The graph to the right gives the percent of persons 25 years and over who have completed four or more years of college. a. Use the method of least squares to obtain the straight line that best fits these data. b. Estimate the percent for the year 2008. c. If the trend continues, when will the percent reach 32?
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.
![**Title: Analysis of College Graduation Trends**
**Introduction:**
The graph depicts the percentage of individuals aged 25 years and older who have completed four or more years of college education across different years.
**Graph Description:**
The bar graph titled "Percentage of College Graduates" presents data for the following years:
- 1985: 19.3%
- 1990: 21.1%
- 1995: 23.3%
- 2000: 25.6%
- 2005: 27.9%
- 2010: 29.6%
**Analysis Instructions:**
a. **Least Squares Method:**
Use this method to determine the equation of the straight line that best fits the data.
- Adjust the year values by counting from 1985.
b. **Estimate for 2008:**
Predict the percentage for the year 2008 using the derived equation.
c. **Future Trend Prediction:**
Determine when the percentage will reach 32% if the trend continues.
**Calculation:**
The least-squares line is given by the equation:
\[ y = \text{[ ]}x + \text{[ ]} \]
(Note: Round the coefficients to the nearest hundredth as needed.)
**Conclusion:**
This analysis enables understanding of the trend in college graduation rates and future projections based on historical data.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F96d5da41-2702-41b3-80a0-15d7da9fac99%2Ffe4107f3-8df5-47f1-8c27-a71fae41c143%2F3j67qcn_processed.png&w=3840&q=75)
Trending now
This is a popular solution!
Step by step
Solved in 3 steps
