AP
Worksheet
Section 3.2
Name ________________________________________hr 5
Date ____/____/____ Score:________/
= ________%
In Exercises 1 and 2, compare the right-hand and left-hand derivatives to show that the function is not differentiable at the
point P. Find all points where f is not differentiable.
1.
2.
In Exercises 3-5, the graph of a function over a closed interval D is given. At what domain points does the function appear
to be a) differentiable? b) continuous but not differentiable? c) neither continuous nor differentiable?
3. y  f ( x ), D : 3  x  2
4. y  f ( x ), D : 3  x  3
5. y  f ( x ), D : 1  x  2
(a)_______________________
(a)_______________________
(a)_______________________
(b)_______________________
(b)_______________________
(b)_______________________
(c)_______________________
(c)_______________________
(c)_______________________
In Exercises 6-8, the function fails to be differentiable at x = 0. Tell whether the problem is a corner, a cusp, a
vertical tangent, or a discontinuity.
6.
tan 1 x ,
y
 1,
x0
x0
7. y  x 
x2  2
8. y  3 x  2 x  1
Find the numerical derivative of the given function at the indicated point. Use h = 0.001. Is the function
differentiable at the indicated point?
9. f ( x )  4 x  x 2 , x  0
10. f ( x)  4 x  x 2 , x  1
11. f ( x )  x 3  4 x , x  2
12. f ( x )  x 2 / 3 , x  0
13. f ( x )  x 2 / 5 , x  0
x 1
 3  x,
14. Let f be the function defined as f ( x)   2
where a and b are constants.
ax  bx, x  1
a) If the function is continuous for all x,
b) Find the unique values for a and b that will
what is the relationship between a and b?
make f both continuous and differentiable.
15. There is another way that a function might fail to be differentiable, and that is by oscillation. Let
1

 x sin , x  0
.
f ( x)  
x

x0
 0,
f (0  h)  f (0)
1
a) Show that f is continuous at x = 0.
b) Show that
 sin .
h
h
c) Explain why
lim f (0  h)  f (0)
does not exist. d) Does f have either a left-hand or right-hand
h0
h
derivative at x = 0?
1
 2
 x sin ,
e) Now consider the function g( x )  
x

0
,

differentiable at x = 0 and that g’(0) = 0.
x0
x0
. Use the definition of derivative to show that g is