Continuously Differentiable Piecewise Function?

In summary, the given piecewise polynomial function is a parabola that turns into a line, with no gaps or corners. The limit of the function as x approaches 1 is 2, and the limit of its derivative as x approaches 1 is also 2. This implies that the function is continuously differentiable at x = 1. However, a calculus teacher may have mistakenly deducted points for lack of differentiability, not taking into account the continuity at the point. This is a common mistake for piecewise functions, as it is important to check for both convergence of slopes and continuity to determine differentiability.
  • #1
peterk
2
0
Here is a piecewise polynomial function:

f(x) = x^2 + 1 if x <= 1
f(x) = 2x if x > 1

I need to prove that this function is differentiable at x = 1?

It's a parabola that turns into a line. It doesn't have any gaps or corners. The limit of f(x) as x approaches 1 is 2, and the limit of f'(x) as x approaches 1 is 2.

This was a problem on a test, but my calculus teacher took points off because she says that the function is not differentiable at x = 1.

Thanks in advance!
 
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  • #2
Does [itex]\lim _{h \to 0}\frac{f(1+h) - 2}{h}[/itex] exist? I.e. do the following two exist and are they equal?

[tex]\lim _{h \to 0^+}\frac{f(1+h) - 2}{h}\mbox{, and }\lim _{h \to 0^-}\frac{f(1+h) - 2}{h}[/tex]
 
  • #3
Looks like your teacher is wrong, the function is continuously differentiable.
 
  • #4
Yes if we look at the limit of the two different slopes as they approach this point they will converge at 1 giving slope 2. However, if you were to graph the function at this point and or checked for continuity you would see a problem.

Lim fx from the right does not equal Lim fx from the left heading toward point 1.

lim x^2 evaluated at 1 = 1 and lim 2x evaluated at 1 = 2. This tells you that there would be a big jump at this point and would not be continious as the function output would not agree from the two sides.

Even though you should use lim [f(x+h) - F(x)]/h with h->0 to check that a function will be differentiable, you must realize this will only tell us if the two slopes will congerve to the same number at the given point. However, we could have a big jump in a graph in respect to height and still have the same slope for that point. The jump implies disscontinious and thus we fail to meet differentiability. We need the same slope and continuity to imply differentiability, as differentiability a implies the function has same slope and height at the given point in question.

If you were to search google you would find that a lot of people neglect to mention this important point for piecewise functions.

Hope that helps,
Kyle Rupps
 
  • #5
rupps98 said:
Yes if we look at the limit of the two different slopes as they approach this point they will converge at 1 giving slope 2. However, if you were to graph the function at this point and or checked for continuity you would see a problem.

Lim fx from the right does not equal Lim fx from the left heading toward point 1.

lim x^2 evaluated at 1 = 1 and lim 2x evaluated at 1 = 2. This tells you that there would be a big jump at this point and would not be continious as the function output would not agree from the two sides.

Even though you should use lim [f(x+h) - F(x)]/h with h->0 to check that a function will be differentiable, you must realize this will only tell us if the two slopes will congerve to the same number at the given point. However, we could have a big jump in a graph in respect to height and still have the same slope for that point. The jump implies disscontinious and thus we fail to meet differentiability. We need the same slope and continuity to imply differentiability, as differentiability a implies the function has same slope and height at the given point in question.

If you were to search google you would find that a lot of people neglect to mention this important point for piecewise functions.

Hope that helps,
Kyle Rupps

er, no, the two one-sided limits at x= 1 are equal - they both equal 2. there is no jump at x = 1.
 
  • #6
statdad said:
er, no, the two one-sided limits at x= 1 are equal - they both equal 2. there is no jump at x = 1.

Yes sorry. I thought the first function was just x^2 and not x^2 + 1. In the x^2 case they would not be continious but would satisfy converging slopes. In the x^2 + 1 case both the slopes converge and the point is continious. My bad, I will try to keep my glasses on rather than off in the future. Looks like that teacher is wrong after all :)

Kyle Rupps
 

1. What is a continuously differentiable piecewise function?

A continuously differentiable piecewise function is a mathematical function that is defined by multiple smaller functions, each of which is continuous and differentiable (meaning it has a well-defined derivative) on a specific interval. These smaller functions are "pieced together" to form a larger function that is continuous and differentiable on its entire domain. This type of function is commonly used in many areas of mathematics, such as calculus and differential equations.

2. How is a continuously differentiable piecewise function different from a regular piecewise function?

A regular piecewise function is defined by multiple smaller functions that are only required to be continuous on their respective intervals, but not necessarily differentiable. This means that the derivative of a regular piecewise function may not exist at certain points where the smaller functions are "joined" together. On the other hand, a continuously differentiable piecewise function requires that the smaller functions be not only continuous but also differentiable on their intervals, ensuring that the larger function is differentiable on its entire domain.

3. What is the significance of a continuously differentiable piecewise function?

Continuously differentiable piecewise functions are often used in mathematical modeling to represent real-world phenomena. By using multiple smaller functions, each of which can accurately represent a specific behavior, a continuously differentiable piecewise function can provide a more precise and realistic representation of a system or process. They are also useful in solving differential equations, as they allow for the use of different functions to describe different behaviors of a system.

4. Can a continuously differentiable piecewise function be non-differentiable at any point?

No, a continuously differentiable piecewise function must be differentiable on its entire domain, which means it cannot have any points where the derivative does not exist. However, the derivative of a continuously differentiable piecewise function may not be continuous at the points where the smaller functions are "joined" together.

5. How do you determine the derivative of a continuously differentiable piecewise function?

To find the derivative of a continuously differentiable piecewise function, you first need to determine the derivative of each of the smaller functions on their respective intervals. Then, you can use these derivatives to piece together a larger function that represents the derivative of the continuously differentiable piecewise function. This process may involve using different derivatives for different intervals and applying the appropriate rules of differentiation, such as the chain rule and product rule.

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