How do you use the Intermediate Value Theorem to show that the polynomial function # 2tanx - x - 1 = 0# has a zero in the interval [0, pi/4]?

Answer 1

#f(x)=2tan(x)-x-1# is continuous on the interval #[0,pi/4]#
and has a value #>0# and a value #<0#
therefore
by the Intermediate Value Theorem
it has ha value #=0#

#f(0) = 2tan(0)-0-1 = -1#
#f(pi/4)=2tan(pi/4)-pi/4-1# #color(white)("XXX")=2(1)-pi/4-1# #color(white)("XXX")=1-pi/4# (and since #pi/4<1#) #f(pi/4) > 0#
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Answer 2

To use the Intermediate Value Theorem to show that the polynomial function ( 2\tan(x) - x - 1 = 0 ) has a zero in the interval ([0, \frac{\pi}{4}]), we need to verify that the function changes sign over that interval.

Evaluate the function at the endpoints of the interval:

  1. At ( x = 0 ): ( 2\tan(0) - 0 - 1 = -1 )
  2. At ( x = \frac{\pi}{4} ): ( 2\tan\left(\frac{\pi}{4}\right) - \frac{\pi}{4} - 1 = 2 - \frac{\pi}{4} - 1 = \frac{7}{4} - \frac{\pi}{4} )

Since ( -1 < 0 < \frac{7}{4} - \frac{\pi}{4} ), and the function is continuous over the interval ([0, \frac{\pi}{4}]), by the Intermediate Value Theorem, there exists at least one value ( c ) in the interval ([0, \frac{\pi}{4}]) such that ( f(c) = 0 ), where ( f(x) = 2\tan(x) - x - 1 ). Therefore, the polynomial function has a zero in the interval ([0, \frac{\pi}{4}]).

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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