Suppose the billboard is 80 feet wide and stands 60 feet from the road, perpendicular to the road. We will call the distance you are from the billboard x and the viewing angle θ. 1. Draw a picture of this situation labeling the distances, x, and θ. 2. Make note of the two right triangles in the picture and the of the angles to θ. 3. Write an equation for θ in terms of x. 4. Find the rate of change of θ with respect to x 5. For what x value is the rate of change equal to 0? This is the location with the maximum viewing angle.
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.
Suppose the billboard is 80 feet wide and stands 60 feet from the road,
perpendicular to the road. We will call the distance you are from the
billboard x and the viewing angle θ.
1. Draw a picture of this situation labeling the distances, x, and θ.
2. Make note of the two right triangles in the picture and the of the angles
to θ.
3. Write an equation for θ in terms of x.
4. Find the rate of change of θ with respect to x
5. For what x value is the rate of change equal to 0? This is the location
with the maximum viewing angle.
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