a) Draw a diagram with all relevant constants and variables labelled. b) Write a function expressing the quantity to be minimized/maximized as a function ofoneother variable. Show all the work required to derive this formula. c) State the domain of the function covering all possible configurations for your problem.
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) Draw a diagram with all relevant constants and variables labelled.
b) Write a function expressing the quantity to be minimized/maximized as a function ofoneother variable. Show all the work required to derive this formula.
c) State the domain of the function covering all possible configurations for your problem.
d) Compute the derivative and determine all critical points of the function.
e) Analyze all critical points and endpoints to determine the minimum/maximum.
Tim and Miguel went to a fortune-teller at the state fair who tipped them off that their next calculus exam would have a problem involving ladders sliding down a wall. Cindy suggests they should replicate this in real life to gain an advantage on the rest of the class, so they all decide to bring a long ladder into their dorm. Tim further adds that they should use as long of a ladder as possible for an optimal experience. Cindy points out the problem that they must maneuver the ladder through a right-angle turn where the hallway constricts from 8 feet down to 5 feet wide. What is the longest ladder the trio can carry horizontally around the corner?
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