EGR 1101 Unit 8 Lecture #1
The Derivative
(Sections 8.1, 8.2 of Rattan/Klingbeil text)
A Little History

1.
2.
3.
4.
Seventeenth-century mathematicians
faced at least four big problems that
required new techniques:
Slope of a curve
Rates of change (such as velocity and
acceleration)
Maxima and minima of functions
Area under a curve
Slope

We know that the slope of a line is defined
as
m 

y
t
(using t for the independent variable).
Slope is a very useful concept for lines.
Can we extend this idea to curves in
general?
40
y1(t) = 3*t + 4
35
30
y
25
20
(4,16)
(3,13)
15
(2,10)
10
(1,7)
5
0
0
0.5
1
1.5
2
2.5
t
3
3.5
4
4.5
5
Derivative


We define the derivative of y with
respect to t at a point P to be the limit of
y/t for points closer and closer to P.
In symbols:
dy
dt
 lim
t  0
y
t
Alternate Notations

There are other common notations for the
derivative of y with respect to t. One
notation uses a prime symbol ():
y ( t ) 
dy
dt

 lim
t 0
y
t
Another notation uses a dot:
y ( t ) 
dy
dt
 lim
t 0
y
t
Tables of Derivative Rules



In most cases, rather than applying the
definition to find a function’s derivative,
we’ll consult tables of derivative rules.
Two commonly used rules (c and n are
constants):
d
(c )  0
dt

d
dt
( t )  nt
n
n 1
Differentiation


Differentiation is just the process of
finding a function’s derivative.
The following sentences are equivalent:



“Find the derivative of y(t) = 3t2 + 12t + 7”
“Differentiate y(t) = 3t2 + 12t + 7”
Differential calculus is the branch of
calculus that deals with derivatives.
Second Derivatives



When you take the derivative of a
derivative, you get what’s called a second
derivative.
Notation:
dy
2
d ( dt )
d y

2
dt
dt
Alternate notations:
y  ( t )
y ( t )
Forget Your Physics
For today’s examples, assume that we
haven’t studied equations of motion in a
physics class.
But we do know this much:




Average velocity:
v avg 
y
t
Average acceleration:
a avg 
v
t
From Average to Instantaneous
From the equations for average velocity
and acceleration, we get instantaneous
velocity and acceleration by taking the limit
as t goes to 0.


Instantaneous velocity:
v (t ) 
dy
dt

Instantaneous acceleration:
a (t ) 
dv
dt
Today’s Examples
1.
2.
Velocity & acceleration of a dropped ball
Velocity of a ball thrown upward
Maxima and Minima

Given a function y(t), the function’s local
maxima and local minima occur at values
of t where
dy
dt
0
Maxima and Minima (Continued)

Given a function y(t), the function’s local
maxima occur at values of t where
dy
0
and
dt

2
d y
dt
2
0
Its local minima occur at values of t where
dy
dt
0
and
2
d y
dt
2
0
EGR 1101 Unit 8 Lecture #2
Applications of Derivatives:
Position, Velocity, and Acceleration
(Section 8.3 of Rattan/Klingbeil text)
Review

Recall that if an object’s position is given by
x(t), then its velocity is given by
v ( t )  lim
t  0

x
t

dx
dt
And its acceleration is given by
a ( t )  lim
t  0
v
t

dv
dt
2

d x
dt
2
Review: Two Derivative Rules

Two commonly used rules (c and n are
constants):

d
(c )  0
dt

d
dt
( t )  nt
n
n 1
Three New Derivative Rules

Three more commonly used rules ( and a
are constants):

d
(sin(  t ))   cos(  t )
dt

d
(cos(  t ))    sin(  t )
dt

d
dt
(e
at
)  ae
at
Today’s Examples
1.
2.
3.
4.
5.
Velocity & acceleration from position
Velocity & acceleration from position
Velocity & acceleration from position
(graphical)
Position & velocity from acceleration
(graphical)
Velocity & acceleration from position
Review from Previous Lecture

Given a function x(t), the function’s local
maxima occur at values of t where
dx
0
and
dt

2
d x
dt
2
0
Its local minima occur at values of t where
dx
dt
0
and
2
d x
dt
2
0
Graphical derivatives



The derivative of a parabola is a slant line.
The derivative of a slant line is a horizontal
line (constant).
The derivative of a horizontal line
(constant) is zero.