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]

=g(b)-g(a)

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">

Read more

Applications of Derivatives in IB Mathematics

Applications of Derivatives in IB

Mathematics-

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})}}

=f'({x_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}}}

Read more