City urbanization City planners model the size of their city usingthe function A(t) = - 1/50t2 + 2t + 20, for 0 < t < 50, where Ais measured in square miles and t is the number of years after 2010.a. Compute A′(t). What units are associated with this derivativeand what does the derivative measure?b. How fast will the city be growing when it reaches a size of38 mi2?c. Suppose the population density of the city remains constantfrom year to year at 1000 people>mi2. Determine the growthrate of the population in 2030
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.
City urbanization City planners model the size of their city using
the function A(t) = - 1/50t2 + 2t + 20, for 0 < t < 50, where A
is measured in square miles and t is the number of years after 2010.
a. Compute A′(t). What units are associated with this derivative
and what does the derivative measure?
b. How fast will the city be growing when it reaches a size of
38 mi2?
c. Suppose the population density of the city remains constant
from year to year at 1000 people>mi2
. Determine the growth
rate of the population in 2030
Trending now
This is a popular solution!
Step by step
Solved in 4 steps
