Je bent je ingevulde velden bij deze pagina aan het verwijderen. Ben je zeker dat je dit wilt doen?
You are erasing your filled-in fields on this page. Are you sure that is what you want?
Nieuwe Versie BeschikbaarNew Version Available
Er is een update van deze pagina. Als je update naar de meest recente versie, verlies je mogelijk je huidige antwoorden voor deze pagina. Hoe wil je verdergaan ?
There is an updated version of this page. If you update to the most recent version, then your current progress on this page will be erased. Regardless, your record of completion will remain. How would you like to proceed?
In Exercises sinecosinegraphfirst - sinecosinegraphlast, graph one cycle of the given function. State the period, amplitude, phase shift and vertical shift of the
function.
\(f(t) = 3\sin (t)\)
Use the Desmos graph below with the settings \(f(t)=\sin (t),\,h=0,\,b=1,\,a=3,\,k=0\)
Use the Desmos graph below with the settings \(f(t)=\sin (t),\,h=\frac {\pi }{4},\,b=-1,\,a=1,\,k=-2\)
Period: \(2\pi \) Amplitude: \(1\) Phase Shift: \(-\frac {\pi }{4}\) (You need to use \(y = -\sin \left ( t + \frac {\pi }{4} \right ) - 2 \) to find this.) (Two cycles of the graph are shown to illustrate the discrepancy discussed
on page ??.) Vertical Shift: \(-2\)
Use the Desmos graph below with the settings \(f(t)=\sin (t),\,h=-\pi ,\,b=-2\pi ,\,a=4,\,k=0\)
Period: \(1\) Amplitude: \(4\) Phase Shift: \(\frac {1}{2}\) (You need to use \(h(t) = -4\sin (2\pi t - \pi )\) to find this.) (This will be the last time we graph two cycles to illustrate the discrepancy
discussed on page ??.) Vertical Shift: \(0\)
In Exercises fitsinecosinefirst - fitsinecosinelast, a sinusoid is graphed. Find a formula for the sinusoid in the form \(S(t) = A \sin (\omega t + \phi ) + B\) and \(C(t) = A \cos (\omega t + \phi ) + B\). Select \(\omega \) so \(\omega > 0\).
Investigate \(\ds {\lim _{t \rightarrow 0} f(t)}\). Does \(\ds {\lim _{t \rightarrow 0} f(t)}\) exist? Why or why not?
\(\ds {\lim _{t \rightarrow 0} f(t)}\) does not exist. We have infinitely many oscillations as \(t \rightarrow 0\).
Determine \(\ds {\lim _{t \rightarrow \infty } f(t)}\) and interpret your answer graphically.
\(\ds {\lim _{t \rightarrow \infty } f(t) = 1}\) since as \(t \rightarrow \infty \), \(\frac {1}{t} \rightarrow 0\). Since cosine is continuous, \(\cos \left (\frac {1}{t}\right ) \rightarrow \cos (0) = 1\). We have a horizontal asymptote \(y = 1\).
Graph \(f(t) = e^{-0.1t} \left ( \cos (2t) + \sin (2t)\right )\) along with \(y = \pm 2 \, e^{-0.1t}\). What do you notice?
The graph of \(f(t) = e^{-0.1t} \left ( \cos (2t) + \sin (2t)\right )\) lies between (We’ll be able to show in Section expandedsinusoid that the graph of \(f\) more perfectly lies between the graphs
of \(y = \pm \sqrt {2} \, e^{-0.1 t}\) …) the graphs of \(y = \pm 2 \, e^{-0.1t}\).
What appears to be \(\ds {\lim _{t \rightarrow \infty } f(t)}\)? Interpret your answer graphically.
\(\ds {\lim _{t \rightarrow \infty } f(t) = 0}\). We have a horizontal asymptote \(y = 0\).
Use the Squeeze Theorem, Theorem squeezeth from Section Sequences to prove your claim.
Show every constant function \(f\) is periodic by explaining why \(f(x + 117) = f(x)\) for all real numbers \(x\). Then show that \(f\) has no period by showing
that you cannot find a smallest number \(p\) such that \(f(x + p) = f(x)\) for all real numbers \(x\).
Said differently, show that \(f(x + p) = f(x)\) for all real numbers \(x\) for ALL values of \(p > 0\), so no smallest value exists to satisfy the definition of
‘period’.
The sounds we hear are made up of mechanical waves. The note ‘A’ above the note ‘middle C’ is a sound wave with
ordinary frequency \(f = 440\) Hertz \(= 440 \frac {\text {cycles}}{\text {second}}\). Find a sinusoid which models this note, assuming that the amplitude is \(1\) and the phase shift is
\(0\).
\(S(t) = \sin \left (880\pi t\right )\)
The voltage \(V\) in an alternating current source has amplitude \(220 \sqrt {2}\) and ordinary frequency \(f = 60\) Hertz. Find a sinusoid which models
this voltage. Assume that the phase is \(0\).
The London Eye is a popular tourist attraction in London, England and is one of the largest Ferris Wheels in the world. It
has a diameter of 135 meters and makes one revolution (counter-clockwise) every 30 minutes. It is constructed so that the
lowest part of the Eye reaches ground level, enabling passengers to simply walk on to, and off of, the ride. Find a sinsuoid
which models the height \(h\) of the passenger above the ground in meters \(t\) minutes after they board the Eye at ground
level.
On page ?? in Section cosinesinebeyond, we found the \(x\)-coordinate of counter-clockwise motion on a circle of radius \(r\) with angular frequency \(\omega \)
to be \(x = r\cos (\omega t)\), where \(t=0\) corresponds to the point \((r,0)\). Suppose we are in the situation of Exercise heightlondoneye above. Find a sinsusoid which models
the horizontal displacement\(x\) of the passenger from the center of the Eye in meters \(t\) minutes after they board the Eye. Here
we take \(x(t) > 0\) to mean the passenger is to the right of the center, while \(x(t) < 0\) means the passenger is to the left of the
center.
In Exercise yoyotrick in Section RadianMeasure, we introduced the yo-yo trick ‘Around the World’ in which a yo-yo is thrown so it sweeps out a
vertical circle. As in that exercise, suppose the yo-yo string is 28 inches and it completes one revolution in 3 seconds. If the
closest the yo-yo ever gets to the ground is 2 inches, find a sinsuoid which models the height \(h\) of the yo-yo above the ground in
inches \(t\) seconds after it leaves its lowest point.
Consider the pendulum below. Ignoring air resistance, the angular displacement of the pendulum from the vertical position, \(\theta \),
can be modeled as a sinusoid. (Provided \(\theta \) is kept ‘small.’ Carl remembers the ‘Rule of Thumb’ as being \(20^{\circ }\) or less. Check
with your friendly neighborhood physicist to make sure.)
The amplitude of the sinusoid is the same as the initial angular displacement, \(\theta _0\), of the pendulum and the period of the motion
is given by
\[T = 2\pi \sqrt {\frac {\ell }{g}}\]
where \(\ell \) is the length of the pendulum and \(g\) is the acceleration due to gravity.
Find a sinusoid which gives the angular displacement \(\theta \) as a function of time, \(t\). Arrange things so \(\theta (0) = \theta _0\).
In Exercise pendulumproblem section RootRadicalFunctions, you found the length of the pendulum needed in Jeff’s antique Seth-Thomas clock to ensure the
period of the pendulum is \(\frac {1}{2}\) of a second. Assuming the initial displacement of the pendulum is \(15^{\circ }\), find a sinusoid which
models the displacement of the pendulum \(\theta \) as a function of time, \(t\), in seconds.
The table below lists the average temperature of Lake Erie as measured in Cleveland, Ohio on the first of the month for each
month during the years 1971 – 2000. (See this website: http://www.erh.noaa.gov/cle/climate/cle/normals/laketempcle.html.)
For example, \(t=3\) represents the average of the temperatures recorded for Lake Erie on every March 1 for the years 1971 –
2000.
Month
Number, \(t\)
1
2
3
4
5
6
7
8
9
10
11
12
Temperature
(\(^{\circ }\) F), \(T\)
36
33
34
38
47
57
67
74
73
67
56
46
Using the techniques discussed in Example sinusoidsunlight, fit a sinusoid to these data.
Graph your model along with the data set to judge the reasonableness of the fit.
Use the model from LakeErieTempData to predict the average temperature recorded for Lake Erie on April \(15^{\text {th}}\) and September \(15^{\text {th}}\) during
the years 1971–2000. (The computed average is \(41^{\circ }\)F for April \(15^{\text {th}}\) and \(71^{\circ }\)F for September \(15^{\text {th}}\).)
Compare your results to those obtained using a graphing utility.
Using the techniques discussed in Example sinusoidsunlight, fit a sinusoid to these data. (You may want to plot the data
before you find the phase shift.)
Graph your model along with the data set to judge the reasonableness of the fit.
Use the model from MoonIllumination to predict the fraction of the moon illuminated on June 1, 2009. (The listed fraction
is \(0.62\).)
Compare your results to those obtained using a graphing utility.
Use a graphing utility to graph \(y = 8.36 \sin (-294.81t - 26.53) + 12.06\). (This is the regression model produced by desmos in Example sinusoidsunlight.) Zoom in, as needed, until
you start to see the wave-like nature of the graph. Use Theorem sinusoidform to determine a window which produces exactly one complete
cycle of this sinusoid and check your answer graphically.
![[Picture]](ExercisesforGraphsofSineandCosine.online-99d192a770ad29921deec1e0cd887941.svg)
![[Picture]](ExercisesforGraphsofSineandCosine.online-7e6337152b730fa8bed5a9754e97b8ce.svg)
![[Picture]](ExercisesforGraphsofSineandCosine.online-627ce234d0bf624638cfd56233199a1c.svg)
![[Picture]](ExercisesforGraphsofSineandCosine.online-fd21f5f8be0a0feeaa4499a733cc7f4b.svg)
