####
Transcript Lesson 7.1

Average Values & Other
Antiderivative Applications
Lesson 7.1
Average Value of a Function
• Consider recording a temperature each
hour and taking an average
1 24
T
f (t )
24 t 1
• If we take it more often and take a limit …
1 n
T lim f (ti )
n n
i 1
2
Average Value of a Function
• Now apply the concept to
a continuous function
1
ba
y lim
f (ti )
b a i 1
n
n
f(x)
a
b
n
b
1
f (t )dt
ba a
3
Application
• A 20,000 L water tank takes 10
min. to drain. After t min the water
left in the tank is V(t) = 200(100 – t2)
• What is the average amount of the water
in the tank while it is draining?
10
1
2
200
100
t
dt
10 0 0
4
Root Mean Square
• Used to calculate the effective current and
effective voltage of alternating current
• Root mean square
b
f rms
1
2
f ( x) dx
ba a
5
Root Mean Square
• If current is given by
i 4t t 2 1
• What is the effective current for the time
interval from 0 < t < 4
4
ieff irms
2
1
2
dt
4
t
t
1
4 0
6
Current and Power
• Current, i defined as the time rate of
change of the charge, q
dq
i
Given in coulombs
dt
Passing through a given point in the circuit
• Total charge transmitted
in a circuit
• Voltage between
terminals of capacitor
with C farads
q i dt
1
Vc i dt
C
7
Current and Power
• An 80-μF capacitor is charged to 100 V. It
is then supplied a current of i=0.04t3 A.
• After what time interval does the capacitor
voltage reach 225 V?
x
1
3
225
0.04
t
dt
6
80 10 0
8
Work
• Work expended in an
electrical system given by
W Pdt
W in joules
P in watts
• Instantaneous power represented by
P V i
P i R
2
2
V
P
R
Any one of these can be
substituted in to the
formula above
9
Work
• The power of a certain electrical
1
P 5t 4
system is charged according to
• What is the formula for energy in this
circuit?
1
4
W 5t dt
5
4
4t k
joules
10
Assignment
• Lesson 7.1
• Page 262
• Exercises 1 – 31 odd
11