58. Distance Function The graph gives a sales representative's distance from his home as a function of time on a certain day. (a) Determine the time intervals on which his distance from home was increasing and those on which it was decreasing. (b) Describe in words what the graph indicates about his travels on this day. (c) Find the net change in his distance from home between noon and 1:00 P.M. Distance from home (miles) 8 A.M. 10 2 4. 6 PM. NOON Time (hours)
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.
Changing Water Levels The graph shows the depth of water
W in a reservoir over a one-year period as a function of the
number of days x since the beginning of the year.
(a) Determine the intervals on which the function W is
increasing and on which it is decreasing.
(b) At what value of x does W achieve a
A
(c) Find the net change in the depth W from 100 days to
300 days.
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