4 2 Mean value theoremMVT Main significance of

Comment: (i) RT guarantees there exists c (a, b) such that the tangent line

(iv) If f stands for the position function, then Rolle’s theorem can be interpreted

Comment: (i) If f stands for the position function, MVT says that: “ If

(iii) RT is the MVT in the special case where f(a) = f(b). (iv)

III. The 0 -derivative theorem Consequences of the MVT: 0 -Derivative Theorem: If f

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§ 4. 2 Mean value theorem(MVT) Main significance of MVT is: Obtaining the information on f(x) using the information from f (x). Topics: I. Rolle’s Theorem II. The mean value theorem III. The 0 -derivative theorem (Consequences of MVT) 1

I. Rolle’s Theorem 2

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

(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

II. The mean value theorem 9

Comment: (i) If f stands for the position function, MVT says that: “ If you averaged 100 km/h during your trip, you must have hit exactly 100 km/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

(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

III. 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