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

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In summary, the function f(x) is defined as 2x+1 for values of x less than or equal to 2, and .5x + k for values of x greater than 2. To find the value of k that would make f(x) continuous at x=2, one must consider both the right and left hand limits. For the function to be continuous, the limit as x approaches 2 from the left must be equal to the limit as x approaches 2 from the right. By plugging in 2 for x in the first equation, we get 5, which must also be the value of the second equation at x=2. Thus, k must equal 4.5 for f(x)
  • #1
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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  • #2
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?
 
  • #3
and remember to think about both right and left hand limits in defining continutity.
 
  • #4
And remember to make the distinction between right and left when discussing differentiability...

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

Daniel.
 

1. How do I find the value of 'k' for a continuous function at x=2?

To find the value of 'k' for a continuous function at x=2, you need to use the limit definition of continuity. Set up the limit as x approaches 2 for the given function and solve for 'k' by equating it to the value of the function at x=2. Once you have solved for 'k', check if the function is continuous at x=2 by plugging in the value of 'k' and verifying if the limit and function value match.

2. What is the significance of finding 'k' for a continuous function at x=2?

Finding 'k' for a continuous function at x=2 helps determine the exact value of the function at that particular point. It also helps in understanding the behavior of the function around the point x=2 and whether or not the function is continuous at that point.

3. Can I use any method to find 'k' for a continuous function at x=2?

No, you need to use the limit definition of continuity to find the value of 'k' for a continuous function at x=2. Other methods like the intermediate value theorem or graphical analysis cannot be used to find 'k' at a specific point.

4. Is it possible for a continuous function to have multiple values of 'k' at x=2?

Yes, it is possible for a continuous function to have multiple values of 'k' at x=2. This could happen if the function has multiple branches or is defined differently on either side of x=2. In such cases, both values of 'k' will satisfy the limit definition of continuity.

5. Can 'k' be a complex number for a continuous function at x=2?

Yes, 'k' can be a complex number for a continuous function at x=2. This is because continuity is defined for all real and complex numbers. However, it is important to check if the function is also defined for complex numbers at x=2 before considering a complex value for 'k'.

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