Graph= A line and several approximate points are in a coordinate plane. The horizontal axis is labeled "t". The vertical axis is labeled "P". The line enters the viewing window in the second quadrant and goes up and right. It crosses the P-axis at approximately P = 16.0. The first point is located at (6, 17.2), and is on the line. The second point is located at (7, 17.5), and is nearly on the line. The third point is located at (8, 17.5), and is on the line. The fourth point is located at (9, 17.8), and is on the line. The fifth point is located at (10, 17.8), and is nearly on the line. The sixth point is located at (11, 18.2), and is nearly on the line. The seventh point is located at (12, 18.4), and is on the line. The eighth point is located at (13, 18.3), and is slightly below the line. The ninth point is located at (14, 18.7), and is on the line. The tenth point is located at (15, 18.4), and is below the line. The eleventh point is located at (16, 19.7), and is above the line. The line exits the viewing window in the first quadrant, after passing near the points (17, 19.2),(18, 19.5), (19, 19.7) 2nd graph= A A line and several approximate points are in a coordinate plane. The horizontal axis is labeled "P". The vertical axis is labeled "t". The line enters the viewing window in the second quadrant and goes up and right. It crosses the t-axis at approximately t = 17.7. The first point is located at (6, 19.4), and is slightly above the line. The second point is located at (7, 19.4), and is on the line. The third point is located at (8, 19.4), and is slightly below the line. The fourth point is located at (9, 19.6), and is slightly below the line. The fifth point is located at (10, 19.9), and is nearly on the line. The sixth point is located at (11, 20.6), and is slightly above on the line. The seventh point is located at (12, 20.9), and is slightly above the line. The eighth point is located at (13, 21.0), and is nearly on the line. The ninth point is located at (14, 20.9), and is nearly on the line. The tenth point is located at (15, 21.3), and is nearly on the line. The eleventh point is located at (16, 21.6), and is on the line. The line exits the viewing window in the first quadrant, after passing near the points (17, 21.8),(18, 22.1), (19, 22.4). 3 graph = A line and several approximate points are in a coordinate plane. The horizontal axis is labeled "t". The vertical axis is labeled "P". The line enters the viewing window in the second quadrant and goes up and right. It crosses the P-axis at approximately P = 16.6. The first point is located at (6, 18.2), and is nearly on the line. The second point is located at (7, 18.4), and is nearly on the line. The third point is located at (8, 18.5), and is on the line. The fourth point is located at (9, 18.7), and is nearly on the line. The fifth point is located at (10, 18.8), and is nearly on the line. The sixth point is located at (11, 19.1), and is nearly on the line. The seventh point is located at (12, 19.4), and is nearly on the line. The eighth point is located at (13, 19.6), and is nearly on the line. The ninth point is located at (14, 19.9), and is on the line. The tenth point is located at (15, 20.3), and is nearly on the line. The eleventh point is located at (16, 20.6), and is nearly on the line. The line exits the viewing window in the first quadrant, after passing near the points (17, 20.7),(18, 20.9), (19, 21.1). 4 graph A line and several approximate points are in a coordinate plane. The horizontal axis is labeled "P". The vertical axis is labeled "t". The line enters the viewing window in the second quadrant and goes up and right. It crosses the t-axis at approximately t = 14.9. The first point is located at (6, 16.6), and is slightly above the line. The second point is located at (7, 16.2), and is slightly below the line. The third point is located at (8, 16.8), and is nearly on the line. The fourth point is located at (9, 16.4), and is below the line. The fifth point is located at (10, 16.4), and is below the line. The sixth point is located at (11, 17.8), and is above on the line. The seventh point is located at (12, 18.0), and is above the line. The eighth point is located at (13, 17.7), and is on the line. The ninth point is located at (14, 17.1), and is below the line. The tenth point is located at (15, 18.8), and is above the line. The eleventh point is located at (16, 18.2), and is nearly on the line. The line exits the viewing window in the first quadrant, after passing near the points (17, 18.6),(18, 18.8), (19, 19.0).
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. Graph=
- A line and several approximate points are in a coordinate plane.
- The horizontal axis is labeled "t".
- The vertical axis is labeled "P".
- The line enters the viewing window in the second quadrant and goes up and right.
- It crosses the P-axis at approximately P = 16.0.
- The first point is located at (6, 17.2), and is on the line.
- The second point is located at (7, 17.5), and is nearly on the line.
- The third point is located at (8, 17.5), and is on the line.
- The fourth point is located at (9, 17.8), and is on the line.
- The fifth point is located at (10, 17.8), and is nearly on the line.
- The sixth point is located at (11, 18.2), and is nearly on the line.
- The seventh point is located at (12, 18.4), and is on the line.
- The eighth point is located at (13, 18.3), and is slightly below the line.
- The ninth point is located at (14, 18.7), and is on the line.
- The tenth point is located at (15, 18.4), and is below the line.
- The eleventh point is located at (16, 19.7), and is above the line.
- The line exits the viewing window in the first quadrant, after passing near the points (17, 19.2),(18, 19.5), (19, 19.7)
2nd graph= A
- A line and several approximate points are in a coordinate plane.
- The horizontal axis is labeled "P".
- The vertical axis is labeled "t".
- The line enters the viewing window in the second quadrant and goes up and right.
- It crosses the t-axis at approximately t = 17.7.
- The first point is located at (6, 19.4), and is slightly above the line.
- The second point is located at (7, 19.4), and is on the line.
- The third point is located at (8, 19.4), and is slightly below the line.
- The fourth point is located at (9, 19.6), and is slightly below the line.
- The fifth point is located at (10, 19.9), and is nearly on the line.
- The sixth point is located at (11, 20.6), and is slightly above on the line.
- The seventh point is located at (12, 20.9), and is slightly above the line.
- The eighth point is located at (13, 21.0), and is nearly on the line.
- The ninth point is located at (14, 20.9), and is nearly on the line.
- The tenth point is located at (15, 21.3), and is nearly on the line.
- The eleventh point is located at (16, 21.6), and is on the line.
- The line exits the viewing window in the first quadrant, after passing near the points (17, 21.8),(18, 22.1), (19, 22.4).
3 graph =
- A line and several approximate points are in a coordinate plane.
- The horizontal axis is labeled "t".
- The vertical axis is labeled "P".
- The line enters the viewing window in the second quadrant and goes up and right.
- It crosses the P-axis at approximately P = 16.6.
- The first point is located at (6, 18.2), and is nearly on the line.
- The second point is located at (7, 18.4), and is nearly on the line.
- The third point is located at (8, 18.5), and is on the line.
- The fourth point is located at (9, 18.7), and is nearly on the line.
- The fifth point is located at (10, 18.8), and is nearly on the line.
- The sixth point is located at (11, 19.1), and is nearly on the line.
- The seventh point is located at (12, 19.4), and is nearly on the line.
- The eighth point is located at (13, 19.6), and is nearly on the line.
- The ninth point is located at (14, 19.9), and is on the line.
- The tenth point is located at (15, 20.3), and is nearly on the line.
- The eleventh point is located at (16, 20.6), and is nearly on the line.
- The line exits the viewing window in the first quadrant, after passing near the points (17, 20.7),(18, 20.9), (19, 21.1).
- 4 graph
- A line and several approximate points are in a coordinate plane.
- The horizontal axis is labeled "P".
- The vertical axis is labeled "t".
- The line enters the viewing window in the second quadrant and goes up and right.
- It crosses the t-axis at approximately t = 14.9.
- The first point is located at (6, 16.6), and is slightly above the line.
- The second point is located at (7, 16.2), and is slightly below the line.
- The third point is located at (8, 16.8), and is nearly on the line.
- The fourth point is located at (9, 16.4), and is below the line.
- The fifth point is located at (10, 16.4), and is below the line.
- The sixth point is located at (11, 17.8), and is above on the line.
- The seventh point is located at (12, 18.0), and is above the line.
- The eighth point is located at (13, 17.7), and is on the line.
- The ninth point is located at (14, 17.1), and is below the line.
- The tenth point is located at (15, 18.8), and is above the line.
- The eleventh point is located at (16, 18.2), and is nearly on the line.
- The line exits the viewing window in the first quadrant, after passing near the points (17, 18.6),(18, 18.8), (19, 19.0).
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