I'm having a tough time figuring out this problem...
S(x) = bracket (piecewise function)
a + b arcsin*(tan x/tan 66) for 0 ≤ x < 66
24 for 66 ≤ x ≤ 90.
Is the function differentiable? Why or why not?
Could someone please help me? My teacher told me the function was not differentiable, but I need to prove why it is not. Thank you!
Oh and in addition,
a= 12,
b= 2/15.
Thanks.
"S(x) = bracket (piecewise function)
a + b arcsin*(tan x/tan 66) for 0 ≤ x < 66
24 for 66 ≤ x ≤ 90. Given a=12, b=2/15.
Is the function differentiable? Why or why not?"
I suppose this is the number of hours of sunshine where x is the latitude in degrees during solstice (summer or winter). I have not checked the validity of the equation.
Yes, your teacher is right, the function is not differentiable, but only at x=66.
Recall that the conditions for differentiability are:
1. the function must be continuous. This condition is satisfied throughout the domain 0≤x≤90, notably even at x≥66.
2. The derivative must exist at all points of the domain [0,90]. This is true.
For [0,66),
f(x)=a + b arcsin*(tan x/tan 66), and
f'(x)=(2*sec((π*x)/180)^2)/(15*tan((11*π)/30)*sqrt(1-tan((π*x)/180)^2/tan((11*π)/30)^2)) ...(1)
exists.
For [66,90], f'(x)=0.
3. For every point c on the domain,
the derivative from the left must equal the derivative from the right, and equal to the derivative at the point c, or
f'(c-)=f'(c)=f'(c+)
This condition is not satisfied at c=66.
To calculate the derivative from the left, we resort to equation (1), which gives the derivative as x->66- to be ∞. As an example, the derivative at 65.999 evaluates to 37.
The derivative at c, and to the right of c is 0.
Therefore the function is not differentiable at x=66.
See:
http://img193.imageshack.us/img193/7569/1289686601.png
Ohh, I understand now! Thank you SO much. Your explanation was clear as day, and I cannot tell you just how much I appreciate it.
To determine if the function S(x) is differentiable, we need to check if it satisfies the conditions required for differentiability.
Differentiability implies that the function must be continuous and have a derivative at every point within its domain.
First, let's consider the continuity of S(x). For the function to be continuous at x = 66, both the left-hand limit and the right-hand limit must exist and be equal.
Limit as x approaches 66 from the left (x → 66^-):
S(x) = a + b arcsin(tan(x)/tan(66))
Since 0 ≤ x < 66, we can substitute x = 66 to find the left-hand limit:
lim (x → 66^-) S(x) = a + b arcsin(tan(66)/tan(66)) = a + b arcsin(1) = a + b(π/2) = a + bπ/2
Now let's consider the function value at x = 66:
S(66) = 24
To ensure continuity, both the left-hand limit and the function value at x = 66 must be the same:
a + bπ/2 = 24
However, this equation is not sufficient to determine the values of 'a' and 'b'. Therefore, the function is not continuous at x = 66 and standard differentiation fails.
Hence, the function S(x) is not differentiable at x = 66.
It's worth noting that in order for a function to be differentiable, it needs to be continuous and have a defined derivative at each point in its domain. In this case, the piecewise function fails the continuity requirement, leading to the conclusion that S(x) is not differentiable.