THE MEAN VALUE THEOREM PAGE

Mean Value Theorem Pt I

The mean value theorem  (MVT ) states that if a function f(x) is continuous on a closed interval [a,b] and differentiable on the open interval (a,b) then there is a c in the interval (a, b) where 
f'(c)=(f(b)-f(a))/(b-a). 
The two initial conditions or hypothesis for the mean value theorem is continuity on the closed interval and differntiability on the open interval. 
f'(c) can be veiwed as the slope of the tangent line to the curve of f at c where c is some point in the interval (a,b).
The ratio (f(b)-f(a))/(b-a). can be viewed as the slope of the secant line or the average rate of change on the interval [a,b]. So in essence the mean value theorem is stating that for a differentiable function that is continuous on the interval including the endpoints, there is a value in the interval where the slope of the tangent line is equal to the slope of the secant line. 


A special case of the mean value theorem is known as Rolle's Theoreom. In this case, the slope of the secant line is 0.  And that implies that there is a c where the slope of the tangent line is 0, a horizontal tangent line. 

VIDEO TUTORIAL
WORKED OUT EXAMPLES ON THE MEAN VALUE THEOREM

POWERPOINTS ON THE MEAN VALUE THEOREM

1.  Extreme Value Theorem Graphical
2.  Extreme Value Theorem Algebraic and Graphical
3.  Connecting f' and f" with the graph of f pt I
4 . Connecting f' and f" with the graph of f pt II
5.  Mean Value Theorem Pt I
6.  Mean Value Theorem Pt II
7.  Optimization and Modelling Part I Rectangular Box Square Cut
8.  Optimizaiton and Modelling  Part III Cylinder Inscribed in a cone
9.  Optimization Distance From a Point to a Curve
10.  Optimization and Modelling Part II Enclosed Cylindrical Can
11.   Optimization and Modelling Fencing Material
12.  Optimization and Modelling Fencing Material Two adjacent pens
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PRACTICE WORKSHEET WITH ANSWERS
AP Calculus Standards Covered in this Section
The student will define and apply the properties of limits of functions. Limits will be
evaluated graphically and algebraically. This will include
a) limits of a constant;
b) limits of a sum, product, and quotient;
c) one-sided limits; and
d) limits at infinity, infinite limits, and non-existent limits. *
*AP Calculus BC will include l’Hopital’s Rule, which will be used to find the limit of
functions whose limits yield the indeterminate forms: 0/0 and ∞/∞.
APC.3 The student will use limits to define continuity and determine where a function is
continuous or discontinuous. This will include
a) continuity in terms of limits;
b) continuity at a point and over a closed interval;
c) application of the Intermediate Value Theorem and the Extreme Value Theorem; and
d) geometric understanding and interpretation of continuity and discontinuity.
APC.4 The student will investigate asymptotic and unbounded behavior in functions. This will
include
a) describing and understanding asymptotes in terms of graphical behavior and limits
involving infinity; and
b) comparing relative magnitudes of functions and their rates of change.