Play with Graphs for JEE Main and Advanced

182 Pages·2018·15.89 MB·english

by  Amit M Agarwal

Skills in
Mathematics for
JEE MAIN &
7
ADVANCED
Play with
H
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arihant
Amit M. Agarwal 5?
i.
Skills in
Mathematics for
JEE MAIN &
ADVANCED
Play with
Graphs
Amit M. Agarwal
jjcarihant
www.jeebooks.in
ARIHANT PRAKASHAN (Series), MEERUT
Skills in Mathematics for
JEE MAIN & ADVANCED
arihant
ARIHANT PRAKASHAN (Series), MEERUT
All Rights Reserved
s ©AUTHOR
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Skills in Mathematics for
J JEE MAIN & ADVANCED
PREFACE
It is a matter of great pleasure and pride for me to introduce to you
this book “play with Graphs”. As a teacher, guiding the Engineering
aspirants for over a decade now, I have always been in the lookout
for right approach to understand various mathematical problems. I
had always felt the need of a book that can develop and sharpen the
ideas of the students within a very short span of time.
The book in your hands, aims to help you solve various
mathematical problems by the use of graphs. Ways to draw different
types of graphs are very easy and can be understood by even an
average student. I feel glad to mention that the use of graphs in
solving various mathematical problems has been well illustrated in
this book.
I would like to take this opportunity to thank
M/s Arihant Prakashan for assigning this work to me.
It is their inspiration that has encouraged me to bring this book in
this present form.
I would also like to thank Arihant DTP Unit for the nice laser
typesetting.
Valuable suggestions from the students and teachers are always
welcome, and these will find due places in the ensuing editions.
Amit M. Agarwal
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Skills in Mathematics for
JEE MAIN & ADVANCED
CONTENTS
1. INTRODUCTION TO GRAPHS 1-50
1.1 Algebraic functions
1. Polynomial function
2. Rational function
3. Irrational function
4. Piecewise functions
1.2 Transcendental functions
1. Trigonometric function
2. Exponential function
3. Logarithmic function
4. Geometrical curves
1.3 Trigonometric inequalities
1.4 Solving equations graphically
2. CURVATURE AND TRANSFORMATIONS 51-137
2.1 Curvature
2.2 Concavity, convexity and points of inflexion
2.3 Plotting of algebraic curves using concavity
2.4 Graphical transformations
2.5 Sketching h(x)= maximum {f(x), g(x)} and h(x)= minimum {f(x), g(x)}
2.6 When f(x), g(x) — f(x) + g(x) = h(x)
2.7 Whenf(x),g(x) — f(x).g(x) = h(x)
3. ASYMPTOTES, SINGULAR POINTS
AND CURVE TRACING 138-165
3.1 Asymptotes
3.2 Singular points
3.3 Remember for tracing cartesian equation
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. HINTS & SOLUTIONS 166-182
r
INTRODUCTION OF
I
f
GRAPHS
fta “
fe-
HCZ
. o-
In this section, we shall revise some basic curves which are given as. *T3'-
•-4-0-
- H»-
Polynomial)
ztoz
Rational )
Algebraic) Modulus ) 7_:S z
Irrational ] - ■u -
Signum ]
Piecewise)
-0-
Greatest integer function)
Fractional part function )
Least integer function)
3^^ Trigonometric J
Exponential )
Logarithmic/lnverse of exponential )
Transcendental —
Geometrical curves )
Inverse trigonometric curves)
lol ALGEBRAIC FUNCTIONS
It Polynomial Function
A function of the form:
f O) = a0 + + a2x2 + ... + anxn ;
where n 6 N and ga0, alt a2,...,an e R.
Then,/is called a polynomial function. “fW is also called polynomial inx”.
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ft
1__L
Some of basic polynomial functions are
(i) Identity function/Graph of /(x) = x
-M A function f defined by f (x) = x for all x e R, is called the
■ W - identity function.
Here, y = x clearly represents a straight line passing through
- the origin and inclined at an angle of 45° with x-axis shown as:
— The domain and range of identity functions are both
!■ equal to R.
t o.q
Fig. 1.1
4(ii) Graph of/(x) = x,2 y y=x2
1. A f■unction given by /(x) = x2 is called the square function.
The domain of square function is R and its range is R+ u {0}
ti ^'_or[°>00)
Clearly y = x2, is a parabola. Since y = x2 is an even x
O
function, so its graph is symmetrical about y-axis, shown as:
(iii) Graph of f(x) = x3
Fig. 1.2
A function given by /(x) = x3 is called the cube function. y = x3
The domain and range of cube are both equal to R.
Since, y = x3 is an odd function, so its graph is symmetrical r
about opposite quadrant, i.e., “origin”, shown as:
(iv) Graph of/(x) = x2n; neN
If n g N, then function/given by /(x) = x2n is an even function.
So, its graph is always symmetrical about y-axis. Fig. 1.3
Also, x2 > x4 > x6 > x8 >... for all x e (-1,1)
and x2 < x4 < x6 < x8 <... for all x g (-«>, -1) u (1, °°)
Graphs ofy = x2, y = x4, y = x6,..., etc. are shown as:
y=*%=x6 y y = x6
y = x2
sy = x
x
O 1
Fig. 1.4
(v) Graph of /(x) = x2"-1; n g N
If n g N, then the function f given by /(x) = x2n-1 is an odd function. So, its graph is
L_LJJ symmetrical about origin or opposite quadrants.
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h | | ~~
u
Here, comparison of values of x, x3, x5,... y=x5 y=x3
‘y ,-.y=*
for
xe(l, oo) x<x3<x5<...
3
xe (0,1) x> x3 > xs >... T
X
- O !
xe(-l, 0) x<x3<x5<...
xe(-oo,-l) x>x3>x5>... c
Graphs of /(x) = x, /(x) = x3, /(x) = x5,... are shown as in :: &
Fig. 1.5 Fig. 1.5
o
2rRational Expression
3
A function obtained by dividing a polynomial by another polynomial is called a rational function.
0
=>
Q(x)
0
Domain e R - {x | Q(x) = 0}
i.e., domain e R except those points for which denominator = 0.
Graphs of some Simple Rational Functions y ■u
(i) Graph of f(x) = —
x
A function defined by f (x) = — is called the reciprocal
x +
o i
function or rectangular hyperbola, with coordinate axis as
asymptotes. The domain and range of f(x) = — is R - {0}.
x
Since, /(x) is odd function, so its graph is symmetrical
about opposite quadrants. Also, we observe Fig. 1.6
lim /(x) = + oo and lim /(x) = - oo .
x —> 0+ x -> 0"
and as x -> ± oo => /(x) -> 0
Thus, /(x) = — could be shown as in Fig. 1.6.
x -H-f
....
(ii) Graph of f(x) = —
x
Hpe__re , /(x) = —1 i* s an even f" uncti•on, so its graph is symmetrical about y-axis.
1 v., j „
X
Domain of f(x) is R - {0} and range is (0, ®o).
Also, as y -> oo as lliimm J/((xx)) or lim /(x) .
x-» 0+ x -> 0"
and y-* 0 as lim /(x).
X —» ± oo
1 O
Thus, /(x) = — could be shown as in Fig. 1.7.
x
Fig. 1.7
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■H—
1
(iii) Graph of f(x) = —-= —ne N
x2n -1 ’ y=—
x.
X3
Here, f (x) = — is an odd function, so its graph is
x
(/)
X symmetrical in opposite quadrants.
V
fi. Also, y -> oo when lim /(x) and
x -> 0+
—i-------- x
6 y —> _ oo when lim /(x).
x -> 0- b\ -1‘
Thus, the graph for f(x) = ; /(x) = -y, ., etc.
will be similar to the graph of /(x) = — which has
x
asymptotes as coordinate axes, shown as in Fig. 1.8 Fig. 1.8
,■r-. (iv) Graph of /(x) = x ; n e N
JL
We observe that the function /(x) = —is an even
x2"
' J function, so its graph is symmetrical about y-axis.
Also, y oo as lliimm //((xx)) or lim /(x)
x-> 0+ x -> O'
and y -> 0 as lim /(x) or lim /(x).
X -»-oo X -» + oo -1 Oo !
The values of y decrease as the values of x increase.
Thus, the graph of /(x) = ; /(x) = etc. will be
Fig. 1.9
x2 x4
similar as the graph of /(x) = which has asymptotes as coordinate axis. Shown as in Fig. 1.9.
x2
3.'Irrational Function
The algebraic function containing terms having non-integral rational powers of x are called
irrational functions.
Graphs of Some Simple Irrational Functions
(i) Graph of f(x) = xV2
Here; /(x) = Vx is the portion of the parabola y2 = x, which
lies above x-axis.
y=Jx
Domain of f(x) eR+ u {0} or [0, <»)
1
and range of /(x) e R+ u {0} or [0, °°).
Thus, the graph of /(x) = x1/2 is shown as; + x
O 1
Note If f(x) = xn and g(x) = x1/n, then f(x)and g(x)are inverse of
Fig. 1.10
each other.
.-. f(x) = x" and g(x) = x1/n is the mirror image about y = x.
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(ii) Graph of f(x) = x'/3
As discussed above, ifg(x) = x3. Then/(x) = x. 1/3 1/3 a
is image of g(x) about y = x. 1
where domain /(x) e R. + r I
O 1
and range of /(x) e R.
Thus, the graph of /(x) = x1/3 is shown in Fig. -T aC
1.11;
4-
Fig. 1.11 ■■I
0
(iii) Graph of f(x) = x'/2n; ne N y* y=/4y=*2 3
Here, fix') = *1/2n is defined for all x e [0, °°) and the values
0
taken by /(x) are positive.
So, domain and range of f (x) are [0, <»). y=x'12
Here, the graph of f (x) = x,1l//22nn is the mirror image of the y=xM 0
graph of /(x) = x2n about the line y = x, when x e [0, «>).
Thus, /(x) = x1/2, _f(x) = x1/4, ...are shown as; o ■1+ X - 0)
-■u
Fig. 1.12
(iv) Graph of f(x) = x1/2n-1, when n e N 0
Here, f(x) = x1/2n-1 is defined for all xe R. So, 4 y=x® y=x3
\ t -y=x
domain of /(x) e R, and range of /(x) e R. Also the
graph of/(x) = x1/2n-1 is the mirror image of the graph y =x1/3
-y= x1/5
of /(x) = x2n-1 about the liney = x when xe R. 1--
Thus, /(x) = x1/3, /(x) = x1/5,..., are shown
1
as; +■ X
O 1
Note We have discussed some of the simple curves
for Polynomial, Rational and Irrational
functions. Graphs of the some more difficult
rational functions will be discussed in
chapter 3. Such as;
Fig. 1.13
x 1 x2 + X + 1 j—
tt
y = 77T y = ZH ’ y = T^~l
4t'Piecewise Functions
As discussed piecewise functions are:
(a) Absolute value function (or modulus function), (b) Signum function.
(c) Greatest integer function. (d) Fractional part function.
(e) Least integer function.
(a) Absolute value function (or modulus function)
x, x > 0
y = 1*1=
-x, x < 0
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