Intermediate Value Theorem
f b
If f is continuous on  a, b and k is any y-value between
k
f  a  and f  b  , then there is at least one x-value c
f a
between a and b such that f  c   k. In other words,
a
f takes on every y-value between f  a  and f  b  .
c b
Definition of the Derivative
Any nonvertical line has the same slope at every point. In Calculus we frequently deal with the
slope of a curve. The slope of a curve is defined to be the same as the slope of the curve’s
tangent line at a given point. To find the slope of a tangent line we use a limit of the slope of a
secant line.
y  f  x
f  x  x 
secant line
tangent line
y  f  x  x   f  x 
f  x
y f  x  x   f  x 

x
x
mtan  lim
x  x
x
msec 
f  x  x   f  x 
x
x  0
x
The slope of a tangent line is called the derivative of the function at a given x-value.
f   x   y 
f  x  x   f  x 
f  x  h  f  x
dy d

f  x   mtan  lim
 lim
x  0
h 0
dx dx
x
h
A vertical tangent line has no slope, so a curve has no derivative at any point where it has a
vertical tangent line. Differentiation is the process of finding derivatives. If a derivative exists at
a point on a curve, the function is said to be differentiable at that point.
Alternate Form of the Limit Definition of the Derivative
(Gives the value of the derivative at a single point.)
y  f  x
secant line
mtan  lim msec
xc
f  x
tangent line
y  f  x   f  c 
f c
c
x
x  x  c
f   c   lim
x c
f  x  f c
xc
Derivative Rules:
d
dx
Power Rule:
x n  nx n 1
Constant Rule: If c is any constant,
d
dx
c  0.
Scalar Multiple Rule: If c is any constant,
Sum Rule:
d
dx
d
dx
 c f  x   c f   x .
 f  x   g  x   f   x   g   x 
Higher-Order Derivatives
Since the derivative of a function is another function, we can repeat the differentiation process to
find the derivative of a derivative. These derivatives are called higher-order derivatives.
Notation:
First Derivative:
y
f  x
dy
dx
d
dx
f  x
Second Derivative:
y
f   x 
Third Derivative:
y
f   x 
d2y
dx 2
d3y
dx3
d2
f  x
dx 2
d3
f  x
dx3
Equation of a Tangent Line:
Since the derivative of a function gives us a slope formula for tangent lines to the graph of the
function, the derivative can be used to find equations of tangent lines.
Sometimes we will want to find a line perpendicular to the
tangent line at a certain point. Such a line is called a normal line.
NONDIFFERENTIABILITY
mnormal 
1
mtangent
(when a derivative does not exist)
Each of these functions has no derivative when x = 1.



hole



jump






vertical
asymptote



sharp turn





vertical
tangent
These five characteristics destroy differentiability:
If a function is not continuous, it is not differentiable (see the first three figures above).
A function may be continuous and still not be differentiable (see the last two figures above).
BC Calculus
Section __________
Name __________________
Date ___________________
Derivatives 2-A worksheet
1.
If y  x 2  x , use the limit definition of the derivative to find y .
2.
If y  t 3  1 , use the limit definition of the derivative to find
3.
dy
.
dt
If f  x   2 x2  4 , use the limit definition of the derivative to find f   x  .
Then find f   4  .
4.
If f  x   2 x2  4 , use the alternate form of the limit definition of the derivative to find
f   4 .
y
x
Use the graph of y  f  x  shown to graph the following.


5.
y  f  x
8.
Find the domain, vertical asymptotes, holes, intercepts, end behavior, and graph for the
function y 
6.
x  x  1
x2 1
.
y  f  x  2  1
7.
y  2 f  x 
Use the graph of y  f  x  for Problems 9-18.

Find the following limits and function values.
9.
12.
15.
lim f  x 
10.
lim f  x 
13.
x  1
x 1
lim f  x 
x  
f  1 11.
lim f  x 
lim f  x 
14.
x  3
lim f  x 
x 
List all nonremovable discontinuities of f  x  .
19. f  x    3 x 2  2   2 x  3
23. f  x   2 x cos x



x 2
18.
9
5x2
x
16. lim f  x 
List all removable discontinuities of f  x  .
y

x 1
17.
21.
y
2 x2  4 x  3
2  3x
20.
y
22.
If f  2   3 and f   2   4
find g  2  when g  x   x 2 f  x  .
24. y   x 2  3
2