Where are the functions f1(x)=|sin(x)| and f2(x) = sin(|x|) differentiable? Use n as an arbitrary integer constant.)
f1(x) = |sin(x)| is differentiable for all x ≠______________
f2(x) = sin(|x|) is differentiable for all x ≠______________
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f1(x) is not at n*pie
f2(x) is not at 0
Ah, the wacky world of differentiability! Now, let me put on my clown nose and take a crack at these functions for you!
For f1(x) = |sin(x)|, this function is differentiable for all x ≠ the times when my jokes fall flat and I can't make anyone laugh! No, just kidding! It's actually differentiable for all x ≠ nπ, where n is an arbitrary integer constant. So, if you're avoiding those funny angles where sin(x) is 0, you're good to go!
As for f2(x) = sin(|x|), this one is differentiable for all x ≠ the times when someone tries to tickle a clown's funny bone and they hit a sore spot! Okay, I'll stop clowning around. It's differentiable for all x ≠ 0, since that's where the absolute value function has a sharp bend.
So, there you have it! Remember, differentiability can be a bit of a clown act, but as long as you steer clear of those special x-values, these functions will put on a smooth performance! Keep on laughing (and differentiating)!
To determine where the functions are differentiable, we need to check for the points where the functions are not differentiable.
1. For f1(x) = |sin(x)|, notice that the function is the absolute value of sin(x). The absolute value function |x| is not differentiable at the points where x = 0. Therefore, f1(x) is not differentiable at x = __0__.
2. For f2(x) = sin(|x|), we again have an absolute value function inside the sine function. The absolute value function |x| is not differentiable at the points where x = 0. Therefore, f2(x) is not differentiable at x = __0__.
In summary:
- f1(x) = |sin(x)| is differentiable for all x ≠ 0.
- f2(x) = sin(|x|) is differentiable for all x ≠ 0.
To determine where the functions f1(x) = |sin(x)| and f2(x) = sin(|x|) are differentiable, we need to identify the values of x where the derivatives exist.
First, let's consider f1(x) = |sin(x)|. The absolute value function |x| is differentiable for all x except at the point where x = 0. However, the sine function sin(x) is differentiable for all real values of x. Thus, the function f1(x) = |sin(x)| is differentiable for all x ≠ 0.
Next, let's examine f2(x) = sin(|x|). The absolute value function |x| is not differentiable at x = 0, as it has a sharp corner at that point. However, the sine function sin(x) is differentiable for all real values of x. Therefore, the function f2(x) = sin(|x|) is differentiable for all x ≠ 0.
In summary:
- The function f1(x) = |sin(x)| is differentiable for all x ≠ 0.
- The function f2(x) = sin(|x|) is differentiable for all x ≠ 0.
Please note that the integer constant "n" mentioned in the question does not affect the differentiability of the given functions.