Finding 'k' for Continuous f(x) at x=2

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Discussion Overview

The discussion revolves around determining the value of 'k' for the piecewise function f(x) to ensure continuity at x=2, and subsequently assessing its differentiability at that point. The scope includes conceptual understanding of continuity and differentiability in the context of piecewise functions.

Discussion Character

  • Exploratory, Conceptual clarification, Homework-related

Main Points Raised

  • One participant suggests starting with the definitions of continuity to understand what is required for f(x) to be continuous at x=2.
  • Another participant emphasizes the importance of considering both right-hand and left-hand limits when defining continuity.
  • A further contribution highlights the need to distinguish between right and left limits in the context of differentiability.
  • There is a hint that the original poster has enough information to proceed with the problem.

Areas of Agreement / Disagreement

Participants generally agree on the importance of limits in discussing continuity and differentiability, but there is no consensus on the specific value of 'k' or the steps to find it.

Contextual Notes

Participants have not explicitly stated all assumptions or definitions related to continuity and differentiability, which may affect the clarity of the discussion.

Who May Find This Useful

This discussion may be useful for students learning about piecewise functions, continuity, and differentiability in calculus.

ashleyk
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Let f(x) be a function defined by:

f(x)= { 2x+1 for x (less than or equal to) 2
.5x + k for x (greater than) 2

A) For what value of 'k' will f(x) be continuous at x=2? Justify your answer.
B) Using the value of 'k' found in part A, determine whether f(x) is differentiable at x=2.


Any help on where to get started would be great. I think I have to plug in the 2 in the first equation...but I'm lost on what to do. Thanks.
 
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The definitions are usually a good place to start when you don't know what to do. What does it mean for this particular function to be continuous at 2?
 
and remember to think about both right and left hand limits in defining continutity.
 
And remember to make the distinction between right and left when discussing differentiability...

I think you got enough clues...:wink:

Daniel.
 

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