QUESTION 1
Function – A function is the relationship that exists between an input and output such that one input has only one possible output. This implies that a function takes an input and produces an output based on the input (Cheney & Kincaid, 2012).
Examples
Linear function
y = 2 x
Quadratic function
y = 2 x2
Third degree polynomial
y = 2 x3
Exponential function
y = e2x
Logarithmic function
y = log 2 x
Periodic function
y = sin x
Domain of a function - The domain of a function is the set of inputs for which the output of the function exists (Cheney & Kincaid, 2012).The domain of linear function is (- ∞ to ∞), quadratic function is (- ∞ to ∞), third degree polynomial function is (- ∞ to ∞), exponential function is, logarithmic function is (- ∞ to ∞), and periodic function is (- ∞ to ∞).
The range of a function – The range of a function is the nature of the outputs of the function (Friedman, 2011). The range of linear function is (- ∞ to ∞), quadratic function is (0 to ∞), third degree polynomial function is (- ∞ to ∞), exponential function is (0 to ∞), logarithmic function is (- ∞ to ∞), and periodic function is (- ∞ to ∞).
QUESTION 2
(a) Describe the relationship you have chosen to investigate and why do you believe the relationship may be periodic
The selected periodic relationship is the position of the sun in the sky at a specific geographic location on the earth. This function is periodic because the position of the sun at the same geographic location will be similar after every 24 hours.
(b) Describe what are your variables – specify dependent and independent.
For the selected periodic function, the dependent variable is the position of the sun in the sky while the independent variable is the time of day.
(c) Describe how you gathered your data.
In order to determine whether the function is periodic data needs to be collected. In order to collect data, both the dependent and independent variables will be defined. In this case, the dependent variable will be morning, noon and evening. The independent variable will be rising, high in the sky and setting. Once the variables have been defined, data will be collected on the position of the sun in the morning, noon and evening, Data will be collected for 5 days, and the results compared in order to determine whether the event is periodic.
(d) Present your data in a table – computer generated screen printout.
Day 1:
Day 2:
Day 3:
Day 4:
Day 5:
The data above indicates that the event is periodic. The position of the sun was the same during a specific time of day. This indicates that the function repeats itself over a 24-hour period.
QUESTION 3
Amplitude – This is the magnitude of a function at a specific point (Cheney & Kincaid, 2012). For the position of the sun, the amplitude of the function is maximum at noon
Frequency – This are the number of times that the function repeats itself in one second (Friedman, 2011). For the position of the sun, the frequency is 1/86,400 = 1.16 x 10-5
Period – This is the amount of time that a periodic function takes to repeat itself (Cheney & Kincaid, 2012). For the position of the sun, the period of the function is 24 hours. In SI units, this is 86,400 seconds.
Average of periodic functions – The average of a period function is given as the mean of all points along the periodic function. It is important to note that average of the entire periodic function is zero.
QUESTION 4
The inverse of a periodic function is defined as function that produces an output that is the reverse of what a period function outputs. This means that a period function will take a specific input and give a specific output. The inverse period function will take the output and give the input (Cheney & Kincaid, 2012). This implies that the composite of a period function and its inverse is one. For example, if f (x) takes the input x and gives an output y then the f-1 (x) will take an input y and give an output x.
For example, sin (x) is a periodic function. Its inverse is given by 1 / sin (x). Therefore,
sin (30) = 0.5
1 / sin (0.5) = 30
In addition, the composite of sin (x) and 1 / sin (x) is equal to 1.
QUESTION 5
Part 1:
What does the Richter Scale/MMS measure?
The Richter/MMS scale measures the intensity of an earthquake. It provides a means through which a signal magnitude value can be assigned to an earthquake (Tussy & Gustafson, 2013).
How does the Richter Scale/MMS measurement compare to the actual magnitude of ground movement?
The number produced by the scale is the log base 10 of the actual magnitude or amplitude of the strongest wave resulting from the earthquake.
What does one point higher on the scale represent, (mathematically) of the magnitude of ground movement?
One point higher on the scale, mathematically represents a ten times increase in the actual magnitude of ground movement. This is mainly because the scale is based on log base 10 (Tussy & Gustafson, 2013).
Part 2: Research two past earthquakes
The two earthquakes selected for this research are the 2010 Haiti earthquake and the 2007 Peru earthquake.
The 2010 Haiti earthquake occurred on 12 January 2010 and had its epicenter in a town named Léogâne in Haiti. The earthquake’s magnitude measured 7 on the moment magnitude scale.
Moment Magnitude Scale: seven
The 2007 Peru earthquake occurred on 15 August 2007 and had its epicenter 50 km south-southeast of Lima, Peru. The earthquake’s magnitude measured 8 on the moment magnitude scale.
Moment Magnitude Scale: eight
Part 3: Mathematically Compute and Compare Earthquakes
Write the equivalent of each Richter scale value in logarithmic form as a basic logarithmic equation.
The logarithmic equation is given as:
f (x) = log10 x
For the 2007 Peru earthquake
8 = log10 100,000,000
For the 2010 Haiti earthquake
7 = log10 10,000,000
Write each logarithmic equation as an exponential equation.
In exponential form, the 2007 Peru earthquake becomes:
100,000,000 = 108
In exponential form, the 2010 Haiti earthquake becomes:
10,000,000 = 107
Write a ratio of exponential expressions, which evolves from the exponential equations and then, compare the intensities of earthquakes.
108 / 107 = 10
References:
Cheney, E. W., & Kincaid, D. (2012). Linear algebra: Theory and applications. Sudbury, MA: Jones & Bartlett Learning.
Friedman, M. (2011). Algebra & functions workbook. Piscataway, NJ: Research & Education Association.
Tussy, A. S., & Gustafson, R. D. (2013). Elementary and intermediate algebra. Belmont, CA: Brooks/Cole, Cengage Learning.
