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Calculus AB: Applications of the Derivative

Problems for "The Mean Value Theorem"

The Mean Value Theorem

Using the First Derivative to Analyze Functions


In problems 1-3, for each of the following functions f defined on [a, b] find the c on [a, b] such that

f'(c) =    

Problem : 1) f (x) = x 2 - 4x on [2, 4]


f'(c) = = 2  
2c - 4 = 2  
c = 3  

Problem : 2) f (x) = sin(x) + cos(x) on [0, 4Π]


    f'(c) = = 0  
    cos(x) - sin(x) = 0  
    x = ,,, or  

Problem : 3) f (x) = on [1, 2]


f'(c)   =  
    = -  
-   = -  
c   = ±  

Problem : 4) On the interval [-5,5], there is no point at which the derivative of f (x) =|x| is equal to zero, even though f (- 5) = f (5) . Is this a contradiction of Rolle's theorem?

No, it isn't a contradiction, since this function is not differentiable on the entire interval (- 5, 5) .

Problem : Find the number c that satisfies Rolle's theorem for f (x) = sin(x) on the interval [0, Π] .

(sin(x))' = cos(x)
cos(x) = 0 at x =

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