The Intermediate Value Theorem:
If the function f(x) is continuous on the interval [a, b], then the value for f(c) must be in between f(a) and f(b).
How to use the Intermediate Value Theorem:
It is usually used for finding zeroes.
This can be done, by saying if f(a) and f(b) are opposite signs, AKA one is positive and one is negative, then there HAS to be a value c that is a zero.
The Mean Value Theorem:
Interval: [a, b]
AKA the derivate of point c is the same as the average rate of change of the interval (a, b)
If f(x) is continuous over [a, b] and differentiable over (a, b), THEN
Some c-value’s instantaneous rate of change is the same as the average rate of change of the interval.
The Extreme Value Theorem:
If the function f(x) is continuous on the interval [a, b], then there HAS to be a maximum value AND a minimum value.
YES, that can be the endpoints too, so check those too.
The Rolle’s Theorem:
Just the same thing as the mean value theorem, but ONLY for horizontal tangents, so kinda boring.
If f(a) = f(b), then f'(c) = 0, HAS to be true at some point c.
AKA if the average rate of change is 0, then the instantaneous rate of change somewhere needs to be 0 too.
The Squeeze Theorem:
If “f(x)” and “g(x)” are the SAME/EQUAL then h(x) is ALSO equal to f(x) and g(x).
ALL three values are the same.
How to use the Squeeze Theorem:
The graph has to be continuous.
- If you get a function, you have to ISOLATE the trigonometric part and the normal part.
- Then find the range of the trig function, AKA get the lower bound and the upper bound R=[l, u]
- Then multiply the upper bound and the lower bound with the normal part of the function.
- Then you have the two functions a and b on the inequality.
- Then evaluate the limit of a and b, and IF they are EQUAL, then c/the whole function is also that value.
Example of Squeeze Theorem: