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Differential Calculus

Blazing goIITian

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5 Sep 2008 16:45:16 IST

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functions (2 easy questions)

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$1)\mbox{Find a formula for a function g(x) satisfying the following conditions }-> \\ \\ a) domain = (- infinity , + infinity) \\ \\ b)range = [-2, 8] \\ \\ c) period = \pi \\ \\ d) g(2) = 3 \\ \\ \mbox{Please explain how to proceed in the above qn } \\ \\ 2)\mbox{A is a point on the circumference of a circle. Chords AB and AC divide the area } \\ \\ \mbox{of the circle into 3 equal parts .If the angle BAC is the root of the equation , f(x)=0, then find f(x)}$

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Comments (5)

Hari Shankar

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5 Sep 2008 18:04:59 IST

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1st one should remind you of SHM.

So, $f(x) = 3 \sin (2x - \phi) + 4 \cos (2x - \phi) + 3$ where $\phi = (\tan^{-1} \frac{4}{3} + 4)$ will serve you admirably well

RyuAmakusa

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5 Sep 2008 18:11:43 IST

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1) heas is what i did.....from the domain and range i thaught that i can use either sin or cos... say i use sin and the period is given to be pi so it cannot be just sinx so i took it to be sin2x......now range bebins from -2 so i took 2sin2x.....now my aim is to find another function such that it's domain is same and range is from [0,6],and the period is pi say i use 6|sin2x| or6| cos2x|...i should also remember that.....i need g(2)=3....so what i thaught was u make the first part ie 2sin2x as 2 sin(2(x-2)) so that when x=2 this becomes 0 and if i choose the other part as 3|cos2(x-2)| now this part will give me 3 when x=2 and it's range os from [0, 3] now all i have to do is add another part such that period is pi domain is satisfied and range is [0,3] and it should give me 0 for x=2 so u can choose that to be 3|sin2(x-2)| ....now add all the parts.....for explaining it is big but for doing it it is very simple....the ans is
2sin(2x-4) + 3|cos(2x-4)| + 3|sin(2x-4|...there can be some other functions also....

RyuAmakusa

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5 Sep 2008 19:35:58 IST

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oh i am sorry there is a simple sol in my method what i was thinking was we should not have any constants in f(x) if we can then it becomes simply 2sin(2x-4)+3|sin(2x-4)|+3 by the same method..

Hari Shankar

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6 Sep 2008 10:37:48 IST

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@ryuamakusa: with the function given by you, for what x is f(x) = -2?

abhishek sinha

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6 Sep 2008 12:02:01 IST

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2. is pretty simple !!

Let the angle be 2@ . Then from symmetry we may write that ,the diameter passing through A makes an equal angle @ with both the chords .

So we may write , 1/2r^2( pi-2@)-1/2r^2sin 2@=1/3pir^2

so we get the fn as ( putting 2@=x = angle BAC)

sin x +x/2 -pi/6 = f(x) =0

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