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Current time:0:00Total duration:3:46

Justification with the intermediate value theorem: equation

AP.CALC:
FUN‑1 (EU)
,
FUN‑1.A (LO)
,
FUN‑1.A.1 (EK)

Video transcript

let G of X equal 1 over X can we use the intermediate value theorem to say that there is a value C such that G of C is equal to 0 and negative 1 is less than or equal to C is less than or equal to 1 if so write a justification so in order to even use the intermediate value theorem you have to be continuous over the interval that you care about and this interval that we care about is from x equals negative 1 to 1 and 1 over X is not continuous over that interval it is not defined when X is equal to 0 and so we could say no because because G of X not defined not defined or I could say let me just say not continuous it's also not defined on every point of the interval but let's say not continuous over the closed interval from negative 1 to 1 and we can even put parentheses not defined not defined at X is equal to 0 all right now let's start asked the second question can we use the intermediate value theorem to say that the equation G of X is equal to 3/4 has a solution where 1 is less than or equal to X is less than or equal to 2 if so write a justification alright so first let's look at the interval if we're thinking about the interval from 1 to 2 well yeah our function is going to be continuous over that interval so we could say G of X is continuous is continuous on the closed interval from 1 to 2 and if you wanted to put more justification there you could say G defined defined for all real numbers real numbers such that X does not equal 0 X does not equal 0 right G of X defined for all real numbers such that X does not equal to 0 and you could say rational functions like 1 over X our continuous our continuous at all points in their domains at all points in their domain that's going really establishing that G of X is continuous on that interval and then we want to see what values does G take over at or what values does G take on at the endpoints or actually these are the endpoints that we're looking at right over here G of 1 is going to be equal to 1 over 1 is 1 and G of 2 is going to be 1 over 2 is equal to 1 over 2 so 3/4 is between is between G of 1 and G of 2 so by the intermediate value theorem there must be an X there must be and X that is that is in the interval from where he's talking with the interval from 1 to 2 1 to 2 such that such that G of X is equal to 3/4 and so yes we can find we can use the intermediate value theorem to say that the equation G of X is equal 3/4 has a solution and we are done
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