1. §4.2 Mean value theorem(MVT)
Main significance of MVT is:
Obtaining the information on f(x) using
the information from f (x).
Topics:
Rolle’s Theorem
The mean value theorem
The 0-derivative theorem (Consequences of MVT) 1

2. Rolle’s Theorem 2

3. (ii) There may be more than one c such that f (c) = 0.
Comment:
(i) RT guarantees there exists c (a,b) such that the tangent line at c is horizontal, or the tangent line // the line through two endpoints: (a,f(a)) and (b,f(b)).
(ii) There may be more than one c such that f (c) = 0.
(iii) RT is the key to the proof of MVT. 3

4. (this happens when the object turns around, or changes direction).
(iv) If f stands for the position function, then Rolle’s theorem can be interpreted as follows:
If an object goes from point A at time a and comes back to A at time b (i.e.. f(a) = f(b)),
then: there exists a time c(a,b) such that the velocity of the object at c is 0 (i.e., f (c)=0)
(this happens when the object turns around, or changes direction). 4

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9. The mean value theorem 9

10. (i) If f stands for the position function,
Comment:
(i) If f stands for the position function,
MVT says that: “ If you averaged 100km/h during your trip, you must have hit exactly 100km/h at one instant during this trip”.
(ii) MVT also says that: “ If the slope of the secant line through (a, f(a)) and (b,f(b)) is m, then there exists c(a,b) such that the tangent line to f at (c,f(c)) has the same slope m”. 10

11. (iii) RT is the MVT in the special case where f(a) = f(b).
(iv) Main significance of MVT is to obtain the information on f from the information on f .
For example,
If f (x)>0 on (a,b), then f is increasing on (a,b).
If f (x)<0 on (a,b),then f is decreasing on (a,b).
(v) MVT asserts the existence of c, but does not tell us how to find it. 11

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18. The 0-derivative theorem Consequences of the MVT:
0-Derivative Theorem: If f (x) = 0 on (a,b), then f must be constant on (a,b). (Read page 287 in the textbook for its proof)
Constant-difference Theorem (CDT):
If f (x) = g(x) on (a,b), then f – g must be constant on (a,b). (Read page 288 in the textbook for its proof)
Comment: (i) These simple results are often used in Math 116 (Integral calculus).
(ii) CDT says that: two functions with equal derivatives on an open interval are the same up to a constant. 18

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