Derivative of Piece-Wise Function

In summary, the first question asks if the function is continuous at x = 1, and the second asks if the derivative is differentiable at x = 1. Neither of these questions have an answer that is clear from the given information.
  • #1
Jeremy
28
0
I am doing my calculus homework and two problems are holding me up.

The first says:

Using one-sided derivatives, show that the function f(x) =

x^3, x_<_1
3x, x>1

does not have a derivative at x=1

Now it is painfully obvious that the function is not continuous at x=1. however, i am not entirely sure that is the answer that the book wants. the slopes from both sides seem to be 3, so how can it not have a derivative (aside from the continuity issue)? A similar (but with continuity) problem follows:



Secondly (i will try to answer the ones i have gotten so far):
let f(x) =

x^2, x_<_1
2x, x>1

a) find f'(x) for x<1......i think this is 2x
b) find f'(x) for x>1......2
c) find lim (x-->1-) f'(x)....2
d) find lim (x-->1+) f'(x)....2
e) does lim (x-->1) f'(x) exist? explain
f) use the def to find the left-hand derivative of f at x=1 if it exists...same as (c)=2
g) use the def...right-hand deriv...same as (d)=2
h) does f'(1) exist? explain

according to the rules of derivatives, if the left and right-hand derivatives are the same at a point, then that point has a derivative (assuming continuous). however, it seems to me that, at x=1, there would be a bit of a "jagged edge," somewhat like an absolute value point. therefore, how could a derivative be found?

thanks in advance. feel free to tell me i am horribly wrong in all aspects of my answer.
 
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  • #2
For the first question, an answer of it isn't differentiable because its not continuous seems to be the answer that would be appropriate to me.

2nd question, yes, it should be differentiable at f'(1). Because the slopes (derivatives) are the same at point x = 1, then it shouldn't have a "jagged egde". If it were graphed it should appear smooth, since the one sided limits of both the functions and derivatives have the same value at x=1.
 
  • #3
sounds good

thanks a bunch
 
  • #4
If there is no derivative at x = 1, which there isn't, then there can be no further derivatives at x = 1. There will be a "hole" in the graph of the derivative there. Therefore, the derivative is not differentiable at x = 1, even if both the left-hand and the right-hand derivatives exist and are the same.
 

What is a piece-wise function?

A piece-wise function is a mathematical function that is defined by different rules or formulas on different intervals of its domain.

How is the derivative of a piece-wise function calculated?

The derivative of a piece-wise function is calculated by taking the derivative of each piece separately and then combining them using the rules of differentiation.

Can a piece-wise function have a derivative at all points?

No, a piece-wise function may not have a derivative at all points if it is not continuous or differentiable at certain points.

How can the derivative of a piece-wise function be used in real life applications?

The derivative of a piece-wise function can be used to analyze and model real-life situations that involve changing rates, such as in physics, economics, and engineering.

Is it possible for a piece-wise function to have a discontinuous derivative?

Yes, a piece-wise function can have a discontinuous derivative if the pieces have different slopes at the point where they meet.

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