The total public debt D (in trillions of dollars) in the United States at the beginning of each year from 2000 through 2008 can be approximated by the model D = 0.032t2 + 0.21t + 5.6, 0 ≤ t ≤ 8 where t represents the year, with t = 0 corresponding to 2000.† (a) Use the model to complete the table to determine when the total public debt reached or surpassed $7 trillion. t D 0 1 2 3 4 5 6 7 8 The public debt reached or surpassed $7 trillion sometime in 2004.The public debt reached or surpassed $7 trillion sometime in 2005. The public debt reached or surpassed $7 trillion sometime in 2006.The public debt reached or surpassed $7 trillion sometime in 2007.The public debt reached or surpassed $7 trillion sometime in 2008. (b) Verify your result from part (a) algebraically and graphically. (Round your answer to two decimal places.) t = (c) Use the model to predict the public debt in 2016. $ trillion
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 total public debt D (in trillions of dollars) in the United States at the beginning of each year from 2000 through 2008 can be approximated by the model
where t represents the year, with t = 0 corresponding to 2000.†
| t | D |
|---|---|
| 0 | |
| 1 | |
| 2 | |
| 3 | |
| 4 | |
| 5 | |
| 6 | |
| 7 | |
| 8 |
(b) Verify your result from part (a) algebraically and graphically. (Round your answer to two decimal places.)
t =
(c) Use the model to predict the public debt in 2016.
$ trillion
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