calculus for the life sciences - san diego state university
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IntroductionDifferentiation of Sine and Cosine
Calculus for the Life Sciences
Lecture Notes – Differentiation of Trigonometric Functions
Joseph M. Mahaffy,〈[email protected]〉
Department of Mathematics and StatisticsDynamical Systems Group
Computational Sciences Research Center
San Diego State University
San Diego, CA 92182-7720
http://www-rohan.sdsu.edu/∼jmahaffy
Spring 2017
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (1/40)
IntroductionDifferentiation of Sine and Cosine
Outline
1 Introduction
Tides
Tidal Forces
Mathematical Model for Tides
2 Differentiation of Sine and Cosine
Basic Differentiation
General Rule of Differentiation
High and Low Tides
Change in Temperature
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (2/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Introduction
Differentiation of Trigonometric Functions
Showed Sine and Cosine Models were good for periodicphenomenon
Trigonometric functions are used to approximate morecomplicated behavior
Joseph Fourier (1768-1830) used series of trigonometricfunctions to approximate other phenomena, such asharmonic motion of vibrating strings
Tidal flow results from the interaction of differinggravitational fields
The complex dynamics are approximated by a short seriesof trigonometric functions with periods related to theastronomical bodies causing the tidal flow
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (3/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Tides – Introduction
Oceans are one of the great remaining frontiers
Over half of the population on this planet lives within 100miles of the oceans
Tidal flow affects the daily dynamics of coastal zones
The Bay of Fundy in Newfoundland has tides rising over 16meters in a 6.25 hour time periodIn San Diego, the tidal flow is not so dramatic
The tides do affect a variety of marine behaviors
Most days there are two high tides (high-high andlow-high) and two low tides (low-low and high-low)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (4/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Tides – Introduction
How are tides predicted and what is the basis for theirvariability?
What causes the changes in amplitude and periodbetween the high and low tides?
The primary forces generating the tides are from thegravity of the sun and the moon
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (5/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Tides – Introduction
Gravitational forces of the Sun and Moon are the primarycauses for generation of tides on Earth
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (6/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Tidal Forces 1
Four Dominant Tidal Forces
Diurnal components about 24 hours (once per day)
K1, the lunisolar forceO1, the main lunar force
Semidiurnal components about 12 hours (twice per day)
M2, the main lunar forceS2, the main solar force
Periodic motion of the moon about the Earth (about 25hours) cause variations
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (7/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Tidal Forces 2
Tidal Forces
When the moon, Earth, and sun align at either a full moonor a new moon, then the tides are at their highest andlowest as the forces of gravity enhance tidal flow
The tides show the least variation when the moon is in itsfirst or last quarter
Other complications include
The elliptical orbit of the moon around the EarthThe elliptical orbit of the Earth around the sunThe influences of other planets
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (8/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 1
Mathematical Model for Tides
What mathematical tools can help predict thetides?
Use a series of trigonometric functions to approximate thebehavior of the tides
Standard programs use 12-14 trigonometric functions
Next Slide are graphs of the high and low tides for SanDiego for September 2002
The model is generated using only four trigonometricfunctions from the four forces described aboveThe data points indicate the actual values of the high andlow tides from standard tide tables
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (9/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 2
First 2 weeks of Tides for San Diego in September 2002Model and Data
1 2 3 4 5 6 7
0
2
4
6
Tid
e H
eigh
t (ft)
Sep
t 1−
7, 2
002
8 9 10 11 12 13 14
0
2
4
6
Tid
e H
eigh
t (ft)
Sep
t 8−
14, 2
002
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (10/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 3
Last 2 weeks of Tides for San Diego in September 2002Model and Data
15 16 17 18 19 20 21
0
2
4
6
Tid
e H
eigh
t (ft)
Sep
t 15−
21, 2
002
22 23 24 25 26 27 28
0
2
4
6
Day
Tid
e H
eigh
t (ft)
Sep
t 22−
28, 2
002
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (11/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 4
Model for Height of Tides, h(t) in feet with t hours frommidnight 1st day of the month
The function h(t) is formed by the sum of four cosinefunctions and a constant
The periods of the cosine functions reflect the periodicnature of the forces
K1, lunisolar diurnal force with period p1 = 23.934O1, main lunar diurnal force with period p2 = 25.819M2, main lunar semidiurnal force with period p3 = 12.421S2, main solar semidiurnal force with period p4 = 12.00
Periods are fixed based on the rotations of the moon andEarth
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (12/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 5
Model for Height of Tides
The amplitudes associated with each force are ai, i = 1, .., 4
The phase shifts associated with each force are φi, i = 1, .., 4
A vertical shift satisfies a0
The parameters, ai and φi, are fit using a least squares bestfit to the high and low tides for the month of September2002
h(t) = a0 +4
∑
i=1
ai cos
(
2π
pi(t− φi)
)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (13/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 6
h(t) = a0 +
4∑
i=1
ai cos
(
2π
pi(t− φi)
)
Best Fitting Parameters
Vertical Shift a0 = 2.937 ft
Force Amplitude Phase Shift
K1 a1 = 0.878 φ1 = 16.246
O1 a2 = 0.762 φ2 = 14.311
M2 a3 = 1.993 φ3 = 6.164
S2 a4 = 0.899 φ4 = 10.857
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (14/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 7
Model and Forces
The strongest force affecting the tides is the semidiurnalmain lunar force
The highest and lowest tides of the month coincide withthe new moon and full moon
New Moon First Quarter Full Moon Last Quarter
September 6 September 13 September 21 September 29
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (15/40)
IntroductionDifferentiation of Sine and Cosine
TidesTidal ForcesMathematical Model for Tides
Mathematical Model for Tides 8
Modelling Low and High Tides
When do the highest and lowest tides occur basedon the mathematical model?
The high and low points of a function are the maxima andminima
This uses differentiation of our model, h(t)
The high and low tides should occur when h′(t) = 0
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (16/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Differentiation of Sine and Cosine 1
Differentiation of Sine and Cosine
The derivative of these functions is found using thedefinition of the derivative and some trigonometricidentities
Derivative of Sine
d
dxsin(x) = cos(x)
Derivative of Cosine
d
dxcos(x) = − sin(x)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (17/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Differentiation of Sine and Cosine 2
Differentiation of Sine Below is the graph of sine and itsderivative
d
dxsin(x) = cos(x)
−2π −π 0 π 2π
−0.5
0
0.5
t
f(t)
and
f ’(t
)
Sine and Its Derivative
f(t) = sin(t)f ’(t) = cos(t)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (18/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Differentiation of Sine and Cosine 3
Differentiation of Cosine Below is the graph of cosine andits derivative
d
dxcos(x) = − sin(x)
−2π −π 0 π 2π
−0.5
0
0.5
t
f(t)
and
f ’(t
)
Cosine and Its Derivative
f(t) = cos(t)f ’(t) = −sin(t)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (19/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Differentiation of Sine and Cosine 4
More Generally, the Differentiation of Sine and Cosine
Derivative of Sine
d
dtsin(ω(t− φ)) = ω cos(ω(t− φ))
Derivative of Cosine
d
dtcos(ω(t− φ)) = −ω sin(ω(t− φ))
Soon learn the chain rule of differentiation for moregeneral cases
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (20/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Example 1
Consider an example of Two Sine Functions
f(t) = 2 sin(2t)− 4 sin(3t)
Properties of trig function give for 2 sin(2t):
Period: T1 = πMaxima occur: tmax = π
4+ nπ, n = ..− 1, 0, 1, ..
Minima occur: tmin = 3π
4+ nπ, n = ..− 1, 0, 1, ..
Properties of trig function give for −4 sin(3t):
Period: T2 = 2π
3
Maxima occur: tmax = π
2+ 2nπ
3, n = ..− 1, 0, 1, ..
Minima occur: tmin = π
6+ 2nπ
3, n = ..− 1, 0, 1, ..
Common Period: Information gives a period for f(t) ofT = 2π
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (21/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Example 2
For the example of Two Sine Functions
f(t) = 2 sin(2t)− 4 sin(3t),
the derivative is easily computed
f ′(t) = 4 cos(2t)− 12 cos(3t).
Find Maxima and Minima by solving f ′(t) = 0 or4 cos(2t) = 12 cos(3t)
This equation is NOT readily solved
Need numerical methods to find Extrema
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (22/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Example 3
The graph of the Two Sine Functions example
f(t) = 2 sin(2t)− 4 sin(3t),
0 5 10 15 20
−6
−4
−2
0
2
4
6
t
f(t)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (23/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Example 4
The Two Sine Functions example
f(t) = 2 sin(2t)− 4 sin(3t),
Graph shows the periodicity, T = 2π
Each period has 3 Relative Maxima and Minima
For t ∈ [0, 2π], numerical Relative Maxima are(1.4604, 4.2206) (3.7117, 5.7786) (5.8283, 2.3366)For t ∈ [0, 2π], numerical Relative Minima are(0.4549,−2.3366) (2.5715,−5.7786) (4.8228,−4.2206)
For t ∈ [0, 2π], the Absolute Maximum and Minimumare(3.7117, 5.7786) and (2.5715,−5.7786)
These Critical Points are readily found with MapleJoseph M. Mahaffy, 〈[email protected]〉
Lecture Notes – Differentiation of Trigonometric— (24/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 1
High and Low Tides
Model and data show the highest and lowest tides in the1st and 3rd weeks (New and Full moon phases)
Use the model to predict the highest high-high tide andlowest low-low tide for the first week
Determine the error between the model and the actualvalues for these tides
The times and heights of the high and low tides use thelocal extrema
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (25/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 2
Model for Tides
h(t) = a0 +
4∑
i=1
ai cos(
2πpi(t− φi)
)
The derivative satisfies:
h ′(t) = −4
∑
i=1
(
2πaipi
)
sin(
2πpi(t− φi)
)
Solving h ′(t) = 0 is too complicated to find the extrema byhand
Easily solved by computers, such as Excel’s Solver or Maple
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (26/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 3
New Moon There was a New moon on September 6, 2002
The graphs show many local extrema for the month ofSeptember
Localize the search for the extrema using the visualinformation from the graph
In the first week, the data show that the highest tide is 6.7ft on Sept. 6, while the lowest tide is −1.0 ft on the sameday
So what does our model using four cosine functions predictto be the highest and lowest tides of this week?
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (27/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 4
Low Tide Prediction for September 6, 2002
Parameters were fit for complete month of September andgiven earlier
Set h ′(t) = 0 and solved with a computer
Low Tide Prediction
The lowest tide of the first week from the model occurswhen tmin = 124.58 hrs with a h(tmin) = −0.86 ftThis corresponds to Sept. 6 at 4:35 amThe actual low-low tide on Sept. 6 is −1.0 ft occurring at3:35 amThe model overshoots the tide height by about 0.14 feetand misses the time by 60 minutes
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (28/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 5
High Tide Prediction for September 6, 2002
Repeat process for High Tide
High Tide Prediction
The highest tide of the first week from the model occurswhen tmax = 142.56 hrs with a h(tmax) = 6.40 ftThis corresponds to Sept. 6 at 10:33 pmThe actual high-high tide on Sept. 6 is 6.7 ft occurring at9:36 pmThe model undershoots the tide height by about 0.3 feetand misses the time by 57 minutes
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (29/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 6
Graph of Tides: Model and Data
0 5 10 15 20
0
2
4
6
Hour
Tid
e H
eigh
t (ft)
September 6, 2002
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (30/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
High and Low Tides 7
Summary of Tide Model for September 2002
The calculations above show that our model introduces amoderate error
Model only uses four cosine functions to try to predict anentire month of high and low tides
This is a reasonable approach to the problem
Obviously, the addition of more trigonometric functionsand more parameters can produce a much more accuratemodel
Actual models use 12-14 trigonometric functions to modeltides
The information line (619-221-8824) for the San DiegoBeach report gives tide information
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (31/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 1
Maximum Change in Temperature The sine function canbe used to approximate the temperature during a day
T (t) = A+B sin(ω(t− φ)),
with constants A, B ≥ 0, ω > 0, and φ ∈ [0, 24) are determinedfrom the data
Suppose that the coolest temperature for a day occurs at3 am and is 56◦F
Assume at 3 pm, the hottest temperature of 82◦F occurs
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (32/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 2
Maximum Change in Temperature The Temperature ismodeled by
T (t) = A+B sin(ω(t− φ)),
Find the constants that best fit the data for thetemperature during the day assuming that the temperaturehas a 24 hour period
Determine the times during the day that the temperatureis rising most rapidly and falling most rapidly
Give the rate of change of temperature at those times
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (33/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 3
Solution: The temperature during a day
T (t) = A+B sin(ω(t− φ))
The average temperature is
A = (56 + 82)/2 = 69◦F
The amplitude of this function is found from the differencebetween the high temperature and the average temperature
B = 82− 69 = 13◦F
The 24 hour periodicity gives
24ω = 2π or ω =π
12
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (34/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 4
Solution (cont): The temperature during a day
T (t) = 69 + 13 sin(
π12(t− φ)
)
The maximum occurs at 3 pm or t = 15
The maximum of the sine function occurs at π2
The phase shift, φ, solves
π
12(15− φ) =
π
2
It follows that
15− φ = 6 or φ = 9 hr
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (35/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 5
Graph of Temperature Model
0 5 10 15 20
60
65
70
75
80
t (in hours)
T(t
) (in
o F)
T(t) = 69 + 13 sin(π(t−9)/12)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (36/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 6
Solution (cont): Graph shows temperature is rising mostrapidly in the morning and falling most rapidly in the evening
Model isT (t) = 69 + 13 sin
(
π12(t− 9)
)
The derivative satisfies
T ′(t) =13π
12cos
(
π12(t− 9)
)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (37/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 7
Graph of Derivative of Temperature Model
0 5 10 15 20
−3
−2
−1
0
1
2
3
t (in hours)
T’(t
) (in
o F/h
r)
T’(t) = (13π/12) cos(π(t−9)/12)
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (38/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 8
Solution (cont): The derivative is
T ′(t) =13π
12cos
(
π12(t− 9)
)
Find the maximum rate of change by properties of T ′(t)
Cosine has maximum when argument is zero, when t = 9
Maximum increase at 9 am with
T ′(9) =13π
12cos(0) =
13π
12≈ 3.4◦F/hr
Minimum increase 12 hours later with t = 21, so
T ′(21) =13π
12cos(π) = −
13π
12≈ −3.4◦F/hr
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (39/40)
IntroductionDifferentiation of Sine and Cosine
Basic DifferentiationGeneral Rule of DifferentiationHigh and Low TidesChange in Temperature
Change in Temperature 9
Alternate Solution: The maximum and minimum rate ofchange occurs when second derivative is zero
T ′′(t) = −13π2
144sin
(
π12(t− 9)
)
The sine function is zero when the argument is nπ
Solveπ
12(t− 9) = nπ, n = 0, 1, ...
Thus,t = 9 + 12n n = 0, 1, ...
This gives same result as before
Joseph M. Mahaffy, 〈[email protected]〉Lecture Notes – Differentiation of Trigonometric— (40/40)