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Mathematics 2
MultivariateCalculus: Small
Changes
1
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Functions of Several Variables
In the first lecture we introduced functions which depended on morethan one variable.
We were introduced to the concept of a partial derivative, e.g. if z(x,!then we can differentiate z w.r.t. x "# .
$oda we%ll loo& at how functions of several variables are affected bchanges to one or more of the independent variables
=
=
++=
y
z
x
z
exyxyyxz y)cos(2),( 2
yexxy
xyy
++
)cos(4
)sin(2 2
2
8/9/2019 Multivar03 Handouts (3)
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Volume of a 'one
$o illustrate the content of toda%s lectureconsider the volume of a cone. $he formula forthe volume of a cone is given opposite.
'learl the volume depends on the height of thecone (h! and the base radius (r! so we could
write V(h,r!. $hree uestions we might want to as& ourselves
could be)*. For a given size of cone, what would the change
in V be for given actual changes in r and h
+. What would the relative change in V be for given
relative changes in r and h-. What would be the rate of change of V for agiven rate of change of r and h
3
hrV 2
3
1=
r
h
8/9/2019 Multivar03 Handouts (3)
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hrV 231=
r
h
erivation from First /rinciples
0et us define the change in r, h and V to be 1r,
1h and 1V respectivel. $hus r r 2 1r, h h 2 1h and V V 2 1V. So doing it the long wa)
V21V 3 *4-5(r 2 1r!+(h 2 1h!
3 *4-5(r+2 +r1r 2 1r+!(h 2 1h!
6 *4-5(r+2 +r1r!(h 2 1h!
6 *4-5(r+h 2 +rh1r 2 r+1h 2 +r1r1h!
6 *4-5r+h 2 +4-5rh1r 2
*4-5r+1h
$hus 1V 6 +4-5rh1r 2*4-5r
+1h
'an ou spot anthing in this formula 2
V 2rh
r 3
V 1r
h 3
=
=
4
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Small Increments We have 1V 6 +4-5rh1r 2
*4-5r+1h which is
actuall)
In general if we have a function of two
variables, z(x,! and we ma&e small changes tothe independent variables, the change to thedependent variable is given b)
2
V 2rh
r 3
V 1 rh 3
=
=
5
hrV 2
3
1=
r
h
yyzx
xzz
+
hh
Vr
r
VV
+
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7xample *
Suppose we have a cone with base radius 8 cm and height *+ cm.Find the approximate increase in volume when r increases b 9.* cmand h decreases b 9.: cm.
We have and
So
So the change in the volume is a
rhVr 3
2=
2
3
1rVh =
6
hh
Vr
r
VV
+
hrrrh 2
3
1
3
2+=
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7xample +
$he flow of slurr along a pipe, V, is given b)
If r increases b ;9.9+p, 1l 3 9.9-l, 1? 3 9
l
prV
8
4
=
ll
VVr
r
Vp
p
VV
+
+
+
7
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7xample + (cont!
$hus we have
l
l
prr
l
prp
l
rV 03.0
8
05.0
8
402.0
8 2
434
+
ll
Vr
r
Vp
p
VV 03.005.002.0
+
+
8
2
4
4
3
8
8
8
4
l
pr
l
r
l
pr
=
=
=
lV
p
V
r
V
l
pr
l
pr
l
prV
803.0
82.0
802.0
444
+
8/9/2019 Multivar03 Handouts (3)
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#ates of change
What if the independent variables in our function arethemselves changing at a particular rate
$he themselves are functions of time@
We want to obtain a formula which will give us the rateof change of the dependent variable with respect totime as follows)
yy
zx
x
zz
+
dz z dx z dy
dt x dt y dt
+
9
t
y
y
z
t
x
x
z
t
z
+
ivideb 1t
0et 1t9
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7xample -For the function h(x, z! 3 cos(x! 2 xz-it is &nown that x isincreasing at 9.; ms>*and z is increasing at 9.* ms>*. Find therate of change of h
dh h dx h dz
dt x dt z dt
+
10
3
2
sin( )
3
x z
xz
+
h
x
h
z
=
=
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Example 4
The force F (N) between two electric chargeswith magnitudes q and (!oulombs)separated b" a distance r (m) is gi#en b"
where $ is a constant% &etermine themaximum percentage error in calculating F ifq is measured to an accurac" of ' to* and r to 2%
11
2
kqQF
r=
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Example 4 continued
12
2 2 32= + kQ kq kqQF q Q r
r r r
2= + F q Q r
F q Q r
2= + +F q Q r
MaxF q Q r
2kqQF
r=
2
2
3
F kQ
q r
F kq
Q r
F kqQ2
r r
=
=
=
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