maths calculus

UECM 1653: Mathematics for Engineering I

TOPIC 2: CALCULUS

Contents:

2.1 Limit.
2.2 Continuity of a function.
2.3 Derivatives.
2.4 Related rates problems.
2.5 Derivatives and the shape of a graph.
2.6 Taylor series approximation of functions.
2.7 Integration.
2.8 Method of integration.
2.9 Applications of integration.


LIMIT

Definition:

If f is a function, then lim f (x) = A if the value of f (x) gets arbitrarily close to A as x
x?a
gets closer and closer to a.


The definition can be stated more precisely as follows:
lim f (x) = A if and only if, for any chosen positive number ?, however small, there
x?a
exists a positive number ? such that
0 < x ? a < , then f (x) ? A < .



f(x)








?

A ?


? ?

x

a



The existence of the limits

Left limit, lim f (x) = A means that f (x) approaches A as x approaches a through
x a?
?
values less than a. In other words, x approaches a from the left.

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UECM 1653: Mathematics for Engineering I

Right limit, lim f (x) = A means that f (x) approaches A as x approaches a through
x a+
?
values more than a. In other words, x approaches a from the right.

A limit, lim f (x) = A exists if and only if lim f (x) = lim f (x) = A .
x?a
x a?
x a+
?
?


Example:
2
x ?1
For f (x) =
, x ? 1, what happens to the values of f (x) as x moves along the
x ?1
x-axis towards 1?

f (x) 1.9 1.95 1.995 1.9999
?
2.0001 2.001 2.05 2.1
x
0.9 0.95 0.995 0.9999
1
1.0001 1.001 1.05 1.1

2
x ?1
2
x ?1
Therefore, lim
= 2, since
? 2 as x ?1.
x 1
?
x ?1
x ?1


f(x)





2




x
1



Example:

Solve the following.
i) lim(x + )
3
x?2

f(x)
5
x

2



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UECM 1653: Mathematics for Engineering I

2
x ? 4 ,x ?
ii)
2
lim f (x) for f (x) = x ? 2
.
x?2
,
2 x = 2

f(x)
4
x

2


x , x ? 0
iii) lim f (x) for f (x) =
.
x?0
x , x < 0

f(x)
1
x
-1
1

-1






When lim f (x) does not exist, f (x) diverges as x approaches a. The following examples
x?a
illustrate how divergence may occur.


Example:
2
x , x ? 2
?
i) Let f (x) =
, since lim f (x) ? lim f (x), so lim f (x) does not exist.
x??2?
x??2+
? 2
x + 2 ,x < 2
x ?
?

f(x)

5

4

3


2

1

x
-4
-3
-2
-1
1

-1

-2

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UECM 1653: Mathematics for Engineering I

1
ii)
If f (x) =
, lim f (x) = +? and lim f (x) = +? .
2?
2
(x ? 2) x
x
+
?
?
So, the limit of f as x tends to a does not exist.


f(x)


4



2



x

-4
-2
2
4
6
8


1
iii) f (x) = sin
oscillate infinitely often between 1 and -1 as x approaches 0.
x
Because the values of f (x) do not approach a unique number as x?0, the limit does
not exist. This kind of function limiting behavior is called divergence by oscillation.


f(x)



1



x



-1




INFINITE LIMITS AND LIMITS AT INFINITY

Infinite Limits

(a) If the values of f (x) increase without bound (infinitely) as x ?a+ or x ?a-, then
lim f (x) = +? or lim f (x) = +? .
x a+
x a?
?
?

(b) If the values of f (x) decrease without bound (infinitely) as x ?a+ or x ?a-, then
lim f (x) = ?? or lim f (x) = ?? .
x a+
x a?
?
?

(c) If lim f (x) = lim f (x) = +? then lim f (x) = +?
x a+
x a?
?
?
x?a

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UECM 1653: Mathematics for Engineering I

(d) If lim f (x) = lim f (x) = ?? then lim f (x) = ??
x a+
x a?
?
?
x?a

Note: The line x = a is called a vertical asymptote of the graph f (x).



Limits at infinity

If the value of f (x) approaches L as x increases without bound (x?+?), then
lim f (x) = L .
x?+?
If the value of f (x) approaches L as x decreases without bound (x?-?), then
lim f (x) = L .
x???

Note: The line y = L is called a horizontal asymptote of the graph f (x).

lim n
x = +?, n = 1, 2,3, 4,...
x?+?
+?, n = 2, 4,6,8,...
lim n
x =
x???
??, n =1,3,5,7,...
lim (
2
c + c x + c x + ...
n
+ c x =
c x
n
lim
n
0
1
2
)
n
x?+?
x?+?
lim (
2
c + c x + c x + ...
n
+ c x =
c x
n
lim
n
0
1
2
)
n
x???
x???
1
1
lim
= lim
= 0
n
n
x?+?
x
x
??? x
f(x)

4

Example:
1
1
2
lim
= +? and lim
= 0 as shown in the figure.
2
2
x?0
x
x
?+?
x


x
-4
-2
2
4



Example:

Solve the following.
i)
8
lim 9x

x?+?
ii)
7
lim 5x
x???
iii)
6
lim 7x
x???
iv) lim 9 8
x + 3 5
x ? 2 4
x + x ?1
x?+?











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UECM 1653: Mathematics for Engineering I

COMPUTATIONS WITH LIMITS

Constant rule lim k = k for any constant k.
x?a
Limit of x rule lim x = a; lim x = + ;
? lim x = ??
x?a
x?+?
x???
Multiple rule lim k[ f (x)] = k lim f (x) for any constant k.
x?a
x?a
Sum and difference rule lim[ f (x) ± g(x)] = lim f (x) ± lim g(x)

x?a
x?a
x?a
Product rule lim[ f (x)g(x)] = lim f (x) lim g(x)
x?a
x?a
x?a
lim f (x)
f (x)
Quotient rule lim
x?a
=
if lim g(x) ? 0
x?a
g(x)
lim g(x)
x?a
x?a
n
Power rule lim[ f (x)]n = lim f (x)
x?a
x?a

Limits of polynomials: lim P(x) = P(a)
x?a

Example:
Solve lim 3
x + 2x ?1.
x 1
?


Example:
5 3
x + 4
i) lim

x?2 x ? 3
2
x ? 9
ii) lim

x 3
? x ? 3
4 2
x ? x
iii) lim

x?+? 2 3
x ? 5
2
x ? 9
iv)
3
lim

x 3
?
x ? 3
2
x + 2
v) lim

x?+? 3x ? 6
2
x + 2
vi) lim

x??? 3x ? 6


Example: (Hybrid or piecewise function)
2
x ? 5, x ? 3
Find lim f (x) for f (x) =
.
x?3
x +13, x > 3




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UECM 1653: Mathematics for Engineering I

LIMITS OF TRIGONOMETRIC FUNCTIONS

sin?
1? cos?
Theorem: lim
=1 , lim
= 0 .
? ?0
?
? ?0
?

Example:

Solve the following.

tan?
sin ?
2
1? cos?
i) lim
ii) lim
iii) lim

??0 ?
??0
?
??0
?


CONTINUITY OF A FUNCTION

- A function f (x) is continuous at x = a if f (a i
) s d
efined and lim f (x e
) xists and
x?a
lim f (x) = f (a) .
x?a

- If one or more of these conditions are not satisfied, then f is called discontinuous at a,
and a is called a point of discontinuity of f .

- A function f (x) is said to be continuous on a closed interval [a, b] if the function is
continuous at all points of [a, b].

- A function f is called continuous from the left at the point a if the conditions are
satisfied: f (a i
) s d
efined and lim f (x e
) xists and lim f (x) = f (a) .
x a?
?
x a?
?

- A function f is called continuous from the right at the point a if the conditions are
satisfied: f (a i
) s d
efined and lim f (x e
) xists and lim f (x) = f (a) .
x a+
?
x a+
?


Theorems:
1. Polynomials are continuous functions on R.
2. The functions sin(x) and cos(x) are continuous on R.
3. A function f is continuous at point a and the function g is continuous at the point
f (a), then g(f (x)) is continuous at point a.
4. If the functions f and g are continuous at a, then
i) f + g is continuous at a.
ii) f - g is continuous at a.
iii) f ?g is continuous at a.
f
iv) i s c
ontinuous a
t a i f g(a) ? a

0 nd i s d
iscontin o
u us a
t a i f g(a) = 0
g





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UECM 1653: Mathematics for Engineering I

Example:
Examin
e the c
ontinuity o
f f unction f (x) = sin(x2 a
) t x = ? .


Example:
2
x +1
Find the discontinuities of f (x) =
.
2
x ? x ? 6

Example:
Examin
e the c
ontinuity o
f f unction f (x)
3
= x + sin(x a
) t x =1.

Example:
Examin
e the c
ontinuity o
f f unction f (x) = xex a
t x = 0.



DERIVATIVES

Increments

The increment ?x of a variable x is the change in x as it increases or decreases from one
value x = a to another value x = b in its domain where, ?x = b? a.

If the variable x is given an increment ?x from x = a and a function f (x) is thereby given
an increment ?y =f (a+?x) ?f (a) from y = f (a), then the quotient

y
?
change i n y



=

x
?
change i n x

is called the average rate of change of the function on the interval between x = a and
x = a+?x. These can be illustrated as a secant line on a graph as shown in the figure.
f(x)
f(b)
f(a)
x
a
b










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UECM 1653: Mathematics for Engineering I

Derivatives
The derivative of a function f with respect to x at the point x = a is defined as
y
?
f (a + x
? ) ? f (a)



lim
= lim

x
? ?0
x
x
?
? ?0
x
?

provided the limit exists. This limit is also called the instantaneous rate of change of y
with respect to x at x = a. On a graph, the derivative gives the gradient of the tangent
line at point x = a.

The notation used to denote the derivative of a function f at a point a is f (?a) . Hence,
the derivative of the function f at a point a from the first principle as

f (a + x
? ) ? f (a)
f (x) ? f (a)



f (?a) = lim
= lim

x
? ?0
x?a
x
?
x ? a


The existence of derivatives

A function is said to be differentiable at a point x = a if the derivative of the function
exists at that point.

f (a + x
? ) ? f (a)
If lim
e
xist sthen f i s d
ifferent a
i ble a
t .
a
?x?0
x
?
Hence, if f is differentiable at point a, then f is also continuous at a.
The converse is false.

Example:
Is f (x) = x differentiable at a=1?

Example:
Find f (?x) of the following functions by first principal.
a)
2
f (x) = x
b)
1
f (x) =
x
c)
( )
r
f x = x













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UECM 1653: Mathematics for Engineering I

DIFFERENTIATION FORMULAS

The process of finding the derivative of a function is called differentiation.

In the following formulas, f (x) and g(x) are differentiable functions of x, while c and n
are constants.

1. A constant function
d
If f (x) = c, then
f (?x) = 0 or
(c) = 0
dx

2. The power rule
d ( nx) n 1
nx ?
=

dx

3. The constant multiple rule
d [
d
cf (x)] = c [ f (x)]
dx
dx

4. The sum and difference rule
d [
d
d
f (x) ± g(x)] =
[ f (x)]± [g(x)]
dx
dx
dx

5. The product rule
d [
d
d
f (x) ? g(x)] = f (x) [g(x)]+ g(x) [ f (x)]
dx
dx
dx

6. The quotient rule
d
d
g(x)
[ f (x)]? f (x) [g(x)]
d
f (x)
dx
dx
=

dx g(x)
[g(x)]2

7. The reciprocal rule
d
?
[ f (x)]
d
1
dx
=

dx f (x)
[ f (x)]2

8. The inverse function rule
dy
1
=
f ?(x)
1
or
=

dx
dx
f ?( y)
dy





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