Example For this example, use the function f(x)=1/2x+1 and an initial value of 4. Note that with each successive iteration, you can use the previous output as your new input to the function. • f (4)=1/2 • 4+1=3 • f^2(4)= f(3)= 1/2 • 3 +1 =2.5 •f^3(4) =f(2.5)=1/2 • 2.5 +1=2.25 A) What happens to the value of the function as the number of iterations increases? B). Choose an initial value that is less than zero. What happens to the value of the function as the number of iterations increases? C) Come up with a new linear function that has a slope that falls in the range -1
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.
Example
For this example, use the function f(x)=1/2x+1 and an initial value of 4. Note that with each successive iteration, you can use the previous output as your new input to the function.
• f (4)=1/2 • 4+1=3
• f^2(4)= f(3)= 1/2 • 3 +1 =2.5
•f^3(4) =f(2.5)=1/2 • 2.5 +1=2.25
A) What happens to the value of the function as the number of iterations
increases?
B). Choose an initial value that is less than zero. What happens to the value of the function as the number of iterations increases?
C) Come up with a new linear function that has a slope that falls in the range -1<m<0. Choose two different initial values. For this new linear function, what happens to the function’s values after many iterations? Are the function’s values getting close to a particular number in each case?
D) Use the function g(x)=-x+2 with initial values of 4, 2, and 1. What happens after many iterations with all three initial values? How do the results of all three iterations relate to each other?
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