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Calculus/Max/Min and implicit differentiation

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Expert: Alon Mandes - 4/20/2009


Question
Hello, I am stuck on two problems and I was wondering if you could help me. I am recently getting back into calculus and some of my trig is rusty.

f(x)=sin^2x + cosx on the interval [-pie,pie]  




Also, d^2y/dx^2 if y-siny = x  

Answer
(1) Find Max/Min on the function f(x)=sin^2x + cosx :
   f'(x)=2sinxcosx-sinx.
   Extremum points are achieved when solving f'(x)=0 , thus
   2sinxcosx=sinx -> cosx=0.5 -> x1=pi/3 & x2=-pi/3 .
   To know if these points are max or min , we derive 2nd time:
   f''(x)=2cos^2x-2sin^2x-cosx
   f''(pi/3)=2*0.25-2*0.866-0.5=-1.732 < 0 Its max
   f''(-pi/3)=-2*0.25+2*0.866-0.5=0.732 > 0 Its min
(2) y-siny = x . Derive this implicit function :
   With implicit functions we derive both sides of the equation
   with respect to x. & for that dx/dx=1 & dy/dx=y' . So,
   dy/dx-dsiny/dx = dx/dx
   y'-cosy*y'=1 -> y'=1/(1-cosy) .
   Now let's derive 2nd time :
   dy'/dx-d[cosy*y']/dx=d1/dx   
   y''-cosy*y''+siny*y'=0
   y''(1-cosy)=-siny*y'
   y''=y'*siny/(1-cosy)
   y''=siny/(1-cosy)^2

Alon.

Calculus

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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Hi-Tech company : GSM4VOIP ; job possition : Algorythm developer.

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M.A in Mathematics & Bs.c in Electronics.

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