2.1 The Derivative and the
Tangent Line Problem (Part 2)
Arches National Park
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
Arches National Park
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington
Objectives
• Understand the relationship between
differentiability and continuity.
Theorem 2.1
If f is differentiable at x = c, then f is continuous at x = c.
Differentiability implies continuity.
If a function is NOT continuous at x=c, then it is NOT
differentiable.
Is the converse true?
No

To be differentiable, a function must be continuous and
smooth.
Derivatives will fail to exist at:
f  x  x
f  x  x
corner
2
3
cusp
1, x  0
f  x  
 1, x  0
f  x  3 x
vertical tangent
discontinuity

Graph with a Sharp Turn
f ( x)  x  2
x2 0
f ( x)  f (2)
f '(2)  lim
 lim
 lim
x 2
x 2
x 2
x2
x2
( x  2)
lim
 1
x 2
x2
x2
lim
1
x 2 x  2
x2
So, lim
DNE
x 2 x  2
x2
x2
f '(2) DNE
(the tangent line is
not unique)
Graph with a Sharp Turn
f ( x)  x
1
3
1
x 3 0
f ( x)  f (0)
 lim
f '(0)  lim
x 0 x  0
x 0
x0
1
1
x 3
 lim 2
 lim
x 0
x 0 x
x 3

The graph is
continuous at x=0,
but f ' (0) DNE.
So f ' (x) does NOT exist (or f is not differentiable)
if the graph has
•a sharp corner or turn,
•a vertical tangent line, or
•a discontinuity.
Most of the functions we study in
calculus will be differentiable.

Homework
2.1 (page 102)
#33,
39-42 all,
61,
67-85 odd