Use the graph, shown below, of a function y = f(x) to answer the following questions. a) What is the average rate of change of f(x) over the interval [-2,3] ? b) Over what intervals is f(x) increasing? c) Over what intervals is f(x) decreasing? d) At what input value does f(x) have a relative maximum value? e) What is the relative maximum output value that corresponds to your answer to (d)?
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.
. Use the graph, shown below, of a function y = f(x) to answer the following questions.
. Use the graph, shown below, of a function y = f(x) to answer the following questions.
- a) What is the average rate of change of f(x) over the interval [-2,3] ?
- b) Over what intervals is f(x) increasing?
- c) Over what intervals is f(x) decreasing?
- d) At what input value does f(x) have a
relative maximum value?
- e) What is the relative maximum output value that corresponds to your answer to (d)?
- The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. Use the table to answer the following questions. (Don’t forget to state units.)
- a) What is the average rate of change of annual sales from 2000 to 2002?
- b) What is the average rate of change of annual sales from 2001 to 2004?
- c) Over what time period was the annual sales increasing?
- d) Over what time period was the annual sales decreasing?
- e) During what year was the annual sales at a maximum?
- f) What was the maximum annual sales that corresponds to your answer to (e)?
- You work for a social media company. In 2016 your platform had 1.5 million users. In 2019 your platform had 7.5 million users.
On average, by how much did your company’s number of users increase from 2016 to 2019?
- a) What is the average rate of change of f(x) over the interval [-2,3] ?
- b) Over what intervals is f(x) increasing?
- c) Over what intervals is f(x) decreasing?
- d) At what input value does f(x) have a relative maximum value?
- e) What is the relative maximum output value that corresponds to your answer to (d)?
- The table below gives the annual sales (in millions of dollars) of a product from 1998 to 2006. Use the table to answer the following questions. (Don’t forget to state units.)
- a) What is the average rate of change of annual sales from 2000 to 2002?
- b) What is the average rate of change of annual sales from 2001 to 2004?
- c) Over what time period was the annual sales increasing?
- d) Over what time period was the annual sales decreasing?
- e) During what year was the annual sales at a maximum?
- f) What was the maximum annual sales that corresponds to your answer to (e)?
- You work for a social media company. In 2016 your platform had 1.5 million users. In 2019 your platform had 7.5 million users.
On average, by how much did your company’s number of users increase from 2016 to 2019?
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