The population in thousands of people of a small town can be modeled by the function P(t) = x 2 - 25x + 630 where x = the number of years since 1960. a. What was the population of the town in 1960? b. In what year did the town reach its minimum population? c. What was the minimum population obtained by the town?
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.
1. The population in thousands of people of a small town can be modeled by the function P(t) = x 2 - 25x + 630 where x = the number of years since 1960.
a. What was the population of the town in 1960?
b. In what year did the town reach its minimum population?
c. What was the minimum population obtained by the town?
d. In what year did the town return to its 1960 population?
e. In what year will the town reach a population of 950 thousand?
2. An artillery shell is fired from the top of a 450-foot ridge with an initial velocity of 860 feet per second. The height of the shell after t seconds is given by the formula:
y=−16?2 +860?+450
a. Find the height of the shell after 6 seconds.
b. What is the maximum height obtained by the shell?
c. When will the shell hit the ground?
3. Given the data table below answer the questions that follow.
| Price per unit | 5 | 10 | 15 | 20 | 25 | 30 |
| weekly profit in thousands | -8256 | -1035 | 485 | 1400 | 820 | -1200 |
a. Find the best fit quadratic model for the data listed above.
b. Predict the weekly profit if the company charges $22 per unit.
c. Predict the price(s) that would cause the company to just break even (to break even means the company would have a profit of $0).
d. According to your model what price would cause the company to have a maximum profit?
e. According to your model what is the company’s predicted maximum profit?
f. The company has decided that as long as they make at least $500 thousand in profit each week they will stay in business. What range of prices will allow them to reach this goal?
4. The table below shows the number, in thousands, of vehicles parked in the central business district of a certain city on a typical Friday as a function of the number of hours after 9 AM.
| hours after 9 AM | Number of vehicles parked in thousands |
| 0 | 6.2 |
| 2 | 7.4 |
| 4 | 7.6 |
| 6 | 6.7 |
| 8 | 4 |
a. Find the best fit quadratic model for the data.
b. Use your model to predict the number of parked cars at 2:30 PM.
c. At what time of day does the city have a maximum number of parked cars?
d. What is the maximum daily amount of parked cars?
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