5), where P is the population, in thousands, and t is the time, in years, after the year 2000 (t > 0).
Clearly label all cases (a, b, c,…). Answers must be clear to read and show all steps. Use appropriate units for your answers.
- Determine x-intercept(s) if it exists.
- Determine y-intercept(s) if it exists.
- Determine the equations of the horizontal asymptote if it exists.
- Determine the equations of the vertical asymptote(s) if it exists.
- State the domain of the entire function
- State the domain for the real-life situation.
- State the range of the entire function
- State the range for the real-life situation.
- Sketch a graph of P versus t for the entire function in a paper-pencil style. Clearly label the axes, scale on both axes, the asymptote(s), and the intercept(s).
- How is the graph of the entire function different from the graph for the real-life situation?
- Determine increasing interval(s) of the function if it exists.
- Determine decreasing interval(s) of the function if it exists.
- Determine the average rate of change between t = 2 years and t = 5 years.
- Determine the instantaneous rate of change at t = 5 years.
- What is the difference between an average rate of change and an instantaneous rate of change?
- What is the population in the year 2000?
- In what year will the population be 30 000?
- Town planners claim that they need not plan for a population above 40 000. Does the model support this conclusion? Explain.
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