In the last section, we discussed how to differentiate implicitly. In this section, we are going to discuss some common theorems from differential calculus including the sandwich theorem, maximum-minimum theorem, intermediate-value theorem, and the mean value theorem.

Let's start with the sandwich theorem. This theorem also goes by the name of the pinching theorem. The **sandwich theorem** starts with some assumptions and those are must be greater than 0, and for all x in the form 0 < |x - c| < , g(x) f(x) h(x). Furthermore, and . What the assumptions are saying is we are considering all x such that x < c + and x > c - , and for those x values, g(x) f(x) h(x). Finally if the two above listed limits hold then, . The result says that the limit of f(x) goes to L if the limits of g(x) and h(x) also go to L.

The **intermediate-value theorem** also has some assumptions. Those are f(x) is continuous on [a,c], and B is a number between f(a) and f(c). If those two criteria are true then there also exists a number b such that f(b)=B. What the theorem is saying is that f takes on all values between f(a) and f(c), and for those values there exists a number in the domain of f called b which is between a and c. We can make this number b take on all the any value in the range of f which is between f(a) and f(c). Well, it may not take on any value in the range just a subset of it because f may be defined elsewhere other than the interval [a,c].

The **maximum-minumum theorem** is a lot simpler. The assumptions are also fewer, and that is f is continuous on [a,b]. If so, f will have a maximum and minimum on [a,b]. Not much to this theorem.

The **mean value theorem**'s assumptions are f is differentiable on (a,c) and continuous on [a,c]. If so, then there is a number b such that . You should recognize the fraction as the slope of the line between (a,f(a)) and (c,f(c)). The theorem is saying the derivative of f will take on the value of the slope of the line between the points (a,f(a)) and (c,f(c)).