l17 the differentials (applications)
TRANSCRIPT
EXAMPLE 1: Use differentials to approximate the change in the area of a square if the length of its side increases from 6 cm to 6.23 cm.Let x = length of the side of the square. The area may be expressed as a function of x, where A= x2. The differential dA is ( ) dxx2dA dxx'fdA ⋅=⇒⋅=
Because x is increasing from 6 to 6.23, you find that Δ x = dx = .23 cm; hence,
( ) ( )2cm76.2dA
cm23.0cm62dA
=
=
( ) .cm 2.8129 is y area in increase
exact the that Note 6.23. to6 from increases length sideits as
cm2.76 ely approximatby increasewill squarethe of area The
2
2
∆
EXAMPLE 2: Use the local linear approximation to estimate the value of to the nearest thousandth.
3 55.26
( )
( ) ( )
( )( )
2.9830.0167-3 55.26 601
355.26 therefore ;327 that less
601
tely approxima be will 55.26 that implies which
0167.0601
10045
271
45.0273
1dy , Hence
0.45dxx then 26.55, to 27 from decreasing is x Because
dx x3
1 dx x
31
dy
xxf dx xf'dy
is dy aldifferenti The 27.x
namely 26.55, to close relatively is and cube perfect a is that xof value
convenient a choose ,xxf is applying are you function the Because
3
33
3
3
2
3
23
2
3
1
3
≈≈
−≈=
−=−=−⋅=−⋅=
−==∆
==
=→=
=
=
−
EXAMPLE 3: If y = x3 + 2x2 – 3, find the approximate value of y when x = 2.01.
2.xof value original an to 0.01dxxof
increment an applyingof result the as 2.01 gconsiderin
are we then 0.01,22.01 write weif that Note dy. y
for solve shall we then value, eapproximat the find to
asked simply are we since but y y is value exact The
===∆
+=+
∆+
( )
( ) ( )
20.1320.013dyy
is ionapproximat required the,therefore
20.001.0812dy
then ,01.0dx and 2x when and
13388y then ,2x when
dxx4x3dy then
3x2xy Since2
23
=+=+
=+===
=−+==+=
−+=
EXAMPLE 4: Use an appropriate local linear approximation to estimate the value of cos 310.
( )( )
( )
( )( ) ( )
( ) 8573.0008725.0866.0dyy
is ionapproximat required the,therefore
008725.001745.05.0dy180
130 sindy
then ,01745.0180
1dx and 30x when and
866.030cosy then ,30x when
dx x sindy then
x cosy Let
0
00
0
00
00
=−+=+
−=−=
π•−=
=
π==
===−=
=
EXAMPLE 4: Use an appropriate local linear approximation to estimate the value of cos 310.
( )( )
( )
( )( ) ( )
( ) 8573.0008725.0866.0dyy
is ionapproximat required the,therefore
008725.001745.05.0dy180
130 sindy
then ,01745.0180
1dx and 30x when and
866.030cosy then ,30x when
dx x sindy then
x cosy Let
0
00
0
00
00
=−+=+
−=−=
π•−=
=
π==
===−=
=