Solving f(x): Finding a & b for Continuity/Differentiability

  • Thread starter Fusilli_Jerry89
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In summary, the function f(x) is defined piecewise, with different expressions for x values less than 2 and greater than or equal to 2. The relationship between a and b is given by 2a+b=2. To find unique values of a and b that make f both continuous and differentiable, one must consider both the continuity and differentiability conditions. The derivative of x+2 at x=2 is 1, while the derivative of ax^2+bx at x=2 is 2a+b. These two values must be equal for f to be differentiable at x=2. However, even if a function is continuous at a point, it does not necessarily mean it is differentiable at that point
  • #1
Fusilli_Jerry89
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Let f be the function defined as:

f(x) = { x+2 , x < 2
{ ax^2+bx, x >(or equal to) 2

a) what is the relationship between a and b?
I got 2a+b=2
b) find the unique values of a and b that will make f both continuous and differentiable.

i substitued (2-2a) into b and got down to y=ax^2+2x-2ax, but now what?
 
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  • #2
I moved this thread to the homework forums. Jerry -- you've used the homework forums before, so I know that you know the rules. This is the 2nd thread of yours that I've had to move tonight. So even though you showed some work in this post (none in the previous post), I'm going to issue some warning points. Please keep homework posts in the HW forums.
 
  • #3
Well, except with the new PF forum changes, I don't see the usual WARN button. Okay, you skated this time -- next time I look harder for the new WARN button placement...
 
  • #4
Well so far, you considered the continuity condition, but what about the differentiability condition?
 
  • #5
Fusilli_Jerry89 said:
Let f be the function defined as:

f(x) = { x+2 , x < 2
{ ax^2+bx, x >(or equal to) 2

a) what is the relationship between a and b?
I got 2a+b=2
b) find the unique values of a and b that will make f both continuous and differentiable.

i substitued (2-2a) into b and got down to y=ax^2+2x-2ax, but now what?

Are you sure you have quoted the problem correctly?
If
"Let f be the function defined as:

f(x) = { x+2 , x < 2
{ ax^2+bx, x >(or equal to) 2"
there doesn't have to be ANY particular relationship between a and b!
HOW did you get 2a+ b= 2?

IF f is continuous at x= 2, then f(2)= 2+ 2= 4= 4a+ 2b so you get 2a+ b= 2 but that isn't given until part b.

What is the derivative of x+ 2 at x= 2? What is the derivative of ax2+ bx at x= 2?


(Note: If a function is differentiable at x= 2, it is not necessarily the case that the derivative is continuous at x= 2. However, any derivative must satisfy the a "intermediate value property" so if the two "one-sided" limits exist, they must be the same. THAT means that the two values above must be the same.)

By the way, the word is "PIECEwise".
 
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  • #6
Also keep in mind, that a continuous function is not necessarily differentiable at a point x. Case in point: f(x) = | x | at x = 0

Here we see that f(x) satisfies the 3 conditions necessary for continuity:

1. f(0) is defined

2. [tex]\lim_{x\rightarrow 0} | x |[/tex] exists

3. [tex]\lim_{x\rightarrow 0} = f(0)[/tex]

Yet, f(x) is not differentiable at x = 0 (differentiable everywhere else).

If f is differentiable at a, then f is continuous at a. Unfortunately, this does not mean if f is continuous at a, f is differentiable at a. As seen above.
 
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Related to Solving f(x): Finding a & b for Continuity/Differentiability

1. What is the definition of continuity in terms of a function?

Continuity of a function f(x) at a point x = a means that the limit of the function as x approaches a from both the left and right sides exists and is equal to the value of the function at that point. In other words, the function is unbroken at x = a.

2. How do you determine if a function is continuous at a given point?

To determine if a function is continuous at a point x = a, you must evaluate the limit of the function as x approaches a from both the left and right sides. If the limit exists and is equal to the value of the function at x = a, then the function is continuous at that point.

3. What is the definition of differentiability?

Differentiability of a function f(x) at a point x = a means that the derivative of the function exists at that point. In other words, the function has a well-defined slope at x = a.

4. How do you find the values of a and b for a function to be both continuous and differentiable?

To find the values of a and b for a function to be both continuous and differentiable, you must set up a system of equations using the definition of continuity and differentiability. Then, solve for the values of a and b that satisfy both equations.

5. Can a function be continuous but not differentiable at a point?

Yes, a function can be continuous but not differentiable at a point. This means that the function has a well-defined value at that point, but its derivative does not exist at that point. This often occurs when there is a sharp turn or corner in the graph of the function at that point.

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