Consider the following scenario. A person is taking a trip across country. They are at a stop light and then turn right to get onto the highway. They merge and then take around 2 minutes to get up to speed. This person will only go 5 miles over the speed limit of 60mph. A. Draw a graph of this scenario. The x-axis should be time in minutes since the driver turned left at the stop light while the y-axis should be the speed of the car. B. Describe your graph in terms of the following: i. increasing/decreasing ii. concave up/concave down C. Considering the scenario above. If we assume that at 10 minutes the driver is going 65mph and at 1minute they were going 20mph, what is the average rate of change? D. Explain what the average rate of change that you found above means in terms of the 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.
Consider the following scenario. A person is taking a trip across country. They are at a stop light and then turn right to get onto the highway. They merge and then take around 2 minutes to get up to speed. This person will only go 5 miles over the speed limit of 60mph.
A. Draw a graph of this scenario. The x-axis should be time in minutes since the driver turned left at the stop light while the y-axis should be the speed of the car.
B. Describe your graph in terms of the following:
i. increasing/decreasing
ii. concave up/concave down
C. Considering the scenario above. If we assume that at 10 minutes the driver is going 65mph and at 1minute they were going 20mph, what is the average rate of change?
D. Explain what the average rate of change that you found above means in terms of the problem.
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