The Mean Value Theorem

Tuesday, 27 March 2018

9:05 PM

This theorem helps in:

  • Identifying whether a function is increasing or decreasing
  • Identifying stationary points
  • Determining the limits of  and

 

Theorem

 

 

For any suitable function, there must be a point at which the instantaneous rate of change is equal to the average (mean) rate of change over a whole interval

 

Example

With x-values ,

 

Machine generated alternative text: Example Apply the M VT to the function f defined by f (x) — x and a the value(s) of c which satisfy the conclusion of the MVT. Solution. Observe that f is continuous and differentiable everywhere. 4. Find 5x4 and, by M VT, there exist at least one c e (—1, 4) such that Now, f' (x) This implies 45 - (-1)5 1025 oc 5 5 We finally obtain c— ±2.53. Since c €- (—1, 4), we take c — 2.53 Remark. One can have more than one solution. E.g., change a to —2 and b to 3 and see what you get. In this case it was not hard to calculate the specific value(s) of c. But the real point of the M VT is that it guarantees the existence of c, even if the necessary calculations are difficult or im ossible. Alina Ostafe (UNSW) Chapter 5: The Mean Value The«em 7/33 Semesær 1 2018

 

 

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