Derivatives of Polynomials
Be sure to show all of your work details for all calculations and explain in detail how the answers for the critical thinking questions were determined.
1.Set up a function that represents the spread of this new computer virus during the first 25 days by choosing a value for A, based on the first letter of your last name, from the table below. (A does not necessarily have to be a whole number.) In your equation, S(t) is the number of machines projected to be infected, and t is the time in days.
|
First Letter of Your Last Name |
Mathematical Model |
Possible Values for A |
|
A–F |
161–180 |
|
|
G–L |
, |
221–240 |
|
M–R |
261–280 |
|
|
S–Z |
321–340 |
State your specific function here:
2.Using the function that you have created in Question 1, find the value of S(15). Show all of your work details.
3.Find S, the derivative of S(t). Show all of your work details, and simplify your answer.
4.Find the value of . Show all of your work details.
5.Interpret the meanings of and in the context of this virus scenario.
6.Generate a graph for S(t). (There are free downloadable programs like Graph 4.4.2 or Mathematics 4.0, online utilities like this site, and many others.) Insert both the function and the graph into your Word document that contains all of your work details and answers. Be sure to label and number the axes appropriately. (Note that some graphing utilities require that the independent variable must be x instead of t.)
7.After how many days is the virus projected to infect the maximum number of machines? What is the projected maximum number of machines to be infected? Show all of your work details for both answers.
8.Functions are increasing on intervals where the derivative is positive, and they are decreasing on intervals where the derivative is negative. Over what intervals is your function increasing and decreasing? What do you think is the reason why the number of infected machines will start to decrease at some point in the domain?
9.Car speed, or velocity, is one example of a derivative, where it is the rate of change in position over a period of time expressed as the following:
This is usually a linear model.
Another function of motion is modeled as a parabola. The following equation represents the height of an object above the ground during a period of time:
h(t) = -16t2 + 112t + 25
where h is the height in feet, and t is time in seconds.
The derivative of this equation will be h’ (t) = -32t + 112.
Discuss the following:
a.Discuss the implication of the time at which the rate of change of the height of the object is equal to 0. What does this point represent?
b.What other examples can you share where a derivative is applied to real-world applications?
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