A toy rocket is shot vertically into the air from a 4-foot-tall launching pad with an initial velocity of 168 feet per second. Suppose the height of the rocket in feet t seconds after being launched can be modeled by the function h(t)=−16t2+v0t+h0, where v0 is the initial velocity of the rocket and h0 is the initial height of the rocket. How long will it take for the rocket to reach its maximum height? What is the maximum height?
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 toy rocket is shot vertically into the air from a
launching pad with an initial velocity of
feet per second. Suppose the height of the rocket in feet t seconds after being launched can be modeled by the function
where
is the initial velocity of the rocket and
is the initial height of the rocket. How long will it take for the rocket to reach its maximum height? What is the maximum height?
Initial height = 4 foot
initial velocity = 168 feet/sec
The height of the rocket in feet t seconds after being launched can be modeled by the function
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