Part II: The Answers
226
Answers
201–300
272.
x = 1
The graph has a point of discontinuity at x = 1, so the graph isn’t differentiable there.
However, the rest of the graph is smooth and continuous, so the derivative exists for
all other points.
273.
differentiable everywhere
Because the graph is continuous and smooth everywhere, the function is differentiable
everywhere.
274.
x = –2, x = 0, x = 2
The graph of the function has sharp corners at x = –2, x = 0, and x = 2, so the
function isn’t differentiable there.
You can also note that for each of the points x = –2, x = 0, and x = 2, the slopes of the
tangent lines jump from 1 to –1 or from –1 to 1. This jump in slopes is another way
to recognize values of x where the function is not differentiable.
275.
x
n
= +
π π
2
, where n is an integer
Because y = tan x has points of discontinuity at x
n
= +
π π
2
, where n is an integer, the
function isn’t differentiable at those points.
276.
x = 0
The tangent line would be vertical at x = 0, so the function isn’t differentiable there.
277.
2
Use the derivative definition with f
 
(x) = 2x – 1 and f (x + h) = 2(x + h) – 1 = 2x + 2h – 1:
lim
(
)
lim
lim
lim
h
h
h
h
x h
x
h
x h
x
h
h
h
→
→
→
→
+ − −
−
=
+ − − +
=
=
0
0
0
0
2
2 1 2 1
2
2 1 2 1
2
2 2
2
=