Logistic Differential Equations (LDE)

A Logistic Differential Equation (LDE) is an ordinary differential Equation whose solution is a logistic function.

LDE(logistic differential equation) include two positive parameters –

i) Growth parameter (growth rate)k: This parameter   plays a role similar to that of r in exponential differential equation.

ii) Carrying capacity (M):The carrying capacity of an organism in a given environment is defined to be the maximum population of that organism that the environment can sustain indefinitely.

If any queries, please drop in comment box.

Understanding Related Rates

This is an important application of  differential calculus. Related rates is nothing but the study  of relations among different variables for the ongoing situation  and how they are changing with respect to another quantities. Here you can find real life word problems on related rates. Process of solving such problems is explained beautifully!

What is the difference between linear approximation and linearization.

There is  no difference between linear approximation and linearization. These are just two different ways to say the same thing.

Linear approximation is an important application of differential calculus.

Applications of integral Calculus

Integral has many applications right from finding areas under the curve to finding volume of solids of revolution, finding arc length, surface area, work done etc and many other applications of physics too.

Here you can find beautiful summary of finding volume by three methods (disk, washer and shell)  as well as  volume by cross sections along with  examples. Finding area under the curve is also discussed


Applications of Differential calculus

Differential calculus  has numerous applications , from finding critical values  to finding maximum  and minimum values , checking concavity, rate of change measures,  linearization  , optimization and many more.

I find these topics very interesting  and i bet you too will find them interesting  once you get the concepts. Here i’m sharing a link where you can find useful notes  on these topics. Some are added already  and some are being added  at regular intervals.

Don’t forget to share your experience  and your valuable suggestions to improve  this service. Thanks !

Interesting Exponential and Logarithmic functions

Exponential function:  Any function represented as f(x)=  is called exponential function, where a>0 and a .

Domain of an exponential function is R, set of all real numbers and range is (0, ) because exponential function attains only positive  values.

For more  on exponential and logarithmic functions, visit here…https://celestialtutors.com/topic/exponential-and-logarithmic-functions/

Polar coordinate System and converting equations from polar to rectangular and viceversa

Polar coordinate system is a plane with Pole(point O) and polar axis  which is horizontal axis from point O. Any point P in this plane is assigned polar coordinates represented as P(r, theta)

we measure theta as positive when moving counterclockwise and negative when moving clockwise.

If r > 0, then P is on the terminal side of  theta . If r < 0, then P is on the terminal side of   (theta + pi).


Complex Numbers (Not so complex)

A complex number is represented as a+ib where a is real part and b is imaginary part. The complex number a+ib can also be represented as ordered pair(a,b) and plotted as a point in Argand plane. In this plane horizontal axis is called real axis and vertical axis is called imaginary axis.

Pascal’s Triangle and Binomial theorem

An algebraic expression containing two terms is called binomial expression. The general form of the binomial expression is (x+a) and the expansion of (x+a)^n, where n is a natural number, is called binomial theorem. It gives a formula for the expansion of the powers of binomial expression.
The coefficients in the binomial expansion follow a specific pattern known as Pascal’s triangle.
Following are some important features of Pascal’s triangle.
-Each row is bounded by 1 on both sides.
-Any entry except first and last, is the sum of two entries in preceding row , one on the immediate left and other on immediate right.
Some important conclusions from the Binomial Theorem:
Total number of terms in binomial expansion is n+1.
The coefficients of terms equidistant from the beginning and end are equal. These coefficients are known as binomial coefficients.
The terms in the expansion of (x-a)^n are alternatively positive and negative first being positive always.
The coefficient of x^r in (r+1)th term in the expansion of (x+a)^n is (_r^n)C.
For more help on this topic, please visit..https://celestialtutors.com/topic/pascals-triangle-and-binomial-theorem-2/

How to solve linear equation in one variable.

The word equation is derived from word ‘equal’. Any expression having equal (=) sign is called equation. If any linear algebraic expression in one variable is equal to some integer or some other algebraic expression, then it is called algebraic equation.
Linear equations can be categorized as follows
1. When variable is only one side.
2. When variable is on both sides

When variable is only on one side: We always solve the equation for an unknown variable using the same operations on both sides so that equation doesn’t become uneven and both sides remain balanced.
When variable is on both sides:
When variables on both sides of equation, we try to bring the variables together either on left or on right side and then move forward with the same process of isolating variable
For detailed explanation on this topic , please reach at…https://celestialtutors.com/topic/how-to-solve-linear-equation-in-one-variable/