EGR 1101: Unit 9 Lecture #1
Applications of Derivatives:
Electric Circuits
(Section 8.4 of Rattan/Klingbeil text)
Review: Some Derivative Rules





d
(c )  0
dt
d n
(t )  nt n 1
dt
d
(sin(t ))   cos(t )
dt
d
(cos(t ))   sin(t )
dt
d at
(e )  aeat
dt
where a, c, n, and  are constants.
Two New Derivative Rules

•
d
dg
df
( f (t ) g (t ))  f (t )
 g (t )
dt
dt
dt
(Product rule)
If f(g) is a function of g and g(t) is a function of t,
df df dg

dt dg dt
(Chain rule)
Today’s Examples
1.
2.
3.
4.
5.
Voltage, current, & power
Current & voltage in an inductor
Current & voltage in an inductor
(graphical and working backwards)
Current & voltage in a capacitor
Current & voltage in a capacitor
(graphical and working backwards)
Review: Some Electrical Quantities
Quantity
Voltage
Current
Charge
Energy
Power
Time
Symbol
Unit
V or v(t)
volt
I or i(t)
ampere
Q or q(t) coulomb
W or w(t)
joule
P or p(t)
watt
t
second
Symbol for
Unit
V
A
C
J
W
s
Review: More Electrical Quantities
Quantity
Symbol
Unit
Symbol for
Unit
Resistance
Inductance
Capacitance
R
L
C
ohm
henry
farad

H
F
Voltage-versus-Current Relations

For resistors,
v(t )  i(t )  R
•
For inductors,
•
For capacitors,
di
v (t )  L
dt
dv
i (t )  C
dt
EGR 1101: Unit 9 Lecture #2
Applications of Derivatives: Beams
(Section 8.5 of Rattan/Klingbeil text)
Some Beam Terminology

Types of beams



Simply supported
Cantilever
Types of load on a beam


Concentrated
Distributed
More Beam Terminology




In addition to the type of beam and load,
a beam’s behavior also depends on its
geometry and the material it is made of.
Its geometry is summarized in a quantity
called the second moment of area (I).
Its material is summarized in a quantity
called the modulus of elasticity (E).
The product of these two (EI) is called
the flexural rigidity.
Some Beam Quantities
Quantity
Symbol
Unit
Modulus of
elasticity
Second moment of
area
Flexural rigidity
E
lb/in2 or
N/m2
in4 or m4
EI
Deflection
Slope
y(x)
(x)
I
lbin2 or
Nm2
in or m
radians
Excellent Online Resource


University of Wisconsin’s online lessons
on Strength of Materials:
http://www3.uwstout.edu/faculty/
scotta/upload/FoleyStaticsStrengths.pdf
See especially Topic 4 (Beams) and
Topic 8.2 (Stress on Incline Planes).
Review

Given a function f(x), the function’s local
maxima occur at values of x where
df
0
dx

and
2
d f
0
2
dx
Its local minima occur at values of x where
df
0
dx
and
2
d f
0
2
dx
Today’s Examples
1.
2.
3.
4.
Deflection of a cantilever beam with
end load
Deflection of a simply supported beam
with central load
Deflection of a simply supported beam
with distributed load
Maximum stress under axial loading