MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics §8.2 Trig Derivatives

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §8.1 → Trigonometric Functions  Any QUESTIONS About HomeWork §8.1 → HW

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 3 Bruce Mayer, PE Chabot College Mathematics §8.2 Learning Goals  Derive and use differentiation formulas for trigonometric functions  Study periodic rate and optimization problems using derivatives of trigonometric functions

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 4 Bruce Mayer, PE Chabot College Mathematics 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)

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 5 Bruce Mayer, PE Chabot College Mathematics Verify Trig Derivs  Use SpreadSheet to Check that

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 6 Bruce Mayer, PE Chabot College Mathematics Trig Deriv Proof  Prove:  Recall Derivate Definition  Use the Trig Sum-Identity  Apply TrigID to Limit

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 7 Bruce Mayer, PE Chabot College Mathematics 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:

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 8 Bruce Mayer, PE Chabot College Mathematics Trig Deriv Proof  Using these Limits  Then Finally

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 9 Bruce Mayer, PE Chabot College Mathematics Example  CoSine Derivative  Find:  SOLUTION: Use the Product Rule

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 10 Bruce Mayer, PE Chabot College Mathematics 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?

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 11 Bruce Mayer, PE Chabot College Mathematics 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 2 nd derivative and 2 nd derivative zeroes.

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 12 Bruce Mayer, PE Chabot College Mathematics Example  Maximizing Microbes  Taking the Derivatives

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 13 Bruce Mayer, PE Chabot College Mathematics Example  Maximizing Microbes  Set to Zero the 2 nd Derivative  The above eqn has infinitely many solutions, but recall that the T-domain Restriction: [0,50].

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 14 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 15 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 16 Bruce Mayer, PE Chabot College Mathematics 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: T03050 dP/dT

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 17 Bruce Mayer, PE Chabot College Mathematics 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)

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 18 Bruce Mayer, PE Chabot College Mathematics tan Trig Deriv Proof  Prove:  Use the Tan definition: Quotient Rule: Previously Proved Trig Derivs:  Then

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 19 Bruce Mayer, PE Chabot College Mathematics tan Trig Deriv Proof  Using the Quotient Rule and Chain Rule  Or  Using another Trig ID →  Find

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 20 Bruce Mayer, PE Chabot College Mathematics Example  tan derivative  Find

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 21 Bruce Mayer, PE Chabot College Mathematics 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?

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 22 Bruce Mayer, PE Chabot College Mathematics 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:

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 23 Bruce Mayer, PE Chabot College Mathematics 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)

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 24 Bruce Mayer, PE Chabot College Mathematics Example  Related Rate & trig  Use implicit differentiation to take derivatives of both sides with respect to time, noting that h is constant:

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 25 Bruce Mayer, PE Chabot College Mathematics 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

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 26 Bruce Mayer, PE Chabot College Mathematics Example  Related Rate & trig  Use x = 216ft after to find sec 2 θ after the 12 second Flite Time:  Now use the Pythagorean identity relating tan and sec  Thus

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 27 Bruce Mayer, PE Chabot College Mathematics 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.

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 28 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §8.2 P → RowBoat Rope Reel-In P → Harmonic Motion

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 29 Bruce Mayer, PE Chabot College Mathematics All Done for Today Trig Deriv Chain Rule

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 30 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer Chabot Mathematics Appendix –

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 31 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 32 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 33 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 34 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 35 Bruce Mayer, PE Chabot College Mathematics

MTH16_Lec-09_sec_7-6_Double_Integrals.pptx 36 Bruce Mayer, PE Chabot College Mathematics