Calculus:

The branch of mathematics which deals with motion and change.

 Real Numbers and Real Line:

The numbers which are represented on real line are called real numbers. They are represented are “R“.

Note: Set of real numbers are union of rational and irrational numbers.

Ordered properties of Real Numbers

(Properties of inequalities)

Note: if a,b,c are element of “R

1)  a < b Approaches to a+c < b+c

2) a < b Approaches to a-c < b-c

3) a < b and c > 0 Approaches to ac < bc

4) a < b and c < 0 Approaches to ac > bc

5) a > 0 Approaches to 1/a > 0

6) If a and b are both positive or both negative then a < b Approaches to 1/a > 1/b

Note:Ordered properties hold only for real numbers and do not for complex numbers.

Finite Interval:

Infinite Intervals:

Example 1:Solve the following inequalities and graph the solution on line number

(a)   2x-1 < x+3

=2x – x < 1+3

=x < 4

Absolute value of Real numbers:

The absolute value of real number ‘x’is denoted by |x| and it is defined as

|x|={+x if x >= 0}

        {- x if x < 0}

|3| = 3

|0| = 0

|-3| = – (-3) = 3

If |x|=3 then x=+- 3 //read as (plus minus 3)

Absolute value properties:

1) |-a| = a

2) |a.b| = |a|.|b|

3) |a/b| = |a| / |b|

4) |a+b| <= |a|+|b|

Inequalities involving absolute values:

|x| <= a <=> -a<= x <=a

Example 2:Solve the inequalities and graph the solution on real line

a)   |2x – 3|<= 1

-1 <= 2x – 3 <= 1

adding +3 on each side

-1 + 3 <= 2x + 3 – 3 <= 1+3

2 <= 2x <= 4

dividing whole equation by 2

1 <= x <= 2

= [1,2] Answer

b) |2x – 3| >= 1

2x – 3 <= -1               2x – 3 >= 1

2x <= 2                      2x >= 4

x <= 1                         x >= 2

Inequalities in Daily Life:

Inequalities are very important in daily life. For Example

Thermostats in cars cause a value to open when the engine gets hot (say more than 95 degree C), allowing the water to circulate and cool the engine down. We can express this condition by using an equality T>95 degree. If the engine is getting too cool (say T < 85 degree), the thermostat closes again reducing the water circulation.

A voltage regulator  in a TV will typically accept a voltage range from 110V to 240V. We could write the range for the voltage V as 110<V<240.

Analytic Geometry

The branch of mathematics which deals with concepts of geometry by the help of algebra.

Cartesian Coordinate System:

Coordinate plan consists of infinite points (x,y) in which x-coordinate(abscissa)   y-coordinate(ordinate)

1) P(4,5) lies in 1st Quadrant.

2) P(-4,5) lies in 2nd Quadrant.

3) P(4,-5) lies in 3rd Quadrant.

4) P(-4,-5) lies in 4th Quadrant.

5) P(4,0) lies on X-axis.

6) P(0,5) lies on Y-axis.

Distance Formula:

The distance between two points A(x1,y1) and B(x2,y2) in a plane is given by:

Inclination of a Line:

An angle made by line with positive direction of x-axis is called inclination of that line.

Slope:

If  ‘a’ is the inclination of a line “L” then it’s slope is given by

m = tan a

If a line of inclination is 45 degree then it’s slope m=45

m = 1         (tan 45 = 1)

When a is zero then it’s slope is zero and when a is 90 degree then it’s slope is undefined.

Physically slope is = rise / run

Analytically slope is m = y2 – y1 / x2 – x1

Parallel and Perpendicular lines:

If two non-vertical lines l1 and l2 have slopes m1 and m2 respectively then,

1-                      l1 || l2     iff     m1 = m2

2-                     l1 _|_ l2     iff     m1.m2 = -1 

Equations of Lines:

X-axis

Y-axis

Point slope form:

When one point on the line and its slope is given then equation of line is:

                    y – y1 = m(x – x1)

Slope Intercept form:

When slope and y-intercept is given then

        y = mx + c

while c can be any constant.

Example 1:Find the equation of a line which passes through (2,4) and having slope 5.

Here (x1,y1) are (2,4)   and m=5

y – y1 = m (x – x1)

y – 4 = 5 (x – 2)

5x – y – 6 = 0

Example 2:Find the equation of a line having slope 5 and y-intercept 6.

Here m=5 and c=-6

y = mx + c

y = 5x + (-6)

5x – y – 6 = 0

 

Next Page:

-Functions

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