multivar03 handouts (3)

8/9/2019 Multivar03 Handouts (3)

1/12

Mathematics 2

MultivariateCalculus: Small

Changes

1


2/12

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


3/12

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


4/12

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


5/12

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

+


6/12

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+=


7/12

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


8/12

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

+


9/12

#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


10/12

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

=

=


11/12

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=


12/12

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

=

=

=

multivar03 handouts (3)

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