MATH1023
Week 9 Lecture 1
Haotian Wu
The University of Sydney
2021 Semester 2
• Recap of Week 8: Partial derivatives and tangent planes
• First-order partial derivatives, fx and fy, of f(x, y). (§8.1–8.3)
• Tangent plane to a surface z = f(x, y) (§8.4):
z f(a, b) = fx(a, b)(x a) + fy(a, b)(y b).
• Week 9: Applications of partial derivatives
• Linear approximation (application of tangent plane). (§9.1)
• Di↵erential (formalising the idea of linear approximation). (§9.1)
• Chain rule and the total derivative. (§9.2)
• Implicit di↵erentiation. (§9.3)
Linear approximation of g(x)
• Let g : I ✓ R! R, x 7! g(x) be di↵erentiable.
• If a 2 I, then the equation of the tangent line to the curve y = g(x) at x = a is
y = g(a) + g0(a)(x a).
• Application of tangent line: If x0 is near a, then g(x0) is approximately
g(x0) ⇡ y|x=x0 = g(a) + g0(a)(x0 a).
if
Y
n
GH)
(Xo
,
91×01)
LH1.¥;ɵ, dixon
ex Let g(x) = lnx. Use g(1) and g0(1) to find an approximate value of ln(1.01).
911 ) = In 11 ) = 0 ,
94×1 = I ⇒ 9
'
(1) = I
i. bill . 01 ) = 811.01) = 911 ) 1- G
'
(1) (1.01-1)
= 0 1- I 10.01 )
= 0.0 /
.
c
'
,
th ( 1.01 ) = 0,01
Rank : Calculator ⇒ hill . d) =
0.00995
. µ,
Linear approximation of f(x, y)
• Consider f : D ✓ R2 ! R, (x, y) 7! f(x, y). Assume fx, fy exist for all (x, y) 2 D.
• The equation of the tangent plane to the surface z = f(x, y) at (x, y) = (a, b) 2 D is
z = f(a, b) + fx(a, b)(x a) + fy(a, b)(y b).
• Application of tangent plane: If (x0, y0) is near (a, b), then f(x0, y0) is approximately
f(x0, y0) ⇡ z|(x,y)=(x0,y0) = f(a, b) + fx(a, b)(x0 a) + fy(a, b)(y0 b).
TEN
to,Yo,U×o,%
))
n
,
2- = fix,y)
⇐it.t"" ""
*" Yo, "
"
\
'
y
ex (a) Find the equation of the tangent plane to the graph of f(x, y) =
p
9 x2 y2 at (2, 2, 1).
(b) Find an approximate value of f(1.95, 2.1).
(a) fxlx,Y) = £19 - ×
'
- y
' )
- I. I - 2x) = - ✗ 19 - ✗
'
- ya )
- I
fylx ,y ) = { ( g-✗Kyi)
-±( -4 ) = - yes - ✗
2- ya)
- É
l
c
'
,
f-✗ ( 2,2 ) =
-2
,
fy 12,2 ) = -2
c
'
,
2- = f- 12,2 ) + fx (2,211×-2) + fy (2,44-2)
I
,
2- = I - 21×-2) - 24-2) It [ ix. y)
(b) f- 11.95.2.1 ) =L 11.95.2.1 )
= / - 2 . (1.95-2) - 2 (2.1-2)
= 0.9
.
""
Di↵erential of g(x)
Definition 1
Let y = g(x), where g is a di↵erentiable function. The di↵erential 1 dy of the
function g is defined by
dy = g0(x)dx.
Remarks:
• dx and dy represent changes in x and y, respectively.
• To find dy, we need both g0(x) and dx.
• For fixed x, dy = g0(x)dx relates the change in x to the change in y.
• Di↵erential and linear approximation:
If dx = x, then dy ⇡ y since
dy = g0(x)dx = g0(x)x ⇡ y = f(x+x) f(x).
1Alternative notation for dy is dg.
⇐ ¥ = fix ,
-
→↳
Mr
M
approx . change in y increment in Y
Visualising dy = g0(x)dx
T.FI
slope = f' 1×0 )
dy = f- ' ix. ) AX I AY
ex Round metal disks of radius 10 cm are manufactured with an error of ±0.1 cm. What is
the approximate error in the area of the disk? What is the relative error as a percentage
of the true size of the disk?
Air ) = Tr ' ⇒ dA=E¥[,dr
If Idrl = I -1-0.11=0.1 , and r = 10 am
,
then / DA1 = 12TH dr / = 21T r / drl
= 21T . to . 0 . / = 21T CM2
.
'
, approx . error in area is INT cm
'
.
i. relative error =
+= ± # =t2% .
IT (f) 2
%
Di↵erential of f(x, y)
Definition 2
Let z = f(x, y), where f is di↵erentiable (so that fx, fy exist). The di↵erential 2
dz of the function f is defined by
dz = fx(x, y)dx+ fy(x, y)dy.
Remarks:
• dx,dy and dz represent changes in x, y and z, respectively.
• To find dz, we need fx, fy, dx and dy.
• For fixed (x, y), dz relates the changes in x and y to the change in z.
• Di↵erential and linear approximation:
If dx = x and dy = y, then dz ⇡ z since
dz = fx(x, y)x+ fy(x, y)y ⇡ z = f(x+x, y +y) f(x, y).
2Alternative notation for dz is df .
-
→↳
- -
approx . change in -2 increment in Z
Visualising dz = fx(x, y)dx+ fy(x, y)dy
"" "
"" """"
"""
soso.dz
= f-
✗
la , b) A✗ + fy la, b) AY I AZ = flat #× , b -1^-1 ) - f / a. b)
ex What is an approximate value of
p
3.92 + 3.12 ?
7 I
fix,y ) = ✗'+yÑ = ( ✗
2 + ya )± @ 4-
b) = 14,3)
• ""÷:)( with ✗ =3 , / ,d -2 = f× ( a. b) dx + fy la, b) dy
,
where f-✗ = f- IX2-1YY
-±
. 4) = ✗ 1×21-44
- ±
,
fy = IH4Y ') - I (4) = yhity')
-±
/
f-
✗ ( 4,3 ) = 45 , fy 14,3) = %
,
dx = A- ✗ =3
.
9- 4 = - 0
. /
,
dy = AY = 3. I -3 = 0 . I .
.
'
,
d -2 = £ to- 1) + £-10.1) = - 0.02
i. f- 13.9
, 3. 1) = f- 14,3 ) -1 AZ If 14,3 ) + dz = 5- 0.02=491
,
Chain rule (the total derivative)
Suppose f : D ✓ R2 ! R, (x, y) 7! f(x, y) is di↵erentiable (so fx, fy exist).
Question: If x = g(t) and y = h(t), then what is dzdt ?
Answer 1: Compose the functions to find z(t) = f(g(t), h(t)), then find dzdt .
ex Suppose z = 2 x2 y2 where x = 1 cos t and y = sin t. Find dzdt .
2- It )= 2- ( 1- cost ) 2- (Sint)
-
'
'
' ¥- = -211- cost ) (Sint ) - 2 Sint cost
= -
zs.int/-2si~-ast-2sintcosti.ddZ---=-2sint
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