Blog Post 2

(A-C)

John is an avid hunter. He uses various bows and arrows in order to figure out which one works the best to painlessly hunt white tail deer in the mountains of Pennsylvania. He compares all his bows by their velocity at exactly three seconds, his grandfather’s favorite number .John needs to know what the velocity at three seconds is of the arrow for his new bow.

X: TIME (sec)	0	1	2	3	4	5	6	7	8	9	10
Y: DISTANCE (yards)	0	10	19	27	34	40	45	49	52	54	55

(E)

I notice that the slope of the secant lines get smaller and smaller the further we get from our original point. In terms of our experiment, this means the further the arrow gets from the bow, the slower the arrow travels.

(G)

The instantaneous rate of change for X=3 is 8 yards per second. In terms of our experiment, this means that at exactly 3 seconds the arrow is traveling at approximately 8 yards per second traveled.

(H)

Since the calculations of part E were continuously getting smaller, the slope of the secant lines going to the right from x=3 are getting closer to the slope of the tangent line. This is evident by the decrease in value of our randomly selected points (Secant lines) comparative to QP.

Blog 2

Blog #2

Part A.

a.) When I left for American University this fall my brother bought me a sunflower as a going away gift.  It began to grow on Day 5. Below is the inches the sunflower grew over time.

b.) I want to solve for the rate of change on exactly day 10.

c.)

d.) 

e.) Rate of Change: (Y2-Y1)/(X2-X1)

(10, 1.5) & (5, .5): (10-5)/(1.5-.5)= 5

(10, 1.5) & (15, 2): (15-10)/(2-1.5)= 10

(10, 1.5) & (20, 3): (20-10)/(3-1.5)= 6.67

f.) 

g.) Points on tangent line: (3,0) & (25,4) 

Slope: (25-3)/(4-0)= 5.5

h.) I used the slope of secant and tangent lines in correspondence with the point of data I recorded on day 10 to find the most exact instantaneous rate of change. This method is effective because by finding the average rates of change between the data points closest to the point I am solving for makes the window smaller and smaller in which the instantaneous rate of change lies. We see this process above in the calculations (Y2-Y1)/(X2-X1) and the solutions to the rates of change in the tangent and secant lines. 

Blog 2 Rafael Cabrera

      a) For this experiment, I have decided to use one of the offered examples. In this case, I will be working with a table that analyzes the change in number of walleyes every year. Walleyes are a species of fish found in Canada and North America.

    b) The problem analyzes the increase or decrease of walleyes for every year that passes. It provides a table of 25 years with values on fish population. One could wonder how what is the difference between certain years. Also, an emerging question could be: How many walleyes were there in year 23? One would then look at the table and see that the amount is 4,989. 

      c)

      

                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               d)  

e) 

f) 

g) 

h) By looking at, for example, the slope between points 2 and 8, and comparing it with points 3 and 8, one can say that the value is the IRC because of how close the numbers are. In this case, one can say that 138.8 and 157.5 are fairly close to each other.

Blog 2

Blog 2

1. In this application, I’m seeking to measure the height of ferns over a growing period of five weeks. The question I am posing is “how tall will a fern get after a five week growing period?” Each week will have a different rate of change, and will be measured by the equation: Y=x²+x

2. Y=x²+x

Time (weeks)	1	2	3	4	5
Height (cm)	2.5	5	10	20	40

***Graph Attached

3. By drawing the secant lines from point “two” to points “three, four, and five” I notice the change of slope per week. From point two’s connection to point five, the change in slope is enormous, more than doubling from a slope of “five” to a slope of “ten”, which means that ferns grow tall relatively quickly: from 5cm to 40cm in five weeks. The other two merely hint at its progress to this point. Point two’s connection to point four has an increase of slope from five to seven point five. And from two to three weeks, the slope is the initial increase of slope: a slope of five.

4.  The way I got my value in part g is by calculating the derivative of the derivative for Y=x²+x and plugged 5 in for the  “a value.” I derived this derivative from a previous problem that required me to find the tangent line for a point of my choosing, which, in this case was point two. I found the slope of the tangent line for point five to detail the difference in growth rates from point two to point five. The slopes of the tangent lines were very close to prospective secant lines. Though they were not exact, they were off by no more than one unit. The work can be found in the document attached.

GS

Bunny Population!

A. Time vs. Population
      I decided to do my example on seeing the growth rate population of bunnies.

B. In this experiment, we will be seeing how much the bunny population grows in a period of 7  
     months.
C.

Months (t)	Population Growth of Bunnies (g)
1	2
2	4
3	8
4	16
5	32
6	64
7	128

D.

 

E. The three secants can be obtained from these 3 points taken from the table/ graph above.

      Points:   D(4,16) E(5,32) F(6, 64)

                 o ROC Point DE (4,16) (5,32)

                        (32-16)/ (4-3) = 16

                 o ROC Point EF (5,32) (6,64) 

                        (64-32)/ (6-5) = 32

       After calculating the secant lines, it seems that the ARC doubles as every month passes. This

       means that, as time passes, the bunny population will double in size every month. The bunny 

       population in this experiment seems to thrive and not fear endangerment.

          

F.

G. (2,4)(3,8)

        (8-4)/(3-2)= 4

This means that from month 2 to month 3, the amount of bunnies went from just 2 bunnies to having 4. It shows that there is a consistent doubling in the population.

H. From this data, we can determine that as the months go by, the population size of the bunnies will double as each month goes by. This shows that as it gets further into the year, the bunny population will significantly increase in size. As the numbers get larger, the values become closer to the secant line.

Rate of Change

A). Boston Months vs. Average Snowfall 

B). In this experiment, I will show how the amount of snowfall varies over the different months of the year. This is useful information for cities because i can be used for preparation to combat severe weather conditions. How does the amount of snow change between months?

C). 

January	38.3
February	18.5
March	1.3
April	0.9
May	0
June	0
July	0
August	0
Septmeber	0
October	0
November	0
December	22

Month	Snowfall (In)
1	38.3
2	18.5
3	1.3
4	0.9
5	0
6	0
7	0
8	0
9	0
10	0
11	0
12	22

D).

E). The ARC between the following months:

Between January (1) and February (2) 

(1, 38.3) and (2, 18.5) = -19.8

Between February (2) and March (3)

(2, 18.5) and (3, 1.3) = -17.2

Between March (3) and April (4)

(3, 1.3) and (4, 0.9) = -0.4

The ARC increases as the warmer months approach and then towards the last month is peaks and significantly increases. I can infer that there is more snow during the colder, winter months like January and February and that there is less snow during warming months if this pattern is continuous.

F). The ARC of the secant lines increase the further the months are from January.

G). Between January (1) and March (3) 

(1, 38.3) and (3, 1.3) --> In January there was 38.3 inches of snow and in March there were 1.3 inches.

(1, 38.3) and (3, 1.3) = -19.8 =-37/2 = -18.5

IRC = -18.5 inches

This amount of won is within the values we calculated earlier and falls somewhere middle to high range of amount of snowfall.

H). I know that this is the IRC because we saw that the slopes increases the further we get from January. Furthermore, it falls within the values I got and is slightly above the smallest value, so we know it follows the tangent.