Heart Rate before and during a Panic Attack 120 Panic 110 Attack 100 90 80 Baseline Relaxation 70 Onset of Panic Attack 60 0 1 2 3 4 5 6 7 8 9 10 11 12 Time (minutes) Heart Rate (beats per minute)
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.
During a diagnostic evaluation, a 33-year-old woman experienced a panic attack a few minutes after she had been asked to relax her whole body. The graph shows the rapid increase in heart rate during the panic attack. Solve,
a. For which time periods during the diagnostic evaluation was the woman’s heart rate increasing?
b. For which time periods during the diagnostic evaluation was the woman’s heart rate decreasing?
c. How many turning points (from increasing to decreasing or from decreasing to increasing) occurred for the woman’s heart rate during the first 12 minutes of the diagnostic evaluation?
d. Suppose that a polynomial function is used to model the data displayed by the graph using (time during the evaluation, heart rate). Use the number of turning points to determine the degree of the polynomial function of best fit.
e. For the model in part (d), should the leading coefficient of the polynomial function be positive or negative? Explain your answer.
f. Use the graph to estimate the woman’s maximum heart rate during the first 12 minutes of the diagnostic evaluation. After how many minutes did this occur?
g. Use the graph to estimate the woman’s minimum heart rate during the first 12 minutes of the diagnostic evaluation. After how many minutes did this occur?
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