Graphs in Rectangular Coordinates System (23... · 2015-07-02 · INTRODUCTION French mathematician...

CONTENTS

Graphs and its applications

Preliminary Definitions

Asymptote.Horizontal Asymptote.Vertical Asymptote.Asymptotes of a Rational function.Examples

Imran Talib Faculty of Mathematics and Statistics (VU).

Graphs in Rectangular Coordinates System

CONTENTS

Asymptote.

Horizontal Asymptote.Vertical Asymptote.Asymptotes of a Rational function.Examples

CONTENTS

Asymptote.Horizontal Asymptote.

Vertical Asymptote.Asymptotes of a Rational function.Examples

CONTENTS

Asymptote.Horizontal Asymptote.Vertical Asymptote.

Asymptotes of a Rational function.Examples

CONTENTS

Asymptote.Horizontal Asymptote.Vertical Asymptote.Asymptotes of a Rational function.

Examples

CONTENTS

INTRODUCTION

French mathematician and philosopher Rene Descartes(1596-1650) devised a simple plan whereby two number lineswere intersected at right angles. This system is called therectangular coordinate system (or Cartesian coordinatesystem)

Graphs are used to summarize and display information in amanner that is easy for most people to comprehend.

Have usage in many academic disciplines, including math,hard sciences and social sciences.

INTRODUCTION

PRELIMINARY DEFINITIONS

ASYMPTOTE

An asymptote is a line to which the graph gets arbitrarilyclose.That means that for any distance named, no matter howsmall, the graph will get within that distance and stay within thatdistance for some section of the graph with infinite length.Moreprecisely

ASYMPTOTE

An asymptote is a line to which the graph gets arbitrarilyclose.

That means that for any distance named, no matter howsmall, the graph will get within that distance and stay within thatdistance for some section of the graph with infinite length.Moreprecisely

ASYMPTOTE

An asymptote is a line to which the graph gets arbitrarilyclose.That means that for any distance named, no matter howsmall, the graph will get within that distance

and stay within thatdistance for some section of the graph with infinite length.Moreprecisely

ASYMPTOTE

An asymptote is a line to which the graph gets arbitrarilyclose.That means that for any distance named, no matter howsmall, the graph will get within that distance and stay within thatdistance for some section of the graph with infinite length.

Moreprecisely

ASYMPTOTE

An asymptote is a line to which the graph gets arbitrarilyclose.That means that for any distance named, no matter howsmall, the graph will get within that distance and stay within thatdistance for some section of the graph with infinite length.Moreprecisely

HORIZONTAL ASYMPTOTE

The graph of y = f (x) has a horizontal asymptote of y = b, ifand only if, either

limx→∞

f (x) = b or limx→−∞

f (x) = b. (1)

limx→∞

f (x) = b. (1)

limx→∞

f (x) = b. (1)

limx→∞

f (x) = b. (1)

VERTICAL ASYMPTOTE

The graph of y = f (x) has a vertical asymptote of x = c , if andonly if, either

limx→c−

f (x) = ±∞ or limx→c+

f (x) = ±∞. (2)

VERTICAL ASYMPTOTE

limx→c−

f (x) = ±∞. (2)

VERTICAL ASYMPTOTE

limx→c−

f (x) = ±∞. (2)

VERTICAL ASYMPTOTE

limx→c−

f (x) = ±∞. (2)

ASYMPTOTES OF A RATIONAL FUNCTIONS

Let r(x) be a rational function with polynomialp(x) = anxn + ... + a0 of degree n in the numeratorand polynomialq(x) = bmxm + ... + b0 of degree m in the denominator

If n < m, then r(x) has a horizontal asymptote of y = 0.

If n > m, then r(x) becomes unbounded for large values of x(positive or negative).

If n = m, then r(x) has a horizontal asymptote of y = anbn

Let r(x) be a rational function with polynomialp(x) = anxn + ... + a0 of degree n in the numerator

and polynomialq(x) = bmxm + ... + b0 of degree m in the denominator

EXAMPLE(1):

Discuss the Asymptotes of the following rational function

r(x) =1

x, (3)

This is defined ∀x 6= 0.Consider the sequence of numbers xn = 1

2 , 13 , 1

4 , 15 , · · · , 1

k fork = 1, 2, · · · , n.These numbers are getting closer and closer to zero.Since r(xn) =

So, r(xn) = 2, 3, 4, · · · , k, · · · for n = 2, 3, 3, · · · , k, · · · ,which is getting larger and larger, so approaching the verticalline x = o.

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

EXAMPLE(1):

r(x) =1

x, (3)

This is defined ∀x 6= 0.

Consider the sequence of numbers xn = 12 , 1

3 , 14 , 1

5 , · · · , 1k for

k = 1, 2, · · · , n.These numbers are getting closer and closer to zero.Since r(xn) =

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

k fork = 1, 2, · · · , n.

These numbers are getting closer and closer to zero.Since r(xn) =

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

k fork = 1, 2, · · · , n.These numbers are getting closer and closer to zero.

Since r(xn) =1xn

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

EXAMPLE(1):

r(x) =1

x, (3)

2 , 13 , 1

4 , 15 , · · · , 1

EXAMPLE(1):

Thus there is a vertical asymptote at x = 0.

EXAMPLE(1):

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Defined ∀x 6= −2

The numerator and denominator are linear functions (degreeof polynomials are the same).

we can see that as x get large then 2 in the denominatorbecomes insignificant.

Thus r(x) ≈ 10xx = 10

Thus horizontal asymptote occurs at x = 10.

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

r(x) =10x

2 + x, (4)

Thus r(x) ≈ 10xx = 10

EXAMPLE(2):

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

Defined ∀x 6= ±2.Clearly the function passes through the origin, so the x andy -intercept is (x , y) = (0, 0).Note that this function is an even function, the edge of thedomain is x = 2, so we see there are vertical asymptotes atx = 2.Thus, for x large r(x) ≈ 4x2

−x2 = −4.Thus, a horizontal asymptote occurs at y = −4.

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

Defined ∀x 6= ±2.

Clearly the function passes through the origin, so the x andy -intercept is (x , y) = (0, 0).Note that this function is an even function, the edge of thedomain is x = 2, so we see there are vertical asymptotes atx = 2.Thus, for x large r(x) ≈ 4x2

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

Defined ∀x 6= ±2.Clearly the function passes through the origin, so the x andy -intercept is (x , y) = (0, 0).

Note that this function is an even function, the edge of thedomain is x = 2, so we see there are vertical asymptotes atx = 2.Thus, for x large r(x) ≈ 4x2

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

Defined ∀x 6= ±2.Clearly the function passes through the origin, so the x andy -intercept is (x , y) = (0, 0).Note that this function is an even function, the edge of thedomain is x = 2, so we see there are vertical asymptotes atx = 2.

Thus, for x large r(x) ≈ 4x2

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

−x2 = −4.

Thus, a horizontal asymptote occurs at y = −4.

EXAMPLE(3):

r(x) =4x2

4− x2, (5)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

Obviously deg(g(x)) < deg(q(x)).

Implies g (x)q(x)→ 0 as x → ±∞.

Thus r(x)→ f (x) as x → ±∞.

Particularly if deg(p(x)) is one more less than deg(q(x)),then f (x) is a linear function whose graph is a non-horizontalline in the plane.

That line is called oblique asymptote.

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

OBLIQUE ASYMPTOTE:

r(x) = p(x)q(x)

= f (x) + g (x)q(x)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

Clearly the quotient is a linear function.

Thus it is our oblique asymptote.

For more understanding see the graph.

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

EXAMPLE(1):

r(x) =x2 − 3

2x + 1,

2x − 1 +

2x + 1. (6)

OBLIQUE ASYMPTOTE:

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

EXAMPLE(2):

r(x) =x3 − 1

x2 − x − 2,

= x + 1 +3x + 1

x2 − x − 2. (7)

OBLIQUE ASYMPTOTE:

ASYMPTOTES IN POLAR COORDINATES

POLAR EQUATION OF A STRAIGHT LINE:

The polar equation of any line is p = r cos(θ − α).where p is the length of the perpendicular from pole to theline.α is the angle which this perpendicular makes with the initialline.

The polar equation of any line is p = r cos(θ − α).

where p is the length of the perpendicular from pole to theline.α is the angle which this perpendicular makes with the initialline.

The polar equation of any line is p = r cos(θ − α).where p is the length of the perpendicular from pole to theline.

α is the angle which this perpendicular makes with the initialline.

The polar equation of any line is p = r cos(θ − α).where p is the length of the perpendicular from pole to theline.α is the angle which this perpendicular makes with the initialline.

POLAR EQUATION OF A STRAIGHT LINE (PROOF):

Let OX be initial line and P be any point on the line whosepolar coordinates are (r , θ).

Therefore, OP = r and ∠POX = θ.

Let us draw a perpendicular OY on the line from P produced.

Now using 4OPY , we have

cosPOY = OYOP = p

p = r cosPOY = r cos(θ − α), (8)

cosPOY = OYOP = p

Now we determine the asymptotes of the curve r = s(θ).

For this we have to determine the constant p and α.

CONSTRUCTION:

Let P(r , θ) be any point on the curve AB whose equation isr = p(θ).

Let us draw a perpendicular OY on the line p = r cos(θ − α)which is MY .

Therefore

PM = LY = OY −OL = p −OP cos(θ − α)

= p − r cos(θ − α). (9)

CONSTRUCTION:

Therefore

= p − r cos(θ − α). (9)

CONSTRUCTION:

Therefore

= p − r cos(θ − α). (9)

CONSTRUCTION:

Therefore

= p − r cos(θ − α). (9)

CONSTRUCTION:

Therefore

= p − r cos(θ − α). (9)

CONSTRUCTION:

Therefore

= p − r cos(θ − α). (9)

CONSTRUCTION:

Now r → ∞ as the point recedes to infinity along thecurve.Let θ → θ1 where r → ∞.

Therefore we have PMr = p

r − cos(θ − α) by (9). whenr → ∞, PM → 0.

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

Now r → ∞ as the point recedes to infinity along thecurve.

Let θ → θ1 where r → ∞.

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

So that

r= PM.

r→ 0 and

r→ 0. (10)

CONSTRUCTION:

cos(θ − α) = 0

cos(θ − α) = cos(π

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

cos(θ − α) = 0

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

cos(θ − α) = 0

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

cos(θ − α) = 0

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

cos(θ − α) = 0

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

cos(θ − α) = 0

θ1 − α =π

2(∵ θ → θ1 as r → ∞)

θ1 = α +π

α = θ1 −π

2. (11)

CONSTRUCTION:

Again from (9), we have when r → ∞

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r cos(θ − θ1 +π

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limr→∞

[r cos(θ − α)]

= limr→∞

[r sin(θ1 − θ)]

= limθ→θ1

[sin(θ1 − θ)

]. . .

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

=limθ→θ1(cos(θ1 − θ))

limθ→θ1(− 1r2

drdθ )

. (13)

Now let u = 1r ,this implies du

dθ = − 1r2

drdθ . Therefore (13)

becomes

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

limθ→θ1(− 1r2

drdθ )

. (13)

dθ = − 1r2

becomes

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

limθ→θ1(− 1r2

drdθ )

. (13)

dθ = − 1r2

becomes

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

limθ→θ1(− 1r2

drdθ )

. (13)

Now let u = 1r ,

this implies dudθ = − 1

r2drdθ . Therefore (13)

becomes

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

limθ→θ1(− 1r2

drdθ )

. (13)

dθ = − 1r2

becomes

CONSTRUCTION:

p = limθ→θ1

[cos(θ1 − θ)

− 1r2

limθ→θ1(− 1r2

drdθ )

. (13)

dθ = − 1r2

becomes

CONSTRUCTION:

p = limθ→θ1−dθ

du. (14)

Hence the asymptotes is

limθ→θ1−dθ

du= r cos(θ − α)

= r cos(θ − θ1 +π

= r sin(θ1 − θ), (15)

CONSTRUCTION:

du. (14)

limθ→θ1−dθ

= r sin(θ1 − θ), (15)

CONSTRUCTION:

du. (14)

limθ→θ1−dθ

= r sin(θ1 − θ), (15)

CONSTRUCTION:

du. (14)

limθ→θ1−dθ

= r sin(θ1 − θ), (15)

CONSTRUCTION:

du. (14)

limθ→θ1−dθ

= r sin(θ1 − θ), (15)

CONSTRUCTION:

du. (14)

limθ→θ1−dθ

= r sin(θ1 − θ), (15)

WORKING RULES:

We have to change r to 1u in the given equation and find out

the limit of θ as u → 0.

Let θ1 be any one of the angle possible limits of θ.

We have to determine then − dθdu and its limit as u → 0 and

θ → θ1.

Let this limit be p.

Then p = r sin(θ1 − θ) is the required asymptote.

WORKING RULES:

θ → θ1.

WORKING RULES:

θ → θ1.

WORKING RULES:

θ → θ1.

WORKING RULES:

θ → θ1.

WORKING RULES:

θ → θ1.

EXAMPLE(1):

Find the asymptotes of the curve

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

r sin(θ) = 2 cos(2θ) (16)

Step1:(16) implies

r =2 cos 2θ

sin θ1

sin θ

2 cos 2θ, (17)

EXAMPLE(1):

according to working rules , we can write (17) as

u =sin θ

2 cos 2θ. (18)

Step2: As u → 0 then θ → nπ, n ∈ Z .

Step3: Now we determine − limθ→nπ(dθdu )

EXAMPLE(1):

u =sin θ

2 cos 2θ. (18)

EXAMPLE(1):

u =sin θ

2 cos 2θ. (18)

EXAMPLE(1):

u =sin θ

2 cos 2θ. (18)

EXAMPLE(1):

u =sin θ

2 cos 2θ. (18)

EXAMPLE(1):

− limθ→nπ

)= lim

θ→nπ

cos 2θ cos θ − sin θ(−2 sin 2θ)

cos2 2θ

=cos2(2nπ)

cos(2π) cos(π)− sin(π)(−2 sin(2π)

cos(nπ)

(−1)n. (19)

EXAMPLE(1):

− limθ→nπ

)= lim

θ→nπ

cos2 2θ

=cos2(2nπ)

cos(nπ)

(−1)n. (19)

EXAMPLE(1):

− limθ→nπ

)= lim

θ→nπ

cos2 2θ

=cos2(2nπ)

cos(nπ)

(−1)n. (19)

EXAMPLE(1):

− limθ→nπ

)= lim

θ→nπ

cos2 2θ

=cos2(2nπ)

cos(nπ)

(−1)n. (19)

EXAMPLE(1):

− limθ→nπ

)= lim

θ→nπ

cos2 2θ

=cos2(2nπ)

cos(nπ)

(−1)n. (19)

EXAMPLE(1):

Step4: This limit be p. That is p = − 2(−1)n . Hence, the

required asymptotes are

r sin(nπ − θ) = − 2

(−1)n

r [(−1)n−1 sin θ] =2

(−1)n

r sin(θ) = 2. (20)

EXAMPLE(1):

r sin(nπ − θ) = − 2

(−1)n

r [(−1)n−1 sin θ] =2

(−1)n

r sin(θ) = 2. (20)

EXAMPLE(1):

r sin(nπ − θ) = − 2

(−1)n

r [(−1)n−1 sin θ] =2

(−1)n

r sin(θ) = 2. (20)

EXAMPLE(1):

r sin(nπ − θ) = − 2

(−1)n

r [(−1)n−1 sin θ] =2

(−1)n

r sin(θ) = 2. (20)

EXAMPLE(1):

r sin(nπ − θ) = − 2

(−1)n

r [(−1)n−1 sin θ] =2

(−1)n

r sin(θ) = 2. (20)

EXERCISES):

Find the asymptotes of the curve r = r tan θ.

Find the asymptotes of the curve r sin nθ = a.

Find the equations of the horizontal asymptote for thefollowing rational curves1−6x3x+7

3x3−44x2−3x+12x2−1

EXERCISES):

Find the asymptotes of the curve r = r tan θ.

Find the asymptotes of the curve r sin nθ = a.

Find the equations of the horizontal asymptote for thefollowing rational curves1−6x3x+7

3x3−44x2−3x+12x2−1

EXERCISES):

Find the equations of the oblique asymptote for the followingrational curves3x3+2x2−x−15x2−3x+1

x+2x2+2x−2(1−x)3

x3−3x2x2−1

Find the equations of the asymptotes of the following curves

(x − y)2(x2 + y2)− 10(x − y)x2 + 12y2 + 2x + y = 0

x2y + xy2 + xy + y2 + 3x = 0.

EXERCISES):

Find the equations of the oblique asymptote for the followingrational curves3x3+2x2−x−15x2−3x+1

x+2x2+2x−2(1−x)3

x3−3x2x2−1

Find the equations of the asymptotes of the following curves

(x − y)2(x2 + y2)− 10(x − y)x2 + 12y2 + 2x + y = 0

x2y + xy2 + xy + y2 + 3x = 0.

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

A function f has an absolute maximum (or global maximum)at c if f (c) ≥ f (x) for all x in D.

Where D is the domain of f .

The number f (c) is called the maximum value of f on D.

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

A function f has an absolute maximum (or global maximum)at c

if f (c) ≥ f (x) for all x in D.

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

MAXIMA AND MINIMA

ABSOLUTE MAXIMUM:

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

A function f has an absolute minimum (or global minimum)at d if f (d) ≤ f (x) for all x in D.

The number f (d) is called the minimum value of f on D.

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

A function f has an absolute minimum (or global minimum)at d

if f (d) ≤ f (x) for all x in D.

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

MAXIMA AND MINIMA

ABSOLUTE MINIMA:

MAXIMA AND MINIMA

LOCAL MAXIMUM OR MINIMUM:

A function f has a local maximum (or relative maximum) at cif f (c) ≥ f (x) when x is near c .This means that f (c) ≥ f (x) for all x in some open intervalcontaining c .Similarly, f has a local minimum (or relative minimum) at d iff (d) ≤ f (x) when x is near c .

MAXIMA AND MINIMA

A function f has a local maximum (or relative maximum) at cif f (c) ≥ f (x) when x is near c .

This means that f (c) ≥ f (x) for all x in some open intervalcontaining c .Similarly, f has a local minimum (or relative minimum) at d iff (d) ≤ f (x) when x is near c .

MAXIMA AND MINIMA

A function f has a local maximum (or relative maximum) at cif f (c) ≥ f (x) when x is near c .This means that f (c) ≥ f (x) for all x in some open intervalcontaining c .

Similarly, f has a local minimum (or relative minimum) at d iff (d) ≤ f (x) when x is near c .

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

f (x) has absolute and local minimum at x = 0.

Having no absolute or local maximum.

The right most diagram showing the function behavior whenx ∈ (−∞, 0) ∪ (0, ∞).

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

MAXIMA AND MINIMA

EXAMPLES:

Let f (x) = x2, x ∈ R

MAXIMA AND MINIMA

LOCAL EXAMPLES:

Let f (x) = f (x) = x4 + x3 − 11x2 − 9x + 18 =(x − 3)(x − 1)(x + 2)(x + 3), x ∈ R

f (x) has the absolute minimum at x ≈ 2.2 and has noabsolute maximum.

It has two local minima at x ≈ −2.6 and x ≈ 2.2.

Having local maximum at x ≈ −0.4.

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

MAXIMA AND MINIMA

LOCAL EXAMPLES:

EXTREME VALUE THEOREM

If f is continuous on a closed interval [a, b],

then f attains anabsolute maximum value f (c) at some number c and an absoluteminimum value f (d) at some number d .

If f is continuous on a closed interval [a, b],then f attains anabsolute maximum value f (c) at some number c

and an absoluteminimum value f (d) at some number d .

If f is continuous on a closed interval [a, b],then f attains anabsolute maximum value f (c) at some number c and an absoluteminimum value f (d) at some number d .

FERMATS THEOREM

If f has a local maximum or minimum at some number c ,

and iff′(c) exists, then f ′(c) = 0.

FERMATS THEOREM

If f has a local maximum or minimum at some number c , and iff′(c) exists,

then f ′(c) = 0.

FERMATS THEOREM

If f has a local maximum or minimum at some number c , and iff′(c) exists, then f ′(c) = 0.

FERMATS THEOREM

If f has a local maximum or minimum at some number c , and iff′(c) exists, then f ′(c) = 0.

FERMATS THEOREM

REMARK(1):

The converse of this theorem is not true.

In other words, when f ′(c) = 0, f does not necessarily have alocal maximum or minimum.

For example, if f (x) = x3, then f′(x) = 3x2 equals 0 at

x = 0.

But, x = 0 is not a point of a local minimum or maximum.

FERMATS THEOREM

REMARK(1):

x = 0.

FERMATS THEOREM

REMARK(1):

x = 0.

FERMATS THEOREM

REMARK(1):

x = 0.

FERMATS THEOREM

REMARK(1):

x = 0.

FERMATS THEOREM

REMARK(2):

Sometimes f′(c) does not exist, but some number f (c) is a

point of a local maximum or minimum.

For example, if f (x) = |x |, then f′(0) does not exist.

But f (x) has its local (and absolute) minimum at x = 0.

FERMATS THEOREM

REMARK(2):

FERMATS THEOREM

REMARK(2):

FERMATS THEOREM

REMARK(2):

FERMATS THEOREM

REMARK(2):

CRITICAL NUMBER

A critical number of a function f is a number c in the domain of fsuch that either f

′(c) = 0 or f

′(c) does not exist.

REMARK(3):

From Fermat’s Theorem it follows that if f has a local maximumor minimum at c , then c is a critical number of f .

CRITICAL NUMBER

A critical number of a function f is a number c in the domain of fsuch that either f

′(c) = 0 or f

′(c) does not exist.

REMARK(3):

From Fermat’s Theorem it follows that if f has a local maximumor minimum at c , then c is a critical number of f .

CRITICAL NUMBER

EXAMPLES:

If f (x) = 2x2 + 5x − 1, x ∈ R, then f′(x) = 4x + 5. Hence

the only critical number of f is x = − 54 .

If f (x) =3√x2, x ∈ R, then f

′(x) = 2

3x−13 . Hence the only

critical number of f is x = 0.

If f (x) = 1x , x ∈ R− {0}, then f

′(x) = − 1

x2. Since x = 0

is not in the domain, f has no critical numbers.

CRITICAL NUMBER

EXAMPLES:

′(x) = 2

If f (x) = 1x , x ∈ R− {0}, then f

′(x) = − 1

x2. Since x = 0

CRITICAL NUMBER

EXAMPLES:

′(x) = 2

If f (x) = 1x , x ∈ R− {0}, then f

′(x) = − 1

x2. Since x = 0

CRITICAL NUMBER

EXAMPLES

CRITICAL NUMBER

EXAMPLES

CRITICAL NUMBER

EXERCISE

Find the critical numbers of f (x) = 2x3 − 9x2 + 12x − 5.

Find the critical numbers of f (x) = 2x + 33√x2.

CRITICAL NUMBER

EXERCISE

Find the critical numbers of f (x) = 2x3 − 9x2 + 12x − 5.

Find the critical numbers of f (x) = 2x + 33√x2.

RELATIVE MAXIMA AND MINIMA

FIRST DERIVATIVE TEAT

If f′(x) > 0 on an open interval extending left from c and

f′(x) < 0 on an open interval extending right from c , then f

has a relative maximum at c .

If f′(x) < 0 on an open interval extending left from c and

f′(x) > 0 on an open interval extending right from c , then f

has a relative minimum at c .

If f′(x) has the same sign on both an open interval extending

left from c and an open interval extending right from c, thenf does not have a relative extremum at c .

Let f (x) = 3x4 − 4x3 − 12x2 + 3.

Differentiable everywhere on [−2.3] , with f′(x) = 0 for

x = −1, 0, 2.By first derivative test f has a relative maximum at x = 0 andrelative minima at x = −1 and x = 2.

Let f (x) = 3x4 − 4x3 − 12x2 + 3.Differentiable everywhere on [−2.3] , with f

′(x) = 0 for

x = −1, 0, 2.

By first derivative test f has a relative maximum at x = 0 andrelative minima at x = −1 and x = 2.

′(x) = 0 for

SECOND DERIVATIVE TEAT

Let c is a critical point at which f′(c) = 0.

Let f′(x) exists in a neighborhood of c .

Let f′′(c) exists.

Then f has a relative maximum value at c if f′′(c) < 0,

and a relative minimum value at c if f′′(c) > 0 .

If f′′(c) = 0 , the test is not informative.

SINGULAR POINTS

A point on the curve at which the curve exhibits anextra-ordinary behavior is called a Singular Point.Mean to say where f

′(x) is not defined but f (x) is defined.

Points of inflection and multiple points are the types ofsingular points.

SINGULAR POINTS

A point on the curve at which the curve exhibits anextra-ordinary behavior is called a Singular Point.

Mean to say where f′(x) is not defined but f (x) is defined.

SINGULAR POINTS

EXAMPLE

Discuss the singular points of f (x) = (x − 1)23 − 3(x − 1) on

[0, ∞].

Look for values x where f′(x) is not defined, but f (x) is

defined.

f′(x) = 2

3(x−1)13− 3.

Notice that the denominator is zero when x = 1, so thatf′(x) is not defined when x = 1, even though f (x) is defined

when x = 1.

Thus, we have a singular point at x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

[0, ∞].

defined.

f′(x) = 2

3(x−1)13− 3.

when x = 1.

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

DEFINITIONS

A Point on the curve through which more than one branch ofthe curve pass is called Multiple Point.

A Point on the curve through which two branches of the curvepass is called Double Point..

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

A Point on the curve through three branches of the curve passis called Triple Point.A Point on the curve through which r branch of the curvepass is called Multiple Point of rth order.A Double Point P on a curve is called a Node if two realbranches of a curve pass through P and two tangents at whichare real and different. Thus the point P shown is a Node.A Double Point Q on a curve is called a Cusp if two realbranches of a curve pass through Q and two tangents atwhich are real and coincident. Thus the point Q shown in theadjoining two figures is a Cusp.

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

A Point on the curve through three branches of the curve passis called Triple Point.

A Point on the curve through which r branch of the curvepass is called Multiple Point of rth order.A Double Point P on a curve is called a Node if two realbranches of a curve pass through P and two tangents at whichare real and different. Thus the point P shown is a Node.A Double Point Q on a curve is called a Cusp if two realbranches of a curve pass through Q and two tangents atwhich are real and coincident. Thus the point Q shown in theadjoining two figures is a Cusp.

SINGULAR POINTS

DEFINITIONS

A Point on the curve through three branches of the curve passis called Triple Point.A Point on the curve through which r branch of the curvepass is called Multiple Point of rth order.

A Double Point P on a curve is called a Node if two realbranches of a curve pass through P and two tangents at whichare real and different. Thus the point P shown is a Node.A Double Point Q on a curve is called a Cusp if two realbranches of a curve pass through Q and two tangents atwhich are real and coincident. Thus the point Q shown in theadjoining two figures is a Cusp.

SINGULAR POINTS

DEFINITIONS

A Point on the curve through three branches of the curve passis called Triple Point.A Point on the curve through which r branch of the curvepass is called Multiple Point of rth order.A Double Point P on a curve is called a Node if two realbranches of a curve pass through P and two tangents at whichare real and different. Thus the point P shown is a Node.

A Double Point Q on a curve is called a Cusp if two realbranches of a curve pass through Q and two tangents atwhich are real and coincident. Thus the point Q shown in theadjoining two figures is a Cusp.

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

A Double Point R on a curve is called a Conjugate Point orIsolated point if there exists no real points of the curve in theneighborhood of R.

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

DEFINITIONS

SINGULAR POINTS

WORKING RULE FOR INVESTIGATING THE NATURE OF THE

DOUBLE POINT AT THE ORIGIN.

Find the tangents at the origin by equating to zero the lowestdegree terms present in the equation of the curve. If origin isa double point, then we shall get tangents real and imaginary.

If the two tangents at the origin are imaginary, then the originis a conjugate point.

If the two tangents at the origin are real and different, thenthe tangent is a node or a conjugate point.

If the two tangents at the origin are real and coincident, thenthe origin is a cusp or a conjugate point.

SINGULAR POINTS

REMARK.

To study the nature of the curve near origin

If the tangents at the origin are y2 = 0, solve the equation ofthe curve for y , neglecting all terms of y having powers abovesecond. If for small non zero values of x , the values of y arereal, then the branches of the curve through the origin arealso real, otherwise they are imaginary.

If the tangents at the origin are x2 = 0, solve the givenequation of the curve for x instead of y and proceed asmentioned in the above point.

In other cases, solve for y or x , whichever is convenient.

SINGULAR POINTS

REMARK.

SINGULAR POINTS

REMARK.

SINGULAR POINTS

REMARK.

SINGULAR POINTS

REMARK.

SINGULAR POINTS

REMARK.

Conditions for the existence of Multiple Points

For Multiple Points of the curve f (x , y) = 0, ∂f∂x = 0 and

∂f∂y = 0.

SINGULAR POINTS

REMARK.

∂f∂y = 0.

SINGULAR POINTS

REMARK.

∂f∂y = 0.

SINGULAR POINTS

EXAMPLE

Determine the nature of the singular point (0, 0) off (x , y) = x3 + y3 − 3xy

According to the last discussions, the tangents at the originare x = 0 and y = 0

Hence the origin is either a node or isolated point.

When x is so small, the equation of the curve is y2 = 3xwhich represent two real branches through the origin.

Hence the origin is a node.

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

Determine the nature of the singular point (0, 0) off (x , y) = (x2 + y2)(2a− x)− b2x .

According to the last discussions, the tangents at the originare x = 0.

Hence the origin is either a cusp or isolated point.

To see whether the origin is a cusp or an isolated point. Solvethe equation of a curve for x .

By neglecting y2, the equation of the curve can also bewritten as

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

−x3 − b2x + 2ax2 = 0

x2 − 2ax + b2 =

x =2a±√

4a2−4(1)(b2)2

x = a±√a2 − b2.

Hence origin is a cusp if a2 − b2 > 0.

SINGULAR POINTS

EXAMPLE

−x3 − b2x + 2ax2 = 0

x2 − 2ax + b2 =

x =2a±√

4a2−4(1)(b2)2

x = a±√a2 − b2.

SINGULAR POINTS

EXAMPLE

−x3 − b2x + 2ax2 = 0

x2 − 2ax + b2 =

x =2a±√

4a2−4(1)(b2)2

x = a±√a2 − b2.

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

EXAMPLE

SINGULAR POINTS

WORKING RULES TO CALCULATE SINGULAR POINTS AND ITS

NATURE USING PARTIAL DERIVATIVES

If f (x , y) be a curve then its singular points are thesimultaneous solutions of f (x , y) = 0, fx (x , y) = 0 andfy (x , y) = 0.The values of tangents at singular points are the roots offyy (y

′(x))2 + 2fxyy

′(x) + fxx = 0.

If fxx , fxy and fyy are not all zero then the point (x , y) will bea double point.If (fxy )2 − fxx fyy > 0 the point (x , y) would be a node.If (fxy )2 − fxx fyy = 0 the point (x , y) would be a cusp.If (fxy )2 − fxx fyy < 0 the point (x , y) would be an isolatedpoint.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

SINGULAR POINTS

If f (x , y) be a curve then its singular points are thesimultaneous solutions of f (x , y) = 0, fx (x , y) = 0 andfy (x , y) = 0.

The values of tangents at singular points are the roots offyy (y

′(x))2 + 2fxyy

′(x) + fxx = 0.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

If fxx , fxy and fyy are not all zero then the point (x , y) will bea double point.

If (fxy )2 − fxx fyy > 0 the point (x , y) would be a node.If (fxy )2 − fxx fyy = 0 the point (x , y) would be a cusp.If (fxy )2 − fxx fyy < 0 the point (x , y) would be an isolatedpoint.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

If fxx , fxy and fyy are not all zero then the point (x , y) will bea double point.If (fxy )2 − fxx fyy > 0 the point (x , y) would be a node.

If (fxy )2 − fxx fyy = 0 the point (x , y) would be a cusp.If (fxy )2 − fxx fyy < 0 the point (x , y) would be an isolatedpoint.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

If fxx , fxy and fyy are not all zero then the point (x , y) will bea double point.If (fxy )2 − fxx fyy > 0 the point (x , y) would be a node.If (fxy )2 − fxx fyy = 0 the point (x , y) would be a cusp.

If (fxy )2 − fxx fyy < 0 the point (x , y) would be an isolatedpoint.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

SINGULAR POINTS

′(x))2 + 2fxyy

′(x) + fxx = 0.

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