ENGINEERING MATHEMATICS II

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ENGINEERING MATHEMATICS - II
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Differential Calculus
Introduction
We have already studied the Cartesian and polar
curves. In this chapter we will learn about
derivative of arcs and radius of curvature in
Cartesian, parametric and polar forms.   In this
chapter we shall discuss, Rolle's theorem,
Lagrange's Mean value theorem. Cauchy's Mean
value theorem, Taylor's theorem, Taylor's and
Maclaurin series expansions of functions both in
single and two variables.   Also we shall discuss
the application of differentiation to
indeterminate forms using L'Hospital rule, and
application of differential calculus to the
determination of a function which are greatest or
least in their neighbour hoods.
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Derivatives of Arc
Derivative of the length of the arc for the
Cartesian curve y f(x). Let y f(x) be the
equation of the curve. A be a fixed point on the
curve. Let P(x, y) and
be two neighbouring points on the
curve such that arc AP s, arc PQ ds and
chord PQ dc. As on the curve
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We have
   
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• If the equation of the curve is x f(y) then
• If the equation of the curve is in parametric
  form x x(t) and y y(t)
• where t is the parameter then

   
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Additional Results
We know that
   
again
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Derivative of Arc Length in Polar Form
Let
be two neighbouring points on the curve
Let arc and chord
. As Q approaches P on the curve
   
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From  DONP since dq is very small,
 
?
?
Since dq is small
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We have NQ OQ - ON
PQ2 PN2 NQ2
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• we have

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?

?
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Radius of Curvature
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Formula for radius curvature in Cartesian form


We have
Differentiate with respect to x.
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Formula for radius of curvature in parametric
form
Let x x(t) and y y(t)


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We have


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Radius of curvature in pedal form
By definition


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We have p r sin f
Differentiating with respect to r,


Comparing (1) and (2)
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Radius of curvature in polar form


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differentiating with respect to q


Dividing by -2r1
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Problem 01
In the ellipse , show
that the radius of curvature at an end of the
major axis is equal to the semi latus rectum.
Solution
,
One end of major axis is (a, 0) in
Differentiating w.r.t x, we get, We have
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Problem 02
Find the radius of curvature to the curve x a
(t - sin t), y a (1 - cos t) at any point t.
Solution
Let x a (t - sin t), y a (1 - cos t)
We have
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Problem 03
Find the radius of curvature for the curve whose
pedal equation is given by pa2 r3.
Solution
Let pa2 r3 .....(1) Differentiate with
respect to p (1)
We have
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Problem 04
Find the radius of curvature for the curve r
aeq cot a
Solution
Let r aeqcot a
We have
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We get, p r sin a as the p-r equation
Differentiate with respect to p
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Problem 05
With usual notations, prove that
Solution
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From (1) and (2)
Differentiate with respect to y
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Rolle's theorem
Statement If a function f(x) is (i) Continuous
in a, b (ii) Differentiable in (a, b)
and (iii) f(a) f(b)
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Geometrical Interpretation
If the graph of f(x) be drawn between x a and x
b having a unique tangent at all points in
the above interval and f(a) f(b), then there
exits at least one point C on the curve
(corresponding to x c between x a and x b),
such that the tangent at C is parallel to x-axis.


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Note 1 There may exists more than one at which
f'(x) vanishes.


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Note 2 The three conditions of Rolle's theorem
are the sufficient conditions (but not
necessary) for f'(x) 0 for some
Note 3 Conclusion of Rolle's theorem does not
hold good for a function which does not satisfy
any of its conditions. Example consider the
function f(x) x in -1, 1

Observe that i) f(x) is continuous in -1,
1 ii) f(-1) 1 f(1)
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But f(x) is not differentiable in (-1, 1) because
   


Since all the three conditions of Rolle's theorem
are not satisfied. Hence the conclusion is not
valid in -1, 1.
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Problem 06
Verify Rolles' Theorem for the function f (x) x
(x 3) e-x/2 in -3, 0.
Solution
(i) f (x) is a product of 2 continuous functions.
\f (x) is a continuous function in -3, 0
is defined for all x in (-3, 0)
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(iii) f (-3) 0 f (0) 0 i.e., f
(-3) f (0) All conditions of Rolles' Theorem
are satisfied. Solve f '(c) 0
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Since
\ Rolles' Theorem is verified.
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Lagrange's Mean value theorem (LMVT)
Let f(x) be a function such that   i) Continuous
in a, b   ii) Differentiable in (a, b)


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Proof Construct a function F(x) such
that   F(x) f(x) - Ax, where A is a constant
such that   F(a) F(b)   i.e,. f(a) - Aa f(b)
- Ab

 

i) Now F(x) is continuous in a,b
f(x), x is continuous   ii) Since f(x), x is
derivable, F(x) is also derivable in (a,
b)   (iii) Also F(a) F(b)
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Hence F (x) satisfies all the conditions of the
Rolle's theorem.
   


Which proves the Lagrange's Mean Value Theorem.
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Another form of LMVT
Let b - a h   We have a lt c lt b



Note Rolle's theorem is a special case of LMVT.
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Geometrical Interpretation of LMVT
Let the graph of f(x) be continuous between A(a,
f(a)) and B(b, f(b)). Let the curve have
tangents at all points between A and B then there
exists C(c, f(c)) on the curve between A and B
such that the tangent at C is parallel to the
chord AB.


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Note c is not unique


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Cor. then
f (x) is strictly increasing Let f(x) satisfy
the conditions of LMVT in a, b. Let x1, x2 be
any two points of a, b such that
x1ltx2 Applying LMVT to x1, x2


Since f(c)gt0
\ f(x) is strictly increasing.
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Problem 07
Verify Lagranges' Mean Value Theorem for the
function
Solution
(i) f (x) is a polynomial. \ It is continuous in
(ii) f '(x) (x - 1) (x - 2) x (x - 1) x (x
- 2) is defined for \ f (x) is differentiable
in (0, ½)
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or
\ LMVT is verified.
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Problem 08
Verify Lagranges' Mean Value Theorem for the
function f (x) log x in 1, e.
Solution
(i) f (x) is continuous in 1, e (as it is
defined for all x in 1, e)
\ f (x) is differentiable in (1, e),
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\ LMVT is verified
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Problem 09
If x gt 0, prove that
Solution
(I) Let f (x) log (1 x) Applying
LMVT in 0, x f (x) f (0) x f '(qx)
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We have, 0 lt q lt 1
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Problem 10
If x gt 0, prove that
Solution
\ f (x) is an increasing function
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\ F (x) is an increasing function
From (1) and (2), we get
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Cauchy's Mean Value theorem
Statement If f(x) and g(x) are any two
functions such that (i) f(x) and g(x) are both
continuous in a, b (ii) f(x) and g(x) are both
differentiable in (a,b) and


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Proof Consider F(x) f(x) - Ag(x) where A is a
constant to be determined such that F(a)
F(b) (i) Since f(x) and g(x) is continuous in
a, b F(x) is also continuous (ii) Since f(x)
and g(x) are derivable in (a, b) F(x) is
also derivable in (a, b) (iii) F(a) F(b)


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F(x) satisfies all the conditions of Rolle's
theorem
   


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Physical Interpretation of CMVT
CMVT interprets that the ratio of actual rates of
increase of f(x) and g(x) at x c is equal to
the ratio of their average rate of increase of
the functions in the interval (a, b).


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Problem 11
Verify Cauchy's Mean value Theorem for the
following function
Solution

• f (x) and g (x) are both continuous in a, b
• (ii) f '(x) ex g '(x) -e-x are defined
• \ f (x) and g (x) are both differentiable in (a,
  b)

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?
e2c ea b
or
? 2c a b
\ CMVT is verified
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Problem 12
Verify Cauchy's Mean value Theorem for the
following function
Solution
(i) f (x) and g (x) are both continuous in 1, e
\ f (x) and g (x) are differentiable in (1, e)
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All conditions of CMVT are satisfied
we get
\ CMVT is verified
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Taylor's Theorem (Statement Only)
Let f(x) be a function such that (i) f, f', f'',
,f(n-1) are continuous in a, b   (ii) f(n-1)
is differentiable in (a, b) then


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Another from of Taylor's


Put a h x   then (2) becomes
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Note 1
If f has continuous derivatives


is called the taylor series expansion of f(x)
about x a.
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Note 2 Suppose x 0


is called the maclaurin series expansion of
f(x). Note 3 If n 1, Taylor's theorem
reduces to LMVT.
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Problem 13
Hence show that
Solution
f (x) log x has continuous derivatives in 0,
2 We will Expand f (x) about the point x
1 We have f(x) log x f(1) 0
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Using the above in the Taylors series
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Put x 1.1
log (1.1) ? 0.0953
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Problem 14
Expand sin x in powers of upto the
term containing
Solution
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Problem 15
Obtain the Maclaurin series of expansion for the
function log (1 tan x) upto term containing
x3.
Solution
Let y log (1 tan x) y (0) 0
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Differentiating (1), we get
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Problem 16
Obtain the Maclaurin series of expansion for the
function sin-1 x.
Solution
Differentiate w.r.t x (1)
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Differentiate w.r.t x (2)
Applying Leibnizs' Rule
Put x 0
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Problem 17
Obtain the Maclaurin series of expansion for the
function log (1 sin x).
Solution
Let y log (1 sin x)
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Problem 18
Obtain the Maclaurin series of expansion for the
function log (sec x).
Solution
y (0) 0
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Problem 19
Obtain the Maclaurin series of expansion for the
function log (1 ex).
Solution
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Indeterminate Forms
Forms like
are called in determinate forms


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Indeterminate form (L'Hospital's Rule)
Let f(x) and g(x) be functions such that f(a) 0
and g(a) 0 or

 

Note If f(a), f"(a),..f(n-1)(a) and g'(a),
g"(a),,g(n-1)(a) are all zero but
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Working Rule
1. Differentiate Numerator and Denominator
separately and put x a. If this reduces
to indeterminate form continue the above
procedure until a finite value is
obtained.   2. Incase where the expansions of
functions, involved in indeterminate form
are known, or some of the standard limits are
known, may be used to simplify the work.

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Problem 20
Solution
Obtain the Maclaurin series of expansion for the
function log (1 ex).
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Problem 21
Solution
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Problem 22
Solution
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Problem 23
Find the value of 'a' so that
Solution
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Problem 24
Solution
which gives -3a b 0 ..(2). Solving (1)
and (2), we get a 1 and b 3.
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Indeterminate form
This form can be converted to form and the
L'Hospital Rule applied.


Note L'Hospital Rule can be applied to the form
without adjusting to the form
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Indeterminate form
 


In this case we can write f (x) . g (x) as
and the form reduces to where L'Hospital
Rule is applicable.
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Indeterminate form


where L'Hospital Rule is applicable.
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Problem 25
Solution
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Indeterminate form 0o
   


Note Similar technique of taking logarithms is
followed in case of the indeterminate forms
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Problem 26
Solution
log y 0
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Problem 27
Solution
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Problem 28
Solution
log a log e
log y log ae y ae
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Problem 29
Solution
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log y 0 y 1
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Problem 30
Solution
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Problem 31
Solution
log a log e
log y log ae y ae
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Problem 32
Solution
log y 0 y 1
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Taylor's Theorem for a Function of Two Variables
Let f(x, y) be a function of two variables
defined over a region R in the x - y plane. Let
P(a, b) and Q(ah, bk) be two neighbouring
points in the region R then


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Cor Put a h x and b k y then


is called the taylor's series expansion of f (x,
y) about (a, b), (as, f (x, y)
possessing continuous partial derivatives of all
order).
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Maclaurin's Expansion
For a 0 and b 0 we get from (2)


is called the Maclaurin's series expansion of
f(x, y).
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Problem 33
Expand sin x cos y in powers of x and y as far as
terms of third degree.
Solution
We have f sinx cos y f (0, 0)
0 fx cosx cosy fx (0, 0 )
1 fy -sinx siny fy (0, 0)
0
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Problem 34
Solution
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Maxima and Minima
Definition A function f(x, y) of two independent
variables is said to have a minimum at (a, b),
if f(a, b) gt f(ah, bk) in the neighbourhood of
(a, b). Note 1 A maximum or minimum value of
a function is called its extreme value. Note
2 A function f(x, y) is said to be stationary
at (a, b) if fx(a, b) 0 and fy(a, b) 0.


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Note 3 A sufficient condition for f(x, y) to
have a maximum at a critical point (a, b) is that
r lt 0 and rt - s2 gt 0 at (a, b).   Note 4 A
sufficient condition for f(x, y) to have a
minimum at a critical point (a, b) is that r gt 0
and rt - s2 gt 0 at (a, b).   Note 5   If rt - s2
lt 0 at (a, b) then (a, b) is called a saddle
point.   Note 6   If rt - s2 0 and/or r 0
at (a, b) than no conclusion can be drawn about
the nature of (a, b).


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Problem 35
Show that the function f (x, y) x3 y3 - 3xy
1 is minimum at (1, 1).
Solution
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We have r 6 gt 0 \ f (x, y) is minimum at
(1, 1).
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Problem 36
Show that the function f (x, y) x y (a - x -
y), a gt 0 is max. at the point
Solution
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Now
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Problem 37
Find the extreme values of f(x,y) sinx siny
sin()x y,
Solution
Let f(x, y) sinx siny sin (x y)
siny sin (2xy)
sinx sin (x 2y)
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Problem 38
Examine the function f (x, y) 1 sin (x2 y2)
for extreme values.
Solution
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Case (i) At (0, 0) p 0 r gt 0
s 0 t 2 Case (ii) At (a, b)
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p 0
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Since rt - s2 0, no conclusion can be made.
Further investigation is required. But
1 1 2 \ Maximum value of f(x,y)
is 2.
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Problem 39
Find the extreme value of the function f(x, y)
x3 3xy2 - 3x2 - 3y2 4.
Solution
We get   fx 3x2 3y2 - 6x    fy 6xy -
6y    fxx 6x - 6    fxy 6y s    fyy 6x -
6    Solving fx 0 and fy 0
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• i.e., 3x2 3y2 - 6x 0 (1) and 6xy - 6y 0
  .(2)
•  
• from (2) we get ? x 1 or y 0
•  
• y 0 in (1)
•  
• ? 3x2 - 6x 0 ? 3x (x - 2) 0
• ? x 0 or x 2
•  
• The critical points (0, 0) (2, 0)
• Put x 1 in (1)
• ? 3y2 - 3 0
• ? y ? 1

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The other critical points are (1, 1), (1,
-1) To examine the nature of the critical points
observe the table.
\ Max. f (x, y) f (0, 0) 4 and   Min. f
(x, y) f (2, 0) 0
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Lagrange's Method of undetermined Multipliers
This method is used to find the stationary values
of a function of several variables subject to a
given condition.   Note The disadvantage of
this method is we cannot determine the nature of
the stationary values.


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Let f(x, y, z) be a function of the variables x,
y ,z subject to the condition.   f(x,y, z)
c (1) Let F f lf (2) where l is a
constant


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Let f(x, y, z) be a function of the variables x,
y ,z subject to the condition.   f(x,y, z)
c (1) Let F f lf (2) where l is a
constant We form the auxillary equations   Fx
0 (3) Fy 0 (4) Fz 0 (5)   Using
(2), (3), (4) and (5) we obtain the stationary
point (x1, y1, z1) and f(x1, y1, z1) is the
stationary value.


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Problem 40
The temperature T at any point (x, y, z) in space
is given by T 400 xyz2. Find the highest
temperature at the surface of a unit sphere x2
y2 z2 1.
Solution
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Problem 41
Find the maximum and minimum distances of the
point (1, 2, 3) from the sphere (x2 y2 z2)
56.
Solution
Let A (1, 2, 3) Let P (x, y, z) be a point on
the sphere (x2 y2 z2) 56 .(1)
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From (2) and (3) from (2) and (4)
Using in (1)
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\ y 4, z 6 for x 2   y -4, z
-6 for x -2   The critical points are (2, 4,
6) and (-2, -4, -6). At (2, 4, 6), f 1 4
9 fmin 14 and the minimum distance is
. At (-2, -4, -6) f 9 36 81 \
fmin 14 and the minimum distance is
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Problem 42
A rectangular box open at the top is to have a
volume of 32 cubit feet. Find its dimension if
the total surface area is minimum.
Solution
Let x, y and z be the dimensions of the
box. Given xyz 32 .(1) \ f(x, y,
z) xyz   Let f xy 2yz 2xz (be the
surface area) Let F f lf xy
2yz 2xz l xyz
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Solve Fx y 2z lyz 0 (2) Fy x 2z
lxy 0 (3) Fz 2y 2x lxy 0 (4)
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From (2) and (3) From (2) and (4) x
y x 2z Using in (1)
x 4 y 4 z 2 units
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Summary
Derivative of arc in Cartesian forms

Derivative of arc in parametric form

Derivative of arc in polar forms
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Radius of curvature
Curvature
Formula for radius of curvature in cartesian form


Formula for radius of curvature in cartesian form
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Formula for radius of curvature in pedal form


Formula for radius of curvature in polar form
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Rolle's theorem If f(x) is i) Continuous in
a, b ii) Differentiable in (a, b) and iii)
f(a) f(b)


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Lagranges Mean Value Theorem If f(x) is i)
Continuous in a, b and ii) Differentiable in
(a, b) then such that


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Cauchy's Mean Value theorem If f(x) and g(x)
are two functions such that i) f(x) and g(x) are
both Continuous in a, b ii) f(x) and g(x) are
both differentiable in (a, b) and iii)


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Taylor's Theorem Let f(x) be a function such
that   i) f, f' , f'', ., f(n-1) are continuous
in a, b ii) f (n-1) is differentiable in (a,
b) then


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Indeterminate form (L'Hospital Rule) If
f (x) and g (x) are such that


L'Hospital Rule can be applied to other
indeterminate forms by converting to the form
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Taylor's Theorem for a function of two variables


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Maclaurin's Expansion f(x, y) f(0, 0) x fx
(0, 0) yfy (0, 0)


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Stationary point (a, b) is called a stationary
point if p fx(a, b) 0 and q fy(a, b) 0


If (a, b) is a stationary point. Then fx(a, b)
and fy(a, b) 0 and the nature of the
stationary point is representating the table.
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Lagrange's Method of undetermined Multipliers
To find stationary values of f (x, y, z)
subject to f(x, y, z) c .(1) Step I F f
lf Step II Auxillary equations Fx 0, Fy
0, Fz 0 are formed. Step III Solve for x, y
and z using f(x, y, z) and auxillary equations.


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• Conclusion
• Differential calculus is used in the studying
  of curves and their properties
• related to derivative of arcs and radius of
  curvature.
• We have studied Mean value theorems and their
  geometrical interpretations.
• Also we have learnt about the expansions of
  functions using Taylor's series
• expansion and Maclaurin series expansion.
•  
• L'Hospital Rule is used to evaluating limits
  in indeterminate forms.
•  
• Partial derivatives are used to obtain
  stationary points and to examine their
• nature.
•  
• Lagrange's Method of Indetermined Multipliers
  are used to obtain the
• stationary values of a function of several
  variable subject to a given
• condition.