3) Imagine a string that is fixed at both ends (for example, a guitar string). When plucked, the string formss a standing wave. The displacement, u(x, t), of the string varies with position r and time t. Suppose (,t)3D2sin(x)sin() for 0 0. At a fixed point in time, the string forms a wave on . Alternatively, if you focus on a point on the string (fix a value of r), that point oscillates up and down in ime. a.) What is the period of the motion in time? За.) b.) Find the rate of change of the displacement with respect to du 3b.) dt time at a constant position (which is the vertical velocity of a point on the string.) 3c.) x = c.) At a fixed time, what point on the string is moving fastest? 3d.) t = d.) At a fixed position on the string, when is the string moving fastest? du Зе.) e.) Find the rate of change of displacement with respect to position at a constant time (which is the slope of the string.) 3f.) x = f.) At a fixed time, where is the slope of the string the greatest?
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.
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