Instantaneous results! Max Rose
Blog submission number 2
Due Date: February 9, 2015 by 11:59 pm Eastern Standard Time
Point value: (20 points)
Learning objectives:
· Recognize, observe, formulate, and understand mathematical connections that functions and their graphs have in everyday life
· Analyze, think critically about, and demonstrate a meaningful understanding of various mathematical functions and their graphs by effectively communicating their thinking and reasoning for their solutions and mathematical connections to real world situations (in other words, rather than simply knowing how to solve any particular problem, students should be able to explain how they to come to a particular solution and as well as explain why it is correct)
· Apply general strategies to more than one type of graph, function, or equation
· Comfortably and effectively utilize a variety of strategies to solve any given mathematical problem involving functions and their graphs
Background:
This blog submission is an exercise in recognizing real world applications of a mathematical concept, instantaneous rates of change or (the derivative at a point). In your book and in lecture, we saw that if one drives 200 miles in 4 hours, the average velocity is 200/4 or 50 miles per hour. But this does not mean that one travels 50mph exactly for the entire trip. Velocity at a given instant (instantaneous velocity) during the trip is shown on the speedometer.
There are many other real world examples/applications where we see the concept of instantaneous rates of change or (IRC or the derivative at a point). One such example is explained in detail in your text in section 2.1 on page 88 (snowball example).
Directions:
(When creating your problem, be sure to take note of these directions as well as the criteria in the rubric) here is a worked example of what you will do on your own using the instructions below.
Part a:
a. Find a real world application OR design your own experimental application relating to rates of change. (in the blog folder you will find plenty of examples to get you started)
My real world example I will be investigating is the Recorded data from bouncing a ball several times in a given interval of time (time vs. height)
b B: Write a narrative or synopsis explaining your application/experiment and include a question. (for example, what is the velocity of the snowball at exactly 2 seconds? Or how can I find the velocity of the baseball at exactly 3 seconds?)
The Table listed below discusses a bouncing ball at a given height and time. With a height and time I will calculate the velocity of the ball. For example, I want to find the velocity of the ball at zero second with a height of 212. I will do this by plugging the information in the instantaneous rate of change formula.
cC: Create a table of values for the data that you have recorded from your application/experiment.
dD: Graph the points using the data from your table of values (connect the dots).
Handmade Graph of Bouncing Ball (Below Picture)
eE: Calculate the slope (ARC) of at least three secant lines originating from the same point on your graph to three different points on your graph (i.e. maybe you want to know what happens exactly at x = 20, so your points might be (20, 62), (20, 56), (20, 50)). Explain what you notice about the ARC of these secant lines and what the calculations mean/represent in terms of your experiment/application.
Some things I noticed from looking at my calculations is that the ARC decreases as it moves closer to zero and then increases again once through that point.
Calculations between three points on Graph (Below is Graph and Work to get to answers)
fF: Sketch an approximation of a tangent line that passes though the same point (P) from part e to which you connected your secant lines (i.e. you would draw a tangent line through the point 20, since that is the same point that you used to calculate your three different secant lines)
A tangent line that passes though the same point (P) from part e to which you connected your secant lines
gG: Choose a second point (Q) on the tangent line, and calculate the slope of the line (PQ). This calculation will be the instantaneous rate of change ((IRC or derivative at a point)…be sure to identify the units correctly). Explain what this calculation means mathematically and in terms of your experiment/application.
The IRC is 825 cm/sec. This calculation means that the ball is
traveling at a velocity of 825cm per 1 second.
Furthermore, the ball is going faster on the way down towards the ground
as gravity plays a role.
hH Explain in detail how you know that the value from part g is the IRC. (i.e. since the values of calculations from part d are getting smaller and smaller, this shows that the slope of the secant is getting closer and closer to the tangent line … or some explanation similar to this). BE DETAILED!!!
The tangent line in Part G is zero. This is because the average of all of the lines is 0 because the point I picked will make all of the IRC zero. However, the instantaneous rate of change is only an average of the rate of change. Since the values of my calculations from part d are getting smaller and smaller, this shows that the slope of the secant is getting closer and closer to the tangent line of zero.
