§4.2 Mean value theorem(MVT)

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Presentation transcript:

§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

Rolle’s Theorem 2

(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

(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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The mean value theorem 9

(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

(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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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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