Definite Integration-Topics in IB Mathematics

Definite Integration

In the previous post, we discussed indefinite integration. Now we shall discuss definite integration

► Definite Integration- We already know that   \int {f\left( x \right){\rm{ }}dx = g\left( x \right) + c}    \leftarrow  this c here is an integral constant. we are not sure about its value. This c is the reason we call this process indefinite integration. But suppose we do our integration between certain limits like:-

\int\limits_a^b {f(x)dx = \left[ {g(x) + c} \right]} _a^b   here a \to  lower limit while b \to  higher limit

\int\limits_a^b {f(x)dx = \left[ {g(b) + c} \right]} - \left[ {g(a) + c} \right]


You can clearly see that this function is independent of ‘c’. Means we can be sure about its value so this type of integration is called  Definite Integration.

►Definite Integration of a function f(x) is possible in [a,b] if f(x) is continuous in the given interval

►If f(x), the integrand, is not continuous for a given value of x then it doesn’t mean that g(x), the integral, is also discontinuous for that value of x.

► Definite integration of a function between given limits like     \int\limits_a^b {f\left( x \right)dx} \Rightarrow         Algebraic sum of areas bounded by the given curve f(x) and given lines x=a and x=b. That’s why the answer for definite integration problems is a single number.

► If \int\limits_a^b {f\left( x \right)dx} = 0 that shows a few things:-

(i) The lines between which area is bounded are co-incident(a=b)

(ii) Area covered above the x-axis=Area covered below the x-axis that means positive part of area and negative part of area is equal

(iii) there must be at least one solution/root to f(x) between x=a and x=b(this is something we study in ROLE’S THEOREM in detail)

► If given function f(x) is not continuous at x=c then we should write

\int\limits_a^b {f\left( x \right)dx} = \int\limits_a^{{c^ - }} {f(x)dx} + \int\limits_{{c^ + }}^a {f(x)dx}

► If given function f(x) > or <0 in any given interval (a,b) then  \int\limits_a^b {f\left( x \right)dx} \Rightarrow  >0 or <0 in given interval (a,b)

► If given function f(x)  \ge  g(x) in the given interval (a,b) then    \int\limits_a^b {f(x)dx \ge } \int\limits_a^b {g(x) \ge } dx 

in the given interval

► If we integrate the given function f(x) in the given interval (a,b) then

\int\limits_a^b {f(x)dx \le } \left| {\int\limits_a^b {g(x) \ge } dx} \right| \le \int\limits_a^b {\left| {f(x)} \right|dx}

<img src="definite integration.jpg" alt="definite integration">

Some More Properties of Definite Integration:- Read more

Increasing and Decreasing Functions

Increasing and decreasing functions

This is my third post in the series of “Applications of derivatives”. The previous two were based on “Tangent and Normal” and “Maxima and Minima”.In this post, we shall learn about increasing and decreasing functions. That is one more application of derivatives.

Increasing and Decreasing Functions- We shall first learn about increasing functions

Increasing Function-

(a) Strictly increasing function- A function f (x) is said to be a strictly increasing function on (a, b) if x1< x2  \Rightarrow f(x1) < f (x2) for all xl, x2 \in (a, b).Thus, f(x) is strictly increasing on (a, b) if the values of f(x) increase with the increase in the values of x.Refer to the graph in below-given figure  \Downarrow <img src="increasing decreasing function.jpg" alt="increasing decreasing function"> Read more

IB Mathematics HL SL-Maxima and Minima

In my previous post, we discussed how to find the equation of tangents and normal to a curve. There are a few more  Applications of Derivatives in IB Mathematics HL SL, ‘Maxima and Minima’ is one of them.

Maxima and Minima:-

1. A function f(x) is said to have a maximum at x = a if f(a) is greater than every other value assumed by f(x) in the immediate neighbourhood of x = a. Symbolically


\left. \begin{array}{l} f(a) > f(a + h)\\ f(a) > f(a - h) \end{array} \right] \Rightarrow x = a   gives maxima for a sufficiently small positive h.

Similarly, a function f(x) is said to have a minimum value at x = b if f(b) is least than every other value assumed by f(x) in the immediate neighbourhood at x = b. Symbolically


\left. \begin{array}{l} f(b) > f(b + h)\\ f(b) > f(b - h) \end{array} \right]  If x = b gives minima for a sufficiently small positive h.


<img src="IB Mathematics HL SL.jpg" alt="IB Mathematics HL SL">

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Applications of Derivatives in IB Mathematics

Applications of Derivatives in IB


In my previous post, we discussed how to find the derivative of different types of functions as well as the geometrical meaning of differentiation. Here we are discussing  Applications of Derivatives in IB Mathematics
There are many different fields for the Applications of Derivatives. We shall discuss a few of them-

Slope and Equation of tangents to a curve- If We draw a tangent to a curve y=f(x) at a given point   ({x_1},{y_1}), then

The gradient of the curve at given point=the gradient of the tangent line  at given  point

and we already discussed that slope or gradient of the tangent at given point   ({x_1},{y_1})

m=  {\frac{{dy}}{{dx}}_{at({x_1},{y_1})}}


Finally to find the equation of tangent we use the slope-point form of equation

y - {y_1} = m(x - {x_1})

The major part of this concept is also discussed in the previous post. We should also remember following points while solving these types of questions.

(i) If two lines are parallel to each other, their slopes are always equal
i.e     {m_1} = {m_2}
(ii) If two lines are perpendicular to each other, the product of their  slopes is always -1

{m_1}.{m_2} = - 1

(iii) If a line is passing through two points   ({x_1},{y_1}) and  ({x_2},{y_2})  then, slope of the line

m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}

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Quadratic equations and Quadratic Functions

Many IB mathematics tutors consider quadratic equations as a very important topic of ib maths. There are following ways to solve a quadratic equation

► Factorization method

►complete square method 

► graphical method

► Quadratic formula method

<img src="ib mathematics tutors.png" alt="ib mathematics tutors>

The quadratic formula is the strongest method to solve a quadratic equation. In this article, I will use …… steps to prove the quadratic formula

Given equation: ax²+bx+c=0

Step-1: transfer constant term to the right side


Step-2: divide both sides by coefficient of x²

Step-3: write (coefficient of x/2)²     that is (b/2a)²=b²/4a²
Step-4: Add this value to both sides
                                x²+bx/a+b²/4a² =-c/a²+b²/4a²
now, take square root on both sides



                                   x = \frac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}

This formula is known as quadratic formula, we can put values of a, b and c  from any equation and find the value of x (the variable) by directly using this formula.

IB Mathematics tutors can also explain the concept of conjugate roots with the help of quadratic formula. In a quadratic equation,
if a, b and c are all rational numbers and one root of the quadratic equation is a+√b then the second root will automatically become a-√b. that can be understood easily as we use one +ve and one -ve sign in quadratic formula.
These types of roots are called Conjugate Roots.
If  a & b  are  the  roots  of  the  quadratic  equation  ax² + bx + c = 0,  then;
(i)  \alpha {\rm{ }} + {\rm{ }}\beta {\rm{ }} = \frac{{--b}}{a}  (ii)  \alpha {\rm{ }}{\rm{.}}\beta {\rm{ }} = \frac{c}{a}     (iii)  \alpha {\rm{ - }}\beta {\rm{ }} = \frac{{\sqrt D }}{a}
Nature  Of  Roots:
(a) Consider the quadratic equation ax² + bx + c = 0  where a, b, c  \in  R & a \ne 0 then
(i) D > 0   \Leftrightarrow  roots  are  real & distinct  (unequal).
(ii) D = 0  \Leftrightarrow  roots  are  real & coincident  (equal).
(iii) D < 0 \Leftrightarrow  roots  are  imaginary
(B) Consider the quadratic equation ax2+ bx + c = 0 where a, b, c  \in  Q & a \ne 0 then
 If  D > 0  &  is a perfect  square , then  roots  are  rational & unequal.
A quadratic  equation  whose  roots  are  a & b  is  (x – a)(x – b) = 0  i.e.   x2 – (a + b) x + a b = 0 i.e.
                          x2 – (sum of  roots) x +  product  of  roots = 0
Consider  the  quadratic  expression , y = ax² + bx + c  , a, b, c  \in  R & a \ne 0  then
(i) The graph between x, y  is always a  parabola.  If a > 0  then the shape of the parabola is concave upwards &  if a < 0  then the shape of the parabola is concave downwards.
Common  Roots  Of  2  Quadratic  Equations  [Only  One  Common  Root]-   Let  \alpha
be  the  common  root  of  ax² + bx + c = 0  &  a’x2 + b’x + c’ = 0
Therefore    \;a{\alpha ^2}{\rm{ }} + {\rm{ }}b\alpha {\rm{ }} + {\rm{ }}c{\rm{ }} = {\rm{ }}0{\rm{ }}\;and{\rm{ }}\;a{\alpha ^2} + {\rm{ }}b\alpha {\rm{ }} + {\rm{ }}c{\rm{ }} = {\rm{ }}0
 If we solve above pair by cramer’s rule we get

                                            \frac{{{\alpha ^2}}}{{bc' - cb'}} = \frac{\alpha }{{ac' - c'a}} = \frac{1}{{ab' - a'b}}

This will give us                    \alpha = \frac{{bc' - cb'}}{{ac' - c'a}} = \frac{{ac' - c'a}}{{ab' - a'b}}

                                            {(ac' - c'a)^2} = (ab' - a'b)(bc' - cb')

Every pair of the quadratic equation whose coefficients fulfils the above condition will have one root in common.

The condition that a quadratic function-  
                                           f(x , y) = ax² + 2 hxy + by² + 2 gx + 2 fy + c  may be  resolved  into  two  linear  factors  is  that     abc{\rm{ }} + {\rm{ }}2{\rm{ }}fgh{\rm{ }} - a{f^2} - b{g^2} - c{h^2}{\rm{ }} = {\rm{ }}0{\rm{ }}\; or
                                                                            \left| {\begin{array}{ccccccccccccccc} a&h&g\\ h&b&f\\ g&f&c \end{array}} \right| = 0
Reducible Quadratic Equations-These are the equations which are not quadratic in their initial condition but after some calculations, we can reduce them into quadratic equations
(i) If the power of the second term is exactly half to the power of the first term and the third term is a constant, these types of equations can be reduced to quadratic equations.
i.e., x + \sqrt x - 6 = 0 , {x^6} + {x^3} - 6 = 0{e^{2x}} + {e^x} - 6 = 0 ,{a^{2x}} + {a^x} - 6 = 0{\left( {2x + \frac{3}{{2x}}} \right)^2} + \left( {2x + \frac{3}{{2x}}} \right) - 6 = 0  all these equations can easily be reduced into quadratic equations by applying the method of substitution.

Example-Solve this equation and find x  {\left( {2x + \frac{3}{{2x}}} \right)^2} + \left( {2x + \frac{3}{{2x}}} \right) - 6 = 0 


let y=  \left( {2x + \frac{3}{{2x}}} \right) then given equation will become


                                   {y^2} + y - 6 = 0  it’s a simple quadratic equation we can be easily factorised it and solve  so y=–3,2

so  \left( {2x + \frac{3}{{2x}}} \right)=-3

                                    \frac{{4{x^2} + 3}}{{2x}} = - 3

                               \begin{array}{l} 4{x^2} + 3 = - 6x\\ 4{x^2} - 6x + 3 = 0 \end{array}

the final equation can be solved using Quadratic formula and the same process can be repeated for  y=2

(ii) If a variable is added with its own reciprocal, then we get a quadratic equation i.e,x + \frac{1}{x} - 6 = 0\frac{{2x + 3}}{{x - 2}} + \frac{{x - 2}}{{2x + 3}} = 0{e^{2x}} + \frac{1}{{{e^x}}} - 6 = 0{a^{2x}} + \frac{1}{{{a^x}}} - 6 = 0 all these equations can be reduced into quadratic by replacing one term by any other variable.


Standard Form of a Quadratic Function-A quadratic function y=ax2+ bx + c can be

reduced into standard form    y = a{(x - h)^2} + k  by the method of completing the square. If we

draw the graph of this function we shall get a parabola with vertex (h,k). The parabola will be upward for a>0 and downward for a<0


Maximum and Minimum value of a quadratic function- If the function is in the form

y = a{(x - h)^2} + k Then ‘h’ is the input value of the function while ‘k’ is its output.

(i) If a>0 (in case of upward parabola) the minimum value of f is f(h)=k

(ii) If a<0 (in case of downward parabola)the maximum value of f is f(h)=k
If our function is in the form of  y=ax² + bx + c then vertex of the parabola V = \left( { - \frac{b}{{2a}},\frac{D}{{4a}}} \right)
The line passing through vertex and parallel to the y-axis is called the axis of symmetry.
The parabolic graph of a quadratic function is symmetrical about axis of symmetry.

 \Rightarrow  f(x) has a minimum value at vertex if a>0 and {f_{\min }} = - \frac{D}{{4a}}  at   x = - \frac{b}{{2a}}


 \Rightarrow   f(x) has a maximum value at vertex if a<0 and   {f_{\min }} = - \frac{D}{{4a}}  at   x = - \frac{b}{{2a}}


In the next post about quadratics, I shall discuss discriminant, nature of roots, relationships between the roots. In the meantime, you can download the pdf and solve practice questions.

 quadratic equation 200 questions.pdf


 quadratic equation and function.pdf

   <img src="demo.png" alt="Smiley face">

IB Mathematics Tutors- types of mathematical function(part-1)

IB Mathematics Tutors should give a fair number of hours in teaching functions.This is my third article on functions in the series of ib mathematics

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here

For the sake of a comprehensive discussion, some standard functions and their graphs are discussed here
1.Constant Function-
Domain= ℝ    

                                            Range    = {k}                                       

<img src="constant-function.jpg" alt="constant-function">


Range    = {k}

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