Find values of x for which the piecewise function g(x): 1) (x+1)^2, x=<0 2) 2x+1, 0<x<33) (4-x)^2, x>=3is differentiable.
I got xer, x cannot equal 3 and 0, but the answer key only has 3. WHy isn't 0 one either since the derivative is different on either sides?
first, g must be continuous
g(0-) = (0+1)^2 = 1
g(0+) = 2*0+1 = 1
g(3-) = 2*3+1 = 7
g(3+) = (4-3)^2 = 1
so, g is continuous at x=0, but not at x=3.
g'(0-) = 2x+2 = 2
g'(0+) = 2
So, g is differentiable at x=0, since the slope is the same from both sides.
So g is differentiable everywhere except at x=3.
To determine if a piecewise function is differentiable at a particular point, we need to check if both the left-hand limit and the right-hand limit of the derivative exist and are equal at that point. Let's consider the three cases for the function g(x):
1) For x ≤ 0: The function g(x) is given by (x + 1)^2.
The derivative of g(x) in this interval is:
g'(x) = 2(x + 1)
To check the differentiability at x = 0, we need to calculate the left-hand limit and the right-hand limit of the derivative.
Left-hand limit:
lim┬(x→0-)〖2(x + 1) = 2(0 + 1) = 2〗
Right-hand limit:
lim┬(x→0+)〖2(x + 1) = 2(0 + 1) = 2〗
Since both the left-hand limit and the right-hand limit of the derivative exist and are equal (both are equal to 2), the function g(x) is differentiable at x = 0.
2) For 0 < x < 3: The function g(x) is given by 2x + 1.
The derivative of g(x) in this interval is:
g'(x) = 2
Since the derivative is a constant, it is continuous throughout the interval (0, 3). Therefore, g(x) is differentiable for all values of x in this interval.
3) For x ≥ 3: The function g(x) is given by (4 - x)^2.
The derivative of g(x) in this interval is:
g'(x) = -2(4 - x) = 2(x - 4)
To check the differentiability at x = 3, we need to calculate the left-hand limit and the right-hand limit of the derivative.
Left-hand limit:
lim┬(x→3-)〖2(x - 4) = 2(3 - 4) = -2〗
Right-hand limit:
lim┬(x→3+)〖2(x - 4) = 2(3 - 4) = -2〗
Since both the left-hand limit and the right-hand limit of the derivative exist and are equal (both are equal to -2), the function g(x) is differentiable at x = 3.
In conclusion, the function g(x) given in the piecewise form is differentiable for all values of x. Hence, the answer key stating only x ≠ 3 is correct, and x = 0 is not excluded because the derivative has a constant value in that interval.