Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

[email protected] • MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1

Bruce Mayer, PE Chabot College Mathematics

Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

§8.2 TrigDerivativ

es



Review §

Any QUESTIONS About• §8.1 → Trigonometric

Functions Any

QUESTIONS About HomeWork• §8.1 → HW-10

8.1



§8.2 Learning Goals Derive and use differentiation formulas

for trigonometric functions

Study periodic rate and optimization problems using derivatives of trigonometric functions



Derivatives for Sine and Cosine For independent variable t measured in

Radians

Use the ChainRule when the sin/cos arguments are a function of t, u(t)

ttdtdtt

dtd sincoscossin

dtduuu

dtd

dtduuu

dtd

sincoscossin



Verify Trig Derivs Use SpreadSheet to Check that

dtdyt

dtdt

tyty

sincossint (°) t (rads) y = sin(t) Δy Δt Δy/Δt cos(t) Error

0 0.000000 0.0000001 0.017453 0.017452 0.017452 0.017453 0.999949 0.999962 -0.0013%2 0.034907 0.034899 0.017447 0.017453 0.999645 0.999657 -0.0013%3 0.052360 0.052336 0.017436 0.017453 0.999036 0.999048 -0.0013%4 0.069813 0.069756 0.017421 0.017453 0.998122 0.998135 -0.0013%5 0.087266 0.087156 0.017399 0.017453 0.996905 0.996917 -0.0013%

89 1.553343 0.999848 0.000457 0.017453 0.026177 0.026177 -0.0013%90 1.570796 1 0.000152 0.017453 0.008726 0.008727 -0.0013%91 1.588250 0.999848 -0.00015 0.017453 -0.008726 -0.008727 -0.0013%

179 3.124139 0.017452 -0.01745 0.017453 -0.999645 -0.999657 -0.0013%180 3.141593 1.23E-16 -0.01745 0.017453 -0.999949 -0.999962 -0.0013%181 3.159046 -0.01745 -0.01745 0.017453 -0.999949 -0.999962 -0.0013%

269 4.694936 -0.99985 -0.00046 0.017453 -0.026177 -0.026177 -0.0013%270 4.712389 -1 -0.00015 0.017453 -0.008726 -0.008727 -0.0013%271 4.729842 -0.99985 0.000152 0.017453 0.008726 0.008727 -0.0013%

358 6.248279 -0.0349 0.017436 0.017453 0.999036 0.999048 -0.0013%359 6.265732 -0.01745 0.017447 0.017453 0.999645 0.999657 -0.0013%360 6.283185 -2.5E-16 0.017452 0.017453 0.999949 0.999962 -0.0013%



Trig Deriv Proof Prove: Recall Derivate

Definition Use the Trig

Sum-Identity Apply TrigID to Limit

tdttd sincos

h

thttdtd

h

coscoslimcos0

sinsincoscoscos

h

ththth

ththh

cossinsincoscoslimcoscoslim00



Trig Deriv Proof Factor the

Limit argument By Limit Properties (c.f. §1.5)

Now Two Limits whose Proof is Beyond the Scope of MTH16:

h

hthth

sinsin1coscoslim0

hht

hht

hhtht

hhh

sinlimsin1coslimcossinsin1coscoslim000

1sinlim&01coslim00

zz

zz

zz



Trig Deriv Proof Using these Limits

Then Finally

1sin0cossinlimsin1coslimcos00

tthht

hht

hh

1sinsinsin1coscoslimcos0

th

hthttdtd

h

ttdtd sincos

Q.E.D.



Example CoSine Derivative Find:

SOLUTION: Use the Product Rule

xedxdxw x cos

xedxdxw x cos

xdxdexe

dxd xx coscos

xexe xx sincos

xxe x sincos



Example Maximizing Microbes An approximate Math Model for the

population of microbes present at temperature T:

• 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?

C50C 0for30

sin08.03)(

TTTTP



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.



Example Maximizing Microbes Taking the Derivatives

30sin08.032

2 TTdTd

dTd

dTPd

3030cos08.00 T

dTd

303030sin T

30sin

30

2 T



Example Maximizing Microbes Set to Zero the 2nd Derivative

The above eqn has infinitely many solutions, but recall that the T-domain Restriction: [0,50].

0max

2

2

T

dTPd

03030

sin2

max

T

030

sin max

T



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

• Where k is any Integer Solving for T find

kTkT 230

or230

kkTkkT 21301

213030

or602

230



Example Maximizing Microbes Then the two branches of solutions in

terms of T: 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

kTkT 6030or60



Example Maximizing Microbes Tabulating the Results

The only negative dP/dT is at T = 30, which then corresponds to the minimum

Then the microbial population at 30 °C:

3030cos08.0

T

dTdP

T 0 30 50dP/dT 0.185 -0.025 0.132

esMegaMicrob 4.530

)30(sin3008.0330

P



Derivatives for tan and sec For independent variable t measured in

Radians

Use the ChainRule when the tan/sec arguments are a function of t, u(t)

tttdtdtt

dtd sectansecsectan 2

dtduuuu

dtd

dtduuu

dtd

tansecsecsectan 2



tan Trig Deriv Proof Prove: Use the

• Tan definition: • Quotient Rule:• Previously Proved Trig Derivs:

Then

tdttd 2sectan

ttdtdtt

dtd sincoscossin

ttt cossintan

xg

dxdgxf

dxdfxg

xgxf

dxd

2

tt

dtdt

dtd

cossintan



tan Trig Deriv Proof Using the Quotient Rule and Chain Rule

Or

Using another Trig ID → Find

t

tttttt

dtdt

dtd

2cossinsincoscos

cossintan

1sincos 22

tttt

dtd

2

22

cossincostan

ttttt

tttdtd 22

2

22

22

secseccos

1cos

1cos

sincostan



Example tan derivative Find

tdtdfand

tttf

dtdf

1tanfor 2

2

22

22

1

tan2tantan21

t

tttdtdtt

dtdf

22

222

1tan2sectan21

tttttt

dtdf

0

1tan2sectan21

22

222

tdtdf



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?



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

ThentheDiagram:



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)

x

h



Example Related Rate & trig Use implicit differentiation to take

derivatives of both sides with respect to time, noting that h is constant:

xdtdh

xh

dtd

dtd

xh

dtd 1tantan

dtdxxh

dtdx

xdxdh

dtd

dtd

dd

22 1sectan

dtdx

xh

dtd

22sec



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

ft 216mi 1

ft 5280sec 3600

hr 1mph 10sec 12ft 40 x



Example Related Rate & trig Use x = 216ft after to find sec2θ after

the 12 second Flite Time:

Now use the Pythagorean identity relating tan and sec

Thus

725

72353

21615tantan

xh

2sec1725

069.1sec 2



Example Related Rate & Trig Now combine all of the values into the

implicit differentiation equation:

After 15 seconds, the angle of inclination to the bird decreases at about 1.59 degrees per second.

dtdx

xh

dtd

22sec

)67.14(21615069.1 2

dtd

secdeg

secrad 59.1 0044.0

dtd



WhiteBoard Work Problems From §8.2

• P8.2-50 → RowBoat Rope Reel-In

• P8.2-57 →HarmonicMotion

http://4.bp.blogspot.com/-iBniwjJhbW4/Te4Wl7OXmrI/AAAAAAAADck/dyCaZMRwWdM/s1600/rowboat.jpg

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=EtWLo1BOpdxt5M&tbnid=6Zpek5smuPKV3M:&ved=0CAUQjRw&url=http://web.ncf.ca/ch865/englishdescr/SHM1.html&ei=pIwSU-2JO5KBogTvr4CwDQ&bvm=bv.62286460,d.cGU&psig=AFQjCNE31Vb6z1ILhJMq9wzU-Bl49aNaCA&ust=1393810760874708



All Done for Today

Trig DerivChainRule

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=36QefxUlMAjl9M&tbnid=I89vUC-fN_VV9M:&ved=0CAUQjRw&url=http://www.artofmanliness.com/2012/11/19/how-throw-a-spiral-football/&ei=Io7VUrvACY-hogSr_IKoDA&bvm=bv.59378465,d.cGU&psig=AFQjCNFpdAGRelnFsLiPrVTkxg_YeEefqw&ust=1389813543017809



Bruce Mayer, PELicensed Electrical & Mechanical Engineer

[email protected]

Chabot Mathematics

Appendix

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Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege

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