A mathematician stands on a beach with their dog at point A. They throw a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve the ball. Assume C is the point on the edge of the beach closest to the tennis ball. (a) Assume the dog runs at speed r and swims at speed s, where r>s and both are measured in meters per second. Also, assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball. (b) Find the value of y that minimizes the time it takes to retrieve the ball. Note: your answer should contain the variables r, s, and x. (c) If the dog runs at 8m/s and swims at 1m/s, what ratio yx produces the fastest retrieving time? (d) A dog named Elvis who runs at 6.4m/s and swims at 0.910m/s was found to use an average ratio of yx=0.144 to retrieve his ball. Does Elvis appear to know calculus?
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 mathematician stands on a beach with their dog at point A. They throw a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve the ball. Assume C is the point on the edge of the beach closest to the tennis ball.
(a) Assume the dog runs at speed r and swims at speed s, where r>s and both are measured in meters per second. Also, assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball.
(b) Find the value of y that minimizes the time it takes to retrieve the ball. Note: your answer should contain the variables r, s, and x.
(c) If the dog runs at 8m/s and swims at 1m/s, what ratio yx produces the fastest retrieving time?
(d) A dog named Elvis who runs at 6.4m/s and swims at 0.910m/s was found to use an average ratio of yx=0.144 to retrieve his ball. Does Elvis appear to know calculus?
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