Assignment 11- Mathematics 1 - AASTMT
Transcript of Assignment 11- Mathematics 1 - AASTMT
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1 | P a g e A A S T M T
ARAB ACADEMY FOR SCIENCE TECHNOLOGY AND
MARITIME TRANSPORT
Problem. 1: (a) Find the first three terms of the Maclaurin series for ln (1 + ex).
(b) Hence, determine the value of .
20
4ln)e1ln(2lim
x
xx
x
Assignment 11- Mathematics 1
MacLaurin series& partial derivatives
Name:
Specialization:
Registration No:
Date: / /2019
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Problem. 2. (a) Find the value of
(b) By using the series expansions for and cos x evaluate
.2sin
lnlim
1
x
x
x
2e x
.cos1
e1lim
2
0
x
x
x
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Problem. 3. (a) (i) Find the first four derivatives with respect to x of y = ln(1
+ sin x)
(ii) Hence, show that the Maclaurin series, up to the term in x4, for y is
y = x –
...12
1
6
1
2
1 432 xxx
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(b) Deduce the Maclaurin series, up to and including the term in x4, for
(i) y = ln(1 – sin x);
(ii) y = ln cosx;
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(iii) y = tan x.
(c) Hence calculate .
x
x
x cosln
)tan(lim
2
0
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Problem. 4. (a) Given that y = ln cos x, show that the first two non-zero terms of
the Maclaurin series for y are
(b) Use this series to find an approximation in terms of for ln 2.
.122
42 xx
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7 | P a g e A A S T M T
Partial Derivatives
Given a function of two variables z = f (x, y). Then
h
yxfyhxfyxf
x hx
),(),(lim),(
respect towith
Derivative Partial
0
h
yxfhyxfyxf
y hy
),(),(lim),(
respect towith
Derivative Partial
0
Notations For Partial Derivatives
Given z = f (x, y).
. respect to with derivative partial),(),( xx
z
x
fyxfyxf
xx
. respect to with derivative partial),(),( yy
z
y
fyxfyxf
yy
),(point at the evaluated sderivative partial
),(
),(
),(
),(
ba
bafdy
z
bafdx
z
y
ba
xba
*Note: In partial differentiation, we treat every variable as a constant except for the one we are
differentiating with respect to.
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Problem. 5 Find the first partial derivatives of the function 221),( yxyxf .
Solution:
█
Problem. 6: Find the first partial derivatives of the function yexxyxyxf 2323 54),( .
Solution:
█
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Problem. 7: Find the first partial derivatives of the function 22
)ln(),( 22 yxeyxyxf .
Solution:
█
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Problem. 8: Find )1,1(xf and )1,1(yf for 22
),(yx
xyyxf
Solution:
█
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Note: We can also differentiate functions of more than 2 variables.
Problem. 9: Find the first partial derivatives of the function
)sin(),,( 22222 yxzyxzyxf .
Solution:
█
Geometric Interpretation of the Partial Derivative
Given a surface z = f (x, y). We want to consider the point ( a, b, f (a, b) )
direction in the )),(,,(point
at the surface the toline tangent theof Slope),(
),( xbafbax
zbaf
bax
direction in the )),(,,(point
at the surface the toline tangent theof Slope),(
),(ybafbay
zbaf
ba
y
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Problem. 10: Find the slope of the surface 22),( yxyxf at the point (-2, 1, 5).
Solution:
█
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Second Order Partial Derivatives
Just as we can find second order derivatives for functions of one variable, we can do the same for
functions of two variables.
Notation for Second Order Derivatives
again respect to then with, respect to with partial Take :),(2
2
2
2
xxz
z
x
f
x
f
dxyxf xx
. respect to then with, respect to with partial Take :),(22
yxxy
z
xy
f
x
f
dyyxf yx
. respect to then with, respect to with partial Take :),(22
xyyx
z
yx
f
y
f
dxyxf xy
again respect to then with, respect to with partial Take :),(2
2
2
2
yyy
z
y
f
y
f
dyyxf yy
Note: In general,
),(),( yxfyxf yxxy
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Problem. 11: Find all of the second derivatives for 22 23),( xxyxyyxf .
Solution:
█
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Problem .12:
a) If 𝒛 = 𝐥𝐧(𝒙𝟐 + 𝒚𝟐) , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝝏𝟐𝒛
𝝏𝒙𝟐 +𝝏𝟐𝒛
𝝏𝒚𝟐 = 𝟎
Solution:
b) If 𝒖 = 𝐭𝐚𝐧−𝟏 𝒚
𝒙 , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕
1. 𝒙𝝏𝒖
𝝏𝒚 − 𝒚
𝝏𝒖
𝝏𝒙 = 𝟏
2. 𝝏𝟐𝒖
𝝏𝒙𝟐 + 𝝏𝟐𝒖
𝝏𝒚𝟐 =0
Solution:
c) 𝒛 =𝒙𝒚
𝒙−𝒚 , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝒙𝟐
𝝏𝟐𝒛
𝝏𝒙𝟐 + 𝟐𝒙𝒚 𝝏𝟐𝒛
𝝏𝒙𝝏𝒚+ 𝒚𝟐
𝝏𝟐𝒛
𝝏𝒚𝟐 = 𝟎
Solution:
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d) If 𝒖 = 𝒇(𝒙𝟐 + 𝒚𝟐), 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝒙𝝏𝒖
𝝏𝒚− 𝒚
𝝏𝒖
𝝏𝒙= 𝟎
Solution:
e) If 𝒖 = 𝒙𝟐𝒚 + 𝒚𝟐𝒛 + 𝒛𝟐𝒙 , 𝒔𝒉𝒐𝒘 𝒕𝒉𝒂𝒕 𝝏𝒖
𝝏𝒙+
𝝏𝒖
𝝏𝒚+
𝝏𝒖
𝝏𝒛= (𝒙 + 𝒚 + 𝒛)𝟐
Solution:
Good luck