Use the intermediate value theorem to show that there is a root of the given equation in the specified interval....tanx = 2x (0,1.4)?


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Oddman answered
The intermediate value theorem says that if the signs of f(a) and f(b) differ, there is a root of f between a and b.

You have the expression
  tan(x) = 2x
This can be formulated as a function f(x) whose root we are seeking.   f(x) = tan(x) - 2x
It is proposed that there is a root between x=0 and x=1.4. We can check the signs.
  F(0) = 0    (this is a root)
  f(1.4) = tan(1.4) - 2*1.4 = 5.7979 - 2.8000 = 2.9979
This check is inconclusive, as the given interval is an open interval that does not include x=0.
We can also check x=.7 (arbitrarily, the midpoint of the interval)
  f(.7) = tan(.7) - 2*.7 = .8423 - 1.4000 = -.5577

There is a change of sign in f(x) between x=0.7 and x=1.4. The intermediate value theorem tells us there is a root in the range (0.7, 1.4), a range that is included in the given range.

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