Chabot Mathematics
§8.2 Trig
Derivatives
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
BMayer@ChabotCollege.edu
Chabot College Mathematics
1
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Review §
8.1
 Any QUESTIONS About
• §8.1 → Trigonometric
Functions
 Any
QUESTIONS
About HomeWork
• §8.1 → HW-10
Chabot College Mathematics
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
§8.2 Learning Goals
 Derive and use differentiation formulas
for trigonometric functions
 Study periodic rate and optimization
problems using derivatives of
trigonometric functions
Chabot College Mathematics
3
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Derivatives for Sine and Cosine
 For independent variable t measured in
Radians
d
sin t   cost 
dt
d
cost    sin t 
dt
 Use the ChainRule when the sin/cos
arguments are a function of t, u(t)
d
du
sin u   cosu 
dt
dt
Chabot College Mathematics
4
d
du
cosu   sin u 
dt
dt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Verify Trig Derivs
 Use SpreadSheet to Check that
y  sin t  
Δy
Δt
Δy/Δt
cos(t)
0.017452
0.017447
0.017436
0.017421
0.017399
0.017453
0.017453
0.017453
0.017453
0.017453
0.999949
0.999645
0.999036
0.998122
0.996905
0.999962
0.999657
0.999048
0.998135
0.996917
-0.0013%
-0.0013%
-0.0013%
-0.0013%
-0.0013%
89 1.553343 0.999848 0.000457 0.017453 0.026177 0.026177
90 1.570796
1 0.000152 0.017453 0.008726 0.008727
91 1.588250 0.999848 -0.00015 0.017453 -0.008726 -0.008727
-0.0013%
-0.0013%
-0.0013%
t (°)
0
1
2
3
4
5
t (rads)
0.000000
0.017453
0.034907
0.052360
0.069813
0.087266
y = sin(t)
0.000000
0.017452
0.034899
0.052336
0.069756
0.087156
179 3.124139 0.017452
180 3.141593 1.23E-16
181 3.159046 -0.01745
Error
-0.01745 0.017453 -0.999645 -0.999657
-0.01745 0.017453 -0.999949 -0.999962
-0.01745 0.017453 -0.999949 -0.999962
-0.0013%
-0.0013%
-0.0013%
269 4.694936
270 4.712389
271 4.729842
-0.99985 -0.00046 0.017453 -0.026177 -0.026177
-1 -0.00015 0.017453 -0.008726 -0.008727
-0.99985 0.000152 0.017453 0.008726 0.008727
-0.0013%
-0.0013%
-0.0013%
358 6.248279
359 6.265732
360 6.283185
-0.0349 0.017436 0.017453 0.999036 0.999048
-0.01745 0.017447 0.017453 0.999645 0.999657
-2.5E-16 0.017452 0.017453 0.999949 0.999962
-0.0013%
-0.0013%
-0.0013%
Chabot College Mathematics
5
y
d
dy
 cost   sin t  
t
dt
dt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Trig Deriv Proof
 Prove: d cos t  dt   sin t
 Recall Derivate d cos t  lim cost  h   cos t
h 0
dt
h
Definition
 Use the Trig cos     cos   cos   sin   sin 
Sum-Identity
 Apply TrigID to Limit
cost  h   cos t
lim
h 0
h
Chabot College Mathematics
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

cos t  cos h  sin t  sin h   cos t
lim
h 0
h
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Trig Deriv Proof
 Factor the
cos t  cos h  1  sin t  sin h
lim
Limit argument h0
h
 By Limit Properties (c.f. §1.5)

cos t  cos h  1  sin t  sin h
cos h  1
sin h
lim
 cos t  lim
 sin t  lim
h 0
h 0
h 0
h
h
h
 Now Two Limits whose Proof is Beyond
the Scope of MTH16:

cos z  1
lim
0
z 0
Chabot College Mathematics
7
z
sin z
& lim
1
z 0
z
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Trig Deriv Proof
 Using these Limits

cos h  1
sin h
cos t  lim
 sin t  lim
h 0
h
h 0
h
 cos t  0  sin t 1
 Then Finally
d
cos t  cos h  1  sin t  sin h
cos t  lim
  sin t 1
h

0
dt
h
d
cos t   sin t
dt
Chabot College Mathematics
8
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  CoSine Derivative
d x
 Find: wx   e  cos x 
dx
 SOLUTION: Use the Product Rule


d x
w x  
e cosx 
dx
 
d x
x d

e  cosx   e  cosx 
dx
dx
 e  x  cosx   e  x   sin x 
 e  x cosx   sin x 
Chabot College Mathematics
9
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 An approximate Math Model for the
population of microbes present at
temperature T:
 T 
P(T )  3  0.08T  sin 
 for 0 C  T  50C
 30 
• Where
– T in Degrees Celsius (°C)
– P in Millions of Microbes (MegaMicrobes, MM)
 What is the population when the microbial
population is decreasing most rapidly?
Chabot College Mathematics
10
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 SOLUTION:
 The population Decreases most rapidly
when the derivative of the population
function; i.e. the GrowthRate dP/dt, is
minimized.
 to minimize the first derivative, find the
critical points, which requires
computation of the 2nd derivative and
2nd derivative zeroes.
Chabot College Mathematics
11
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 Taking the Derivatives
d 2P
d d


2
dT  dT  dT
d

dT

 T  
3  0.08T  sin  30  

 


 T   
0  0.08  cos 30   30 




 T       
  sin 
  
 30   30   30 
 
 T 
   sin 

 30 
 30 
2
Chabot College Mathematics
12
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 Set to Zero the 2nd Derivative
d 2P
dT 2 T
0
max
 Tmax    
 sin 
   0
 30   30 
 Tmax 
sin 
0
 30 
2
 The above eqn has infinitely many
solutions, but recall that the T-domain
Restriction: [0,50].
Chabot College Mathematics
13
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 The simplest solutions to sin(θ)=0
are 0 and π. However, any solution that
is a multiple of 2π away from either
solution is also a solution. Thus

T
30
 2k
or
T
30
   2k  
• Where k is any Integer
 Solving for T find
30 2k
T
 60k
 2
Chabot College Mathematics
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or
30  1  2k 

 301  2k 
30 
1
T
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 Then the two branches of solutions in
terms of T: T  60k or T  30  60k
 The only solutions for T on the interval
[0,50] are 0 and 30
 Need to consider both critical points, as
well as the endpoints 0 (0 is also a
critical point) and 50, then note which
input corresponds to the smallest (most
negative) value of dP/dT
Chabot College Mathematics
15
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Maximizing Microbes
 Tabulating the Results
dP
 T  
 0.08  cos

dT
 30  30
T
dP/dT
0
0.185
30
-0.025
50
0.132
 The only negative dP/dT is at T = 30,
which then corresponds to the minimum
 Then the microbial population at 30 °C:
  (30) 
P30   3  0.0830   sin 
  5.4 MegaMicrob es
 30 
Chabot College Mathematics
16
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Derivatives for tan and sec
 For independent variable t measured in
Radians
d
tan t   sec 2 t 
dt
d
sect   tan t   sect 
dt
 Use the ChainRule when the tan/sec
arguments are a function of t, u(t)


d
du
2
tan u   sec u  
dt
dt
Chabot College Mathematics
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d
du
secu   secu   tan u 
dt
dt
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
tan Trig Deriv Proof
 Prove: d tan t  dt  sec 2 t 
 Use the
• Tan definition: tan t  sin t cos t
• Quotient Rule:
d  f x 

dx  g x  
g x 
df
dg
 f x 
dx
dx
g 2 x 
• Previously Proved Trig Derivs:
d
d
sin t   cost 
cost    sin t 
dt
dt
 Then d tan t 
dt
Chabot College Mathematics
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d  sin t 
dt  cos t 
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
tan Trig Deriv Proof
 Using the Quotient Rule and Chain Rule
d
d  sin t 
tan t  


dt
dt  cos t 
 Or
d
tan t
dt

cos t  cos t  sin t   sin t 
cos 2 t
cos 2 t  sin 2 t
cos 2 t
2
2
cos


sin
 1
 Using another Trig ID →
 Find
d
cos t  sin t
1
 1 
2
2


tan t 



sec
t

sec
t
2
2


dt
cos t
cos t  cos t 
2
Chabot College Mathematics
19
2
2
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  tan derivative
2
tan t
df
for f t   2
and
t 1
dt t 
d
2
2
t  1  2 tan t  tan t   2t  tan t
df
dt

2
2
dt
t 1
2
2
2
df t  1 2 tan t  sec t  2t  tan t

2
2
dt
t  1
 Find df
dt



df
dt



2
 1  2 tan   sec 2   2  tan 2 
t 
Chabot College Mathematics
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


2

1
2
0
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & trig
 A birdwatcher observes a bird flying
overhead away from her. She
estimates that the bird is flying parallel
to the ground at 10 mph and is initially
40 feet away horizontally and 15 feet
above the birdwatcher’s line of sight.
 How quickly is the angle between the
birdwatcher’s light of sight and the
location of the bird changing after 12
seconds?
Chabot College Mathematics
21
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & trig
 A diagram REALLY helps in this case.
 Let
• W ≡ the initial location of the birdwatcher
• B ≡ the current position of the bird
 Then
the
Diagram:
10 mph
h =15 ft
q
Chabot College Mathematics
22
x = 40 ft
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & trig
 SOLUTION:
 First find the rate of change in θ with
respect to time. The relationship
between the angle and the given
distances can be represented by the
tangent function (opposite over adjacent)
h
q
Chabot College Mathematics
23
x
h
tanq =
x
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & trig
 Use implicit differentiation to take
derivatives of both sides with respect to
time, noting that h is constant:
d 
h
d
d h
d 1
tan       h   
tan    

dt 
x
dt
dt  x 
dt  x 
d
d
d
d  1  dx
dx
2
2
tan    sec    h      h   x 
d
dt
dt
dx  x  dt
dt

d
h dx
sec  
 2 
dt
x dt
2
Chabot College Mathematics
24
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx

Example  Related Rate & trig
 To find dθ/dt replace all of the other
variables with their values at the time
when the bird has been flying for 12
seconds.
 First, the value of x is initially 40 ft, but
after 12 seconds flying at 10 mph, the
horizontal distance increases to
1 hr
5280 ft
x  40 ft  12 sec   10 mph  

 216 ft
3600 sec 1 mi
Chabot College Mathematics
25
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & trig
 Use x = 216ft after to find sec2θ after
the 12 second Flite Time:
h
tan  
x
15
35
5
 tan  


216 3  72 72
 Now use the Pythagorean
tan2 q +1= sec2 q
identity relating tan and sec
 Thus
 5 
2
   1  sec 
 72 
sec 2   1.069
Chabot College Mathematics
26
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Example  Related Rate & Trig
 Now combine all of the values into the
implicit differentiation equation:
d
h dx
sec  
 2 
dt
x dt
d
15
1.069    2  (14.67)
dt
216
d
 0.0044 rad sec  1.59
dt
2
deg
 After 15 seconds, the angle of
inclination to the bird decreases at
about 1.59 degrees per second.
Chabot College Mathematics
27
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
sec
WhiteBoard Work
 Problems From §8.2
• P8.2-50 →
RowBoat Rope
Reel-In
• P8.2-57 →
Harmonic
Motion
Chabot College Mathematics
28
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
All Done for Today
Trig Deriv
Chain
Rule
Chabot College Mathematics
29
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot Mathematics
Appendix
r  s  r  s r  s 
2
2
Bruce Mayer, PE
Licensed Electrical & Mechanical Engineer
2a
–
Chabot College Mathematics
30
BMayer@ChabotCollege.edu
2b
Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
Chabot College Mathematics
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx
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Bruce Mayer, PE
BMayer@ChabotCollege.edu • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx