Your task is to estimate how far an object traveled during the time interval 0≤t≤80≤t≤8, but you only have the following data about the velocity of the object. time (sec) 0 1 2 3 4 5 6 7 8 velocity (feet/sec) -2 -1 -2 -4 -1 -2 4 2 1 To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below. (Image attached for left end point approximation)
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.
Your task is to estimate how far an object traveled during the time interval 0≤t≤80≤t≤8, but you only have the following data about the velocity of the object.
| time (sec) | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
| velocity (feet/sec) | -2 | -1 | -2 | -4 | -1 | -2 | 4 | 2 | 1 |
To get an idea of what the velocity function might look like, you pick up a black pen, plot the data points, and connect them by curves. Your sketch looks something like the black curve in the graph below.
(A) Using the left endpoint Riemann sum, find approximately how far the object traveled. Your answers must include the correct units.
Total displacement =
Total distance traveled =
Total displacement =
Total distance traveled=
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