Notes Nov 30, 2020 Vocabulary and Theorems



  • Mean Value Theorem 
  • Critical Number/Critical Point 
  • Extreme (Absolute/relative max/min)
  • Extreme Value Theorem (MVT)
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Mean Value Theorem (MVT)

Conditions: If 

     (1) f(x) is continuous on [a,b]

     (2) f(x) is differetiable on (a,b)

Then: there exists an x=c where  

      - f'(c) = f(b) - f(a)/ b-a 

      - Instantaneous rate of change = average rate of change

- Doesn't tell us what C is. It only tells us that there is at least one number C that will satisfy the conclusion of the theorem.

* if f'(x)=0 for all x in an interval (a,b), then on f(x) is constant on (a,b)

*f'(x) = g'(x) for all x in an interval (a,b) then is this interval we have f(x)=g(x)+c where C is some constant 

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Critical Number (Critical Point)

"C" (f(c) is definded) 

is an x-value on an interval where either 

      - f'(c) = 0 or 

      - f'(c) is undefined

we say that x=c is s critical point of the function f(x) if f(c) exists is either of the following are true. 

*We require that f(c) exists in order to actually be a critical point. 

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Extreme: Minimum or Maximum

relative (local) - x=c is a local min 

   if f(c) is lower than or equal to all points in its neighborhood 

   absolute (global)

 x=c is an absolute max/min if f(c) is (larger/smaller) then /or equal to) all points in the domain 

If x=c is a relative extreme, then, x=c is the critical number 

False: if x=c is a critical number, then x=c is a relative extreme 

*Extreme value: give the y-value or height 

(1) we say that f(x) has an absolute (or global) maximum at x=c if f(x)-< f(c) for every x in the domain we are working on 

(2) we say that f(x) has a relative (or local) maximum at x=c if f(x) -< f(c) for every x in some open interval around x=c 

(3) We say that f(x) has an absolute (or global) minimum at x=c if f(x)-> 

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Extreme Value Theorem

If f(x) is continuous and defined on [a,b]

Then there will exist at least one absolute max and min 

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