iclicker quiz (1) i have completed at least 50% of the reading and study-guide assignments...
Post on 21-Dec-2015
213 views
TRANSCRIPT
iClicker Quiz
(1) I have completed at least 50% of the reading and study-guide assignments associated with lecture 1-4, as indicated on the course schedule.
a) True b) False
Calculus is the art of cutting things up into little pieces!
Summing the pieces is called “integration”. Taking ratios of pieces is called “differentiation”.
Use your imagination. But in any given situation, one type of cut will usually be more convenient that others.
xx
dx
d
eedx
d
xxdx
d
xxdx
d
xnxdx
d
xx
nn
1)ln(
)sin()cos(
)cos()sin(
1
x
axa
dx
d
aeedx
d
xaaxadx
d
xaaxadx
d
xanxadx
d
xaxa
n
nn
n
nn
)ln(
)sin()cos(
)cos()sin(
1
Let f(x) and g(x) be functions of x
dx
dg
dg
dfxgf
dx
d
dx
dg
g
f
dx
df
gg
f
dx
d
dx
dgf
dx
dfggf
dx
d
dx
dg
dx
dfgf
dx
d
constdx
d
))((
1
0
2
Sum rule
Quotient rule
Chain rule
Product rule
Constant rule
)ln(1
)sin()cos(
)cos()sin(
1
1
xdxx
edxe
xdxx
xdxx
n
xdxx
xx
nn
)ln(1
])[sin(])[cos(
])[cos(])[sin(
1
1
axdxax
b
edxe
b
axbdxaxb
b
axbdxaxb
n
axbdxaxb
xbxb
nn
)(
)()()(
)(
1)(
)()()(
)()(
ab
aFbFdxxf
abxf
aFbFdxxf
dxxfxF
b
a
b
a
Definite integral
Average
Indefinite integral
drrdA 2
Circular area
da
daadA
2
2
)(
2
drdrdA 22 xay
dydxdA
Spherical Coordinates
)sin()sin()sin(
)cos()sin()cos(
)cos(
)sin(
222222
ry
rx
rz
r
rzzyx
Spherical Area
ddrdrddA )sin(2
d
rd
rz
)cos(
)sin(
rz
r
Spherical Area
dda
daddA
)sin(2
sphericalpatchwork
dza
dza
dzdA
2
)sin()sin(2
)sin(2
tomatoslices
cones
da
addA
)sin(2
22
orangeslices
dadA 22
22)sin( zaa
az
Spherical Volume
drrdddV
drdrddV2)sin(
drrddV 2)sin(2 drrddV 22 dzzadV )( 22
sphericalpatchwork
conesorangeslices
tomatoslices
22)sin( zrr rz
dr
iClicker Quiz
Which of the following area differentials will yield the surface area of a sphere of radius a?
above theof all)
2)
2)
)sin(2)
)sin()
2
2
2
e
dzad
dac
dab
ddaa
Conical Volume
dzzL
adV
L
zazr
dzzrdV
22
2
2
)(
)(
dLa
daLa
dV
6
322
???
dzdzL
adV
L
zazr
dzdzrdV
22
2
2
2
1
)(
)(2
1
Conical Area
dzzL
LaadA
L
dz
H
dsL
zazr
dszrdA
2
22
2
)(
)(2
dLaa
dA
LaH
adHdA
2
2
22
22
dzdzL
LaadA
dsdzrdA
2
22
)(
???
iClicker Quiz
The charge density of a disc of radius a is described by the function = Cr3. The best differential area to use in this case is:
dydxe
dad
rdrc
rdrb
drdra
)
)
2)
)
)
221
Uniform Density
dAdx
dxdQ
xQ
dAdQ
AQ
C/m C/m2
dVdQ
VQ
C/m3
dV
Non-uniform Density
dA
dx
dxxdQQ
dxxdQ
)(
)(
dV
dAxdQQ
dAxdQ
)(
)(
dVxdQQ
dVxdQ
)(
)(
Key concept
When integrating a density to obtain a complete total (e.g. total mass or total charge), you must choose a differential over which the density is constant!
iClicker Quiz
The charge density of a disc of radius a is described by the function = Cr3. The best differential area to use in this case is:
dydxe
dad
rdrc
rdrb
drdra
)
)
2)
)
)
221
