Let B, C and D be fixed points and let A be a moveable point such that line segment AB is parallel to line segment CD. Let a be the distance between C and D and let h be the distance between AB and CD. Find x, the distance between A and B, such that the sum of the areas of A ABE and A CDE is minimized. You may move the slider in the figure below to see the effect of moving point A. (a) Let y equal the altitude of A ABE, measured from E to line segment AB. Write the function A(x) that gives the sum of the areas of A ABE and A CDE. Make sure your function is written in terms of a , h , , and not y . A(x) (b) What is the open interval on which A(x) is defined? (c) A’ (x) = ш
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.
Let B, C and D be fixed points and let A be a moveable point such that line segment AB is parallel to line segment CD. Let a be the distance between C and D and let h be the distance between AB and CD. Find x, the distance between A and B, such that the sum of the areas of Δ ABE and Δ CDE is minimized.
(a) Let y equal the altitude of Δ ABE, measured from E to line segment AB. Write the function A(x) that gives the sum of the areas of Δ ABE and Δ CDE. Make sure your function is written in terms of a , h , x , and not y .
(b) What is the open interval on which A(x) is defined?
(c) A'(x)=?
Trending now
This is a popular solution!
Step by step
Solved in 4 steps with 8 images
