CALCULUS : IMPORTANT REVIEW TOPICS

8/14/2019 CALCULUS : IMPORTANT REVIEW TOPICS

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IMPORTANT REVIEW TOPICS ON CALCULUS

FUNCTIONS

There are two fundamental processes of Calculus, Differentiation and Integration . Theseprocesses are applied to functions. To carry them out, one has to be familiar withfunctions.

A function is a Rule. Usually we use small letters f, g, h ...... to denote different functionsor different rules, such as f x

` a= x @ 1 , g x

` a= 10 x 2 , h x

` a= x 2 + 3 x @ 2 etc.

Definition: Let A and B be two non-empty sets. A function f from A to B is a rule that associates, with each value of x in set

A, exactly one value f(x) in set B. The function is indicated by the notation f : A Q B . f(x) is read “ f of x”. We usuallyconsider functions for which sets A and B are sets of real numbers.

f(a) is the value of f(x) when x=a. If f x` a= 2 x @ 3 then f 2` a= 2.2 @ 3 = @ 1

The set A, which contains all possible values of the variable x is called the Domain , and the set B which contains the corresponding values of f(x) or y is called the Range of thefunction (Range consists only of those elements of B which are actually paired withelements of A).

We call x the independent variable because we choose it first, and from that we calculate y, which is called dependent variable.

Example: Find the Domain & Range of the function.(a) f(x) = Sin x. (b) f x` a= 2 x 2 @ 8 x + 2

Solution :a) Here x can be any real number, so the Domain is the set of real numbers R or

x 2 RP Q. All values of Sin x lie between -1 and 1. So, Range is @ 1 ≤ y ≤ 1 or

y 2 R : @ 1 ≤ y ≤ 1R S.

b) It can be rewritten as



y = f x` a= 2 x 2 @ 8 x + 2 = 2 x 2 @ 4 xb c+ 2

= 2 x 2 @ 4 x + 4b c@ 8 + 2 = 2 x @ 2` a2

@ 6

Here x can be any real number, so the Domain is the set of real numbers R or x 2 RP Q.As the value of square of any real number cannot be negative therefore, x @ 2

` a2

≥ 0 .

Thus y ≥ @ 6 . So, Range is the interval @ 6 , 1B cor y 2 R : y ≥ @ 6R S.Example : Let f x` a= 2 x 2 @ 5 x + 3. Evaluate f 2` a , f @

12ffff g , f ap wwwwwwwwwwwwwwwwwww b c ,

f 1 + h` a@ f 1` ahfffffffffffffffffff fffffffffffffffffff fffffffff

Solution : Substituting the values of x in f(x) we get f 2` a= 2 A2 2 @ 5.2 + 3 = 8 @ 10 + 3 = 1

f @12ffff g= 2 A @

12ffff g

2

@ 5 @12ffff g+ 3 = 1

2fff + 5

2fff + 3 = 6

f ap wwwwwwwwwwwwwwwwwww b c= 2 ap wwwwwwwwwwwwwwwwwww b c2

@ 5 ap wwwwwwwwwwwwwwwwwww b c+ 3 = 2a @ 5 ap wwwwwwwwwwwwwwwwwww + 3

f 1 + h` a@ f 1` ahffffffffff fffffffffffffffffff ffffffffffffffff ff =

2 1 + h` a2

@ 5 1 + h` a+ 3B C@ 2 A12 @ 5.1 + 3B Ch

ffffff fffffff ffffff ffffff fffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fff

=2 1 + 2h + h 2b c@ 5 1 + h` a+ 3D E@ 2 @ 5 + 3@ A

hfffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff

=2 + 4h + 2h 2 @ 5 @ 5h + 3B C@ 0

hfffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ff

= @ h + 2h 2

hffffffffffffffffffffffffffffff = @ 1 + 2h

A function can also be defined differently for different sets of values of x.

Example : Find f @ 1` aand f 2` awhere the function f(x) is defined as follows :

f x` a=2 x + 1 for x <0

1 @ 2 x 2 for x ≥ 0X\Z

Solution :

Since @ 1<0 , using f x` a= 2 x + 1 we get f @ 1` a= 2 @ 1` a+ 1 = @ 1

Since 2 ≥ 0 , using f x` a= 1 @ 2 x 2 we get f 2` a= 1 @ 2.2 2 = 1 @ 8 = @ 7



GRAPHS OF FUNCTIONS

The graph of f is the set of all points ( x,y) such that y = f(x)Or

The graph of f is the graph of the equation y =f(x)

To draw the graph of y = f x` awe make a table consisting two columns : one for x andone for y [or f(x)]. We take different values of x and calculate the corresponding valuesof y [or f(x)]. Then we plot the points on a graph paper and join them with a smooth

curve to get the graph of the function .

Example : Draw the the graph of f x` a= x 3

Solution : We first make a table. We take different values of x and calculate thecorresponding values of y [or f(x)]. Then we plot the points on a graph paper and jointhem with a smooth curve to get the graph

x : -2 @32fff -1 @

12fff 0 1

2fff 1 3

2fff 2

y or

f(x) :

-8 @27

8

fffffff -1 @1

8

fff 0 1

8

fff 2 27

8

fffffff 8





(a,b ) is called an open interval where a,bb c= x | x 2 R and a < x<bR S

[a,b ] is called an closed interval where a,bB C= x | x 2 R and a ≤ x ≤ bR S

[a,b ) is an interval open at one end and closed at the other, wherea,b

B c= x | x 2 R and a ≤ x<b

R S

Some more commonly used intervals are :a, + 1@ a= x | x 2 R and x ≥ aR S

a, + 1` a= x | x 2 R and x >aR S

@ 1 ,a` A= x | x 2 R and x ≤ aR S

@ 1 ,a` a= x | x 2 R and x <aR S

Example : Interval @ 2,1

b c= x | x 2 R and @ 2< x<1

R S

Interval @ 2 , + 1B c= x | x 2 R and @ 2 ≤ xR S

STRAIGHT LINESSLOPE OF A LINE

The slope of a line is a number which indicates the direction or the slant of the line. It isgenerally represented by “m”. For a non-vertical line passing throughP x 1 , y1` aand Q x 2 , y 2` athe slope of the line segment PQ is

Slope = m = y2

@ y1

x2@ x1

fffffffffffffffffffffff =Δ yΔ x

fffffffff =Change iny @ coordinateChange in x @ coordinate

ffffff fffffff ffffff ffffff fffffff ffffff ffffff fffffff ffffff fffffff ffffff ffffff fffff

The slope of a horizontal line is 0, because as we move along the line the x-coordinate

changes, but the y coordinate does not change. Δ y = 0 here ASo, m =Δ yΔ x

fffffffff =0

Δ xfffffffff = 0 A

The slope of a vertical line is not defined, because as we move along the line the x-

coordinate does not change, but the y coordinate changes. Δ x = 0 here ASo, m =Δ yΔ x

fffffffff =Δ y0fffffffff

which is not defined (a symbol with denominator 0 does not signify a number).



For other lines, if the slope is positive, then it slopes up i.e. with increase in x-coordinate, the y-coordinate increases. If the slope is negative, then it slopes down i.e.with increase in x-coordinate, the y-coordinate decreases.

Parallel & Perpendicular lines:

If two lines l1 and l2 have slopes m1 and m2 :The lines are parallel or l 1 || l2 if and only if m 1

= m2

The lines are perpendicular or l 1? l2 if and only if m 1

B m2= @ 1

In the above picture the lines l3 ? l4 as their slopes are 1 and -1 andm1

B m2= 1 A @1` a= @ 1

Example : Find the slope of the line joining points A (1,-3) and B (-2,1).

Solution :Taking x 1

= 1 , y1= @ 3 , x 2

= @ 2 , y 2= 1

Slope of AB = m = y2

@ y1

x2 @ x1

fffffffffffffffffffffff =1 @ @ 3` a@ 2` a@ 1

ffffffffffffffffffffffffffff =4

@ 3fffffffffff = @

43ffff

Example : Find the slope of a line perpendicular to the line joining A(-3,1) and B(2,3).

Solution :Slope of AB = m1

=3 @ 1

2 @ @ 3` affffffffffffffffffffffffffff =

25fff

If the slope of the line perpendicular to it is m 2

then m 1B m2

= @ 1

[25fff B m2

= @ 1 multiply both sides with52fff

[ m2= @

52fff A # Slope of a line perpendicular to AB is @

52fff A



DIFFERENT FORMS OF EQUATIONS FOR STRAIGHT LINES

General or Standard form:

A linear equation of x and y represents a straight line and is called the general form or thestandard form of the equation of the line. This is written as

Ax + By + C = 0 where A,B,C are real numbersand both A and B are not 0 at the same time

Thus 2 x @ 3 y @ 5 = 0 , 3p wwwwwwwwwwwwwwwww x + y = 0 , 3 x @ 2 = 0 , y @ 4 = 0 are equations of straight lines.In 2 x @ 3 y @ 5 = 0 , A = 2 , B = @ 3 , C = @ 5 A In 3p wwwwwwwwwwwwwwwww x + y = 0 , A = 3p wwwwwwwwwwwwwwwww , B = 1 , C = 0 A

Many a times we are required to find the general form of equation of a line from its slope,a few known points on the line or its intercepts on the axes. There we take help of thefollowing forms :

Point-Slope form :If the line passes through the point x1 , y1` a and its slope is m , then the equation of the

line can be written in the form y @ y1

= m x @ x1` aIntercept Form :

If a line intersects the x-axis at A( a,0 ) and the y-axis at B( 0,b ) then a is called the x-intercept and b the y-intercept. Then the equation of the line can be written in the form

xaffff +

ybfffff = 1 where a ≠ 0 , b ≠ 0



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CALCULUS : IMPORTANT REVIEW TOPICS