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Examples and Samples

Functions Assignment

Question 1. A function assigns a unique output value to every input value. This definition implies that a vertical line cannot intersect a function in more than one place (Stewart). The set of input values is known as the domain of a function. The set of output values is known as the range of a function.
An example of a linear function is,
f left (x right ) = 4x+3
where 4 and 3 are the slope and vertical intercept of the straight line f(x). The domain of this linear function is all values x can take on, namely, all real numbers. The range is also all real numbers. A parabola is a good example of a quadratic function,
f left (x right ) = {x} ^ {2}
The vertex of the parabola coincides with the origin of a Cartesian coordinate system. The domain of this function is all real numbers and its range is all positive real numbers, f(x)≥0. An example of a third degree polynomial is,
f left (x right ) = {x} ^ {3} +3 {x} ^ {2} +2x+1
The domain and range of this polynomial are all positive real numbers. The exponential function is defined as,
f left (x right ) = {e} ^ {x}
The domain of the exponential function is all real numbers and its range corresponds to all positive real numbers. The natural logarithm is a good example of a logarithmic function and is defined as,
f left (x right ) = ln {x}
The domain of the natural logarithm is all positive real numbers, x>0. The range is all real numbers. The trigonometric functions are good examples of periodic functions. For instance, the sine function,
f left (x right ) = sin {x}
It repeats every 2π radians. The domain of the sine function is all real numbers. The range is -1 ≤ f(x) ≤ 1.
Question 2. The electrical activity of the human heart can be monitored by means of an Electrocardiogram (ECG). An example of ECG signal is shown in Figure 1. The signal was retrieved from National Instruments website (“National Instruments: Test, Measurement, and Embedded Systems”). The electrical heartbeats are represented by the voltage amplitude (dependent variable) over time (independent variable). The figure shows that heartbeats exhibit nearly periodic behavior. Strong beats repeat approximately every 1.6 seconds.

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Figure 1. Example of heart’s electrical activity.

Question 3. The displacement of a piston in an internal combustion engine is represented by a periodic function,
f left (t right ) = A sin {ωt}
where ω is referred to as the angular frequency and has units of radians per second (rad/s). The independent variable is time t (s). A stands for the amplitude and describes the magnitude of the maximum displacement of the piston. The amplitude is a constant value. One cycle takes place when the crack (a linkage connecting the engine shaft and piston) turns through 2π radians. In other words, the sine function repeats itself when ωT=2π, where T is the period, so this time is defined as,
T= {2π} over {ω}
The frequency is the inverse of the period and it is measured in s-1 or hertz (hz)
f= {1} over {T}
For example, a period of 10s means a frequency of 1 every 10 seconds. The average value of function f(x) over an interval [a,b,] is defined as,
F = {1} over {b-a} int from {a} to {b} {f(x)dx}
The above expression can be used to calculate the average displacement over one cycle of the piston [0,T],
F = {1} over {T} int from {0} to {T} {A sin {{2πt} over {T}} dt=} {1} over {T} {left [- {T} over {2π} cos {{2πt} over {T}} right ]} rsub {0} rsup {T} =- {1} over {2π} left [cos {2π} - cos {0} right ] =0
Question 4. The inverse of a function ( f -1) takes on a value from the range of f and assigns a single value from its domain. The inverse of a function exist only if a horizontal line does not intersect the function f in more than one place. This is not satisfied for the case of periodic functions like the sinusoidal displacement of a piston,
y=f left (t right ) = A sin {ωt}
The domain of sin ωt has to be restricted in order to define the inverse function. It can be worked out as follows,
{sin} ^ {-1} {left ({y} over {A} right ) =ωt}
{{f} ^ {-1} (y) = t = {1} over {ω} sin} ^ {-1} {left ({y} over {A} right )}
The inverse function holds as long as,
- {π} over {2} ≤ {y} over {A} ≤ {π} over {2}
In a given cycle of the piston, the inverse function provides the time for which the piston is displaced y length units.
Question 5. Part 1. The Richter scale is a scientific tool to measure the intensity of an earthquake (“Khan Academy | Khan Academy”). The following facts summarize the main features of the Richter scale:
The Richter scale measures the earthquake magnitude. The intensity or amplitude of the ground motion is collected at the measuring stations. The maximum ground motion detected at a measuring station can be plotted against the distance to the epicenter.
The Moment Magnitude Scale (MMS) was calibrated to the Richter scale. MMS is related to the energy released by earthquakes.
Unfortunately, strong and weak earthquakes do not fit on the same plot if a linear scale is used. The Richter scale employs a logarithmic scale in order to present the data on the same chart.
The Richter scale can be used to compare the magnitude of two earthquakes. For instance, the 5.8 earthquake that took place on the east coast of US in 2011and the 9.0 earthquake off the coast of Japan in the same year. Both magnitudes are reported on a logarithmic scale so the difference between these two earthquakes is not 3.2 but 103.2 (3.2 powers of 10). It means the earthquake in Japan was about 1600 times stronger than the east coast earthquake.
Part 2. Table 1 summarizes the facts about the earthquakes chosen to draw a comparison.

Date Location of epicenter Richter magnitude scale (M)
25 September 2013 Southwestern Pakistan 7.7
27 February 2010 Central Chile (near city of Concepción) 8.8

Table 1. Earthquake information.

Part 3. Quantitative comparison of the two earthquakes. The equation for Richter magnitude can be written as,
M = {log} rsub {10} {A}
where A is the amplitude in millimeters directly measured from a seismograph. The equation may also include a distance correction factor originally used by Richter. It is omitted for the sake of clarity. The Richter magnitude of Pakistan’s 2013 earthquake can be related to its amplitude by,
7.7 = {log} rsub {10} {{A} rsub {Pakistan}}
The equation above can be written as an exponential equation, that is, the amplitude read in the seismograph,
{A} rsub {Pakista n} = {10} ^ {7.7}
Similarly, the magnitude of Chile’s 2010 earthquake can be mathematically written as,
8.8 = {log} rsub {10} {{A} rsub {Chile}}
It can also be recast as an exponential equation,
{A} rsub {Chile} = {10} ^ {8.8}
The ratio of exponential expressions can be used to compare the intensities of earthquakes,
{{A} rsub {Chile}} over {{A} rsub {Pakistan}} = {{10} ^ {8.8}} over {{10} ^ {7.7}} =12.6
It can be concluded that Chile’s earthquake was 12.6 times as intense as the recently ground motion in Pakistan.
Task 1. The exponential function ex can be written as an infinite Taylor series about x=0. The Taylor series is,
{e} ^ {x} =1+x+ {{x} ^ {2}} over {2!} + {{x} ^ {3}} over {3!} + {{x} ^ {4}} over {4!} +… {{x} ^ {n}} over {n!} +…
It can be used to approximate the value of e1 up to n= 10. The results are shown in Table 2. The approximation value for e1 up to 10 decimal places is obtained when n = 15,
e = 2.7182818285

n Approximation for e
1 1
2 2
3 2.5
4 2.666666667
5 2.708333333
6 2.7166666667
7 2.7180555556
8 2.7182539683
9 2.7182787698
10 2.7182815256

Table 2. Approximation value for e.

Works Cited

“Khan Academy | Khan Academy.” N. p., n.d. Web. 4 Dec. 2013.
“National Instruments: Test, Measurement, and Embedded Systems.” N. p., n.d. Web. 3 Dec. 2013.
Stewart, J. Calculus. Brooks/Cole, 2009.

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