In the previous Problem Set question, we started looking at the position function s (t), the position of an object at time t. Two important physics concepts are the veloocity and the acceleration. If the current position of the object at time t is s (t), then the position at time h later is s (t+ h). The average velocity (speed) during that additional time (s(t+h)-s(t)) h is . If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h → 0, i.e. the derivative s' (t). Use this function in the model below for the velocity function v (t). The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a (t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a (t) = (t) = s'" (t). Problem Set question: A particle moves according to the position function s (t) = elt sin (2t). Enclose arguments of functions in parentheses. For example, sin (2t). (a) Find the velocity function.
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.
A particle moves according to the position function s(t)=e7tsin(2t).
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