It includes all the basics of calculus. It also includes all the formulas of derivatives and how to carry it out. It also includes function definition and different types of function along with relation.

1. CALCULUS
&
Analytical Geometry
Presented by: Group-E (The Anonymous)
BSc.CSIT 1st Semester
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2. Contents
 Relation and Function
 Relation
 Types of function
 Meaning of limit
 Types of limit
 Limits and continuity
 Types of discontinuity
 Intermediate value theorem
 Derivative
 Application of derivatives
 Applying derivative for curve sketching
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3. Function
■ Cartesian product : let A and B be any two sets , the
Cartesian product of two set A and B is defined as the
set of all ordered pair (x ,y)such that x 𝜖 A and y 𝜖 B i.e
A×B ={(x,y):x𝜖A and y 𝜖 B}
■ Note that A×B≠B×A
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4. Relation
■ Relation :- Let A and B be any two non-empty set then any subset R of
Cartesian product A*B is called a relation from A to B. It is denoted by
R:A→B.
■ Function :- let A and B be any two non-empty set then a relation
R:A→B is called a function from A to B if every element of set A
associates with a unique element of set B.A function from set A to set B
is denoted by f:A→B
■ Domain of the function :- The set A is known as the domain of the
function is denoted by Dom(f)={x:x 𝜖 A}
■ Range of the function:-The set of values of y=fx 𝜖 B for every x 𝜖 A is
known as range of the function .
■ It is denoted by Range(f) and Range(f) ={y:y 𝜖 B, y=f(x) for all x 𝜖 A}
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5. Types of Function
■ One to one function:- A function f:A→B is said to be one to one
(injective) function if distinct element in A have distinct image in B.
i.e x₁≠x₂→f(x₁)≠f(x₂) for all x₁,x₂ 𝜖 A
■ Onto function:-a function f:A→B is said to be onto (surjective)
function if for each element y 𝜖 B there is at least one element xЄA
such that f(x)=y i.e f(A)=B
■ Composite function :-Let f:A→B and g:B→C be any two functions
then, there exists a new function h:A→C defined by h(x)=gf for all
xЄA. This function h is called the composite function of f and g and is
denoted as gof or simply gf.
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6. ■ Even and odd function:A real y=f(x) is said to be even , if f(-
x)=f(x) for xЄR.
A real function y=f(x) is said to be odd , if f(-x)= -f(x) for all
XЄR
■ One to one corresponding:-If a function f:A→B is both one to
one and onto , then it is called one to one corresponding or
bijective function.
■ Absolute value:-let x denote any real number .the absolute
value of x is written as lxl , is a non-negative real number
defined by
lxl={x if x ≥ o or –x if x<0
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7. Meaning of limit
A restriction on the size or amount of something possible
(literally)
A fundamental concept in calculus and analysis of the
behavior of any function near a particular input
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8. Types of Limit
Right hand limit:
A function f(x) is said to be right hand limit L at a and written as
lim
𝑥→𝑎+
𝑓(𝑥) = 𝐿 if every number 𝜀 >0, there exists a corresponding
number 𝛿>0 such that for all x, a < x < (a + 𝛿)
→ |f(x)-L| < 𝜀
Left hand limit:
A function f(x) is said to be left hand limit L at a and written as
lim
𝑥→𝑎−
𝑓(𝑥) = 𝐿 if every number 𝜀 >0, there exists a corresponding
number 𝛿 >0, such that for all x, (a - 𝛿) < x < a → |f(x)-L| < 𝜀
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9. Definition of limit
Let f(x) be defined on an open interval about a, we say that f(x)
approaches the limit L as x approaches a and written as
lim
𝑥→𝑎
𝑓(𝑥) = 𝐿 if every number 𝜀 >0, there exists a corresponding
number 𝛿>0 such that for all x, a < x < (a + 𝛿)
→ |f(x)-L| < 𝜀
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Verification:
lim
𝑥→1
3𝑥 + 2 = 5 Here, f(x) = 3x+2, a=1, L=5 then,
𝑓 𝑥 − 𝐿 < 𝜀 Choosing, 𝛿 =
𝜀
3
[∴ 𝑥 − 1 < 𝛿]
= 3𝑥 + 2 − 5 < 𝜀 Again, 𝑥 − 1 < 𝛿
= 3𝑥 − 3 < 𝜀
= 𝑥 − 1 <
𝜀
3
= 3𝑥 − 3 < 3𝛿
= 3𝑥 + 2 − 5 < 3
𝜀
3
= 𝑓(𝑥) − 5 < 3
𝜀
3
Hence, lim
𝑥→1
(3𝑥 + 2) = 5

10. Limit
A function f(x) has a limit as x approaches (if and only if it has left hand and right
hand limits there and these one-sided limits are equal. (The limit exists)
Continuity
a. Interior point:
A function y=f(x) is continuous at an interior point c of its domain if
lim
𝑥→𝑐
𝑓(𝑥) = 𝑓(𝑐)
b. End point:
i. A function f is continuous at a left end point x=a of its domain if lim
𝑥→𝑎+
𝑓(𝑥) = 𝑓(𝑎)
, it is also known as continuous from right.
ii. A function f is continuous at right end point x=b of its domain if lim
𝑥→𝑏−
𝑓(𝑥) =
f(x)=f(b), it is also known as continuous from left.
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11. Graphically
.
. .
Interior point
ca b
Continuous
from the left
Continuous
from right
𝑎+ 𝑏−

12. Types of discontinuity
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A function y=f(x) may have any one of the following types of discontinuity.
1. Jump discontinuity:
If lim
𝑥→𝑎−
𝑓(𝑥) ≠ lim
𝑥→𝑎+
𝑓(𝑥), then f(x) is said to have a jump or an ordinary
discontinuity.
2. Removable discontinuity :
If lim
𝑥→𝑐−
𝑓(𝑥) = lim
𝑥→𝑐+
𝑓(𝑥) ≠ f(c) or f(c) is not defined then f(x) is said to have
removable discontinuity.
3. Infinite discontinuity:
If one or both of lim
𝑥→𝑐−
𝑓(𝑥) and lim
𝑥→𝑐+
𝑓(𝑥) tends to +∞ to -∞, the f(x) is said to
have an infinite discontinuity.

13. The Intermediate Value Theorem
Suppose f(x) is continuous on an interval ‘I’ and ‘a’ and ‘b’ are any
points of I. Then if yo is a number between f(a) and f(b), then
there exist a number c between ‘a’ and ‘b’ such that f(c) = yo.
( Without proof)
13
bca
f(a)
f(c) =yo
f(b)

14. 14
Example:
f(x) = 3x + 2, it is continuous in an interval [a , b]. Let a=3 , b=5
f(3) = 3 × 3 + 2 = 11
f(5) = 3 × 5 + 2 = 17
Let y=12, to verify intermediate value theorem we find c є (a , b) such that
y = f(c)
Now,
f(c) = 3c + 2
or, f(c) = y
or, 3c + 2 = 12
or, c = 3.3 є (3,5)

15. 15
Let y = f(x) be a function , if the limit 𝑙𝑖𝑚
𝑥→0
𝑓 𝑥+ℎ −𝑓(𝑥)
ℎ
exists and is finite, we
call it the derivative of the function f with respect to x and denoted by f’(x).
Right hand and left hand derivatives:
The right hand limit 𝑙𝑖𝑚
𝑥→0+
𝑓 𝑥+ℎ −𝑓(𝑥)
ℎ
exists , its called the right hand
derivative of f(x) at x=c and is denoted by R f’(c). Rf’(c)= 𝑙𝑖𝑚
ℎ→𝑐+
𝑓 𝑐+ℎ −𝑓(𝑐)
ℎ
Similarly, the left limit exists is called left derivative of f(x) at x=c and is
written as L f’(c). L f’(c) = 𝑙𝑖𝑚
ℎ→𝑐−
𝑓 𝑐+ℎ −𝑓(𝑐)
ℎ
Derivatives

16. Theorem
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If f has derivatives at a point x=c then it is continuous at x=c.
The converse may not hold.
Proof: Given that f’(c) exists, we must show that, 𝑙𝑖𝑚
𝑥→0+
𝑓(𝑥) = f c or equivalently
that 𝑙𝑖𝑚
𝑥→0
𝑓(𝑐 + ℎ) = 𝑓(𝑐)
Now, f(c + h) - f(c) =
𝑓 𝑐+ℎ −𝑓(ℎ)
ℎ
× h
𝑙𝑖𝑚
𝑥→0+
[𝑓(𝑐 + ℎ) − 𝑓 𝑐 ] = 𝑙𝑖𝑚
ℎ→0
[
𝑓 𝑐+ℎ −𝑓(𝑐)
ℎ
× ℎ]
= 𝑙𝑖𝑚
ℎ→0
[
𝑓 𝑐+ℎ −𝑓(𝑐)
ℎ
× 𝑙𝑖𝑚
ℎ→0
h
= f’(c) × 0
𝑙𝑖𝑚
ℎ→0
[f(c+h) – f(c)] = 0
𝑙𝑖𝑚
ℎ→0
f(c+h) – f(c) = 0
𝑙𝑖𝑚
ℎ→0
f(c+h) = f(c)
By the definition of continuity, it follows that f(x) is
continuous at x=c. The converse of the above theorem is
in general, not true. The function f(x)=|x| is continuous at
x=0 but f(0) does not exist ,i.e , f(x)is not differentiable at
x=0.

17. Application of Derivative
a. (Global) Absolute maximum:
Let f be a function with domain D. Then f has an absolute maximum value on D at a
point c if f(a)≤f(c) for all x in D.
b. Absolute minimum:
Let f be a function with domain D. Then f has absolute minimum value on D at c if
f(x)≥f(c) for all x in D.
c. Local (Relative) maximum:
A function y = f(x) is said to have the local maximum value at x=c , if f(c) ≥ f(c+h) for
sufficiently small positive value of h.
d. Local (Relative) minimum:
A function y = f(x) is said to have the local minimum value at x=c if f(c) ≤ f(c+h) for
sufficiently small +ve value of h.
e. Stationary (or critical) point:
The point x=c where f’(c) = 0 or f’(0) does not exist, is called a stationary point of the
function.
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18. Applying derivatives
for sketching a curve
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19. First derivative test
Suppose that c is a critical point of a continuous function f and
that f is differentiable at every point in some interval containing
c, moving across c from left to right.
i) If f’ changes from –ve to +ve at c, then f has a local minimum at c.
ii) If f’ changes from +ve to –ve at c, then f has a local maximum at c
iii) If f’ does not changes sign at c, then f has no local extrema at c.
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Also,
i) If f’(x)>0 for all x 𝜖(𝑎, 𝑏)  f(X) is increasing on (a,b)
ii) If f’(x)< for all x 𝜖(𝑎, 𝑏)  f(X) is decreasing on (a,b)

20. Example: y = 𝑥5/3 − 5𝑥2/3
First Derivative test
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For stationary points, y’ = 0
So, either x=0 Or, x=2
Intervals (-∞, 𝟎) (0,2) (2,∞)
Sign of f’ +ve -ve +ve
Nature of curve Increasing Decreasing Increasing
Since f’(x) changes from –ve to +ve at x=2 f‘(x) has minimum value at
x=2 and maximum value at x=0
∴f(2) =
3
25 - 5×
3
23 = -3
3
4 f(0) = 0

21. Second derivative test
Suppose f” is continuous on an open interval that contains x=c
i) If f’(c) = 0 and f”(c)<0, the f has local maximum at x=c.
ii) If f’(c) = 0 and f”(c)>0, then f has local minimum at x=c.
iii)If f’(c) = 0 and f”(c)=0, then the test fails, the function f may
have a local maximum or local minimum or neither.
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Also,
i) If f’’(x)>0 throughout (a,b)  f is concave up on [a,b]
ii) If f’’(x)<0 throughout (a,b) f is concave down on [a,b]

22. For concavity ,
Second derivative test
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For point of inflection, y” = 0
On solving either x= -1 or x= 0
Intervals (-∞,-1) (-1,0) (0,∞)
Sign of y” -ve +ve +ve
Nature of curve Concave down Concave up Concave up
So, the point of inflection is (-1,-6).

23. (2,-3
3
4 )
Final graph
23
(0,0)
-1,-6

24. Reference
Calculus and Analytical Geometry (Binod Prasad Dhakal)
www.google.com
www.Wikipedia.com
Teachers note
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25. Queries
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