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# Calculus

A brief explanation of calculus, which gives a brief introduction to derivatives

by

Tweet## Kate Middleton

on 20 May 2011#### Transcript of Calculus

Calculus Differential Integral Is used when finding the gradient(m) of the tangent to the curve at a certain value (which is represented as x.) Another way of talking about it is finding the derivitive of

a function. A function is written as f(x), and you use it when writing the rule for a line. So instead of writing y=3x+2 we write f(x)=3x+2. This is so that when we are given a value of x, e.g. 3 we insert it into ALL of the "x" in the equation. So it would be f(3)= 3(3)+2. We would say it as f of 3 equals three times three plus two. Intergration is an importa There is a long history of calculus and it dates back to the ancient Greeks. Some expressions often used in calculus are;

lim

h >0

Lim translates to Limit. This means a number as close to the number you have, but never actually reaching it. A limit of a function is a number a function will approach but never quite reach.

h >0 means as h approaches 0. One of the biggest aims of differential Calculus is to find the derivitive or gradient function. These are represented as f'(x) or

dy

dx

To find the gradient of a curve at a point, you must find the derivative first. You use this formula;

f(x)=ax

f'(x)=nax Once you have found the derivative you THEN substitute the x-value into the equation to find the gradient. Lets just start by finding the derivative though remembering the rule;

f(x)=ax

f'(x)=nax

Okay so for f(x)=4X find the derivative.

f(x)=4x

f'(x)=24x

Now that you've seen how to do it. Do this question;

Find the derivative of f(x)=8x The correct answer is f'(x)=24x Now you have a basic undersatnding of the basis of differential calculus.

WE will move on to the very basics of integral calculus. INtegral calculus is often used to find the area between the curve and the

x-axis. There is a symbol VERY often used in integral calculus is This sign means "sum" and is often said to be a big, streched

"s." It also means add the area. The "sum" symbol is also often written like this:

b

f(x)dx

a This means "find the area between the curve f(x) and the x-axis between the points (a,f(a)) and (b,f(b)) a b This formula will help you with the area fo polynomials: f(x) dx=ax

. n+1 +c It helps to remember that 1=1x Thats as far as we're going to go into it, because it gets VERY complicated after that.

But before we finish, Lets take a quick look back, to the history, to the roots of this tree that is calculus. Calculus was discovered by two people at the same time.

Liebniz, who was German, and Newton, who was English. Both discovered it around 1676, but did

not publish their methods immediately. Newton waited nearly twenty years to publish. They ended up hating each other, in one of the

most famous feuds in science history. Newton's notation of calculus lead to our writing of f'(x)

Leibniz however gave us the sum of symbol and dy

. dx On Newton's tomb in westminster abbey was written;

"If i have seen further than others it is becauseI have stood on the shoulders of giants." On Newton's tomb in Westminster Abbey was written;

"If I have seen further than others it is because I have stood on the shoulders of giants." This was a clear shot at Leibniz who was a short man. References

www.sosmath.com/calculus/calculus.html

www.calculus.org

en.wikipedia.org/wiki/Differential_calculu

sydney.edu.au/stuserv/documents/maths.../differentialcalculus.pdf

www.math.about.com

www.google.com.au/images

And the mathematics genius' that are Mrs Royce, Mr Middleton and Mrs Middleton By Katherine Middleton

and Owen Metcalfe

Full transcripta function. A function is written as f(x), and you use it when writing the rule for a line. So instead of writing y=3x+2 we write f(x)=3x+2. This is so that when we are given a value of x, e.g. 3 we insert it into ALL of the "x" in the equation. So it would be f(3)= 3(3)+2. We would say it as f of 3 equals three times three plus two. Intergration is an importa There is a long history of calculus and it dates back to the ancient Greeks. Some expressions often used in calculus are;

lim

h >0

Lim translates to Limit. This means a number as close to the number you have, but never actually reaching it. A limit of a function is a number a function will approach but never quite reach.

h >0 means as h approaches 0. One of the biggest aims of differential Calculus is to find the derivitive or gradient function. These are represented as f'(x) or

dy

dx

To find the gradient of a curve at a point, you must find the derivative first. You use this formula;

f(x)=ax

f'(x)=nax Once you have found the derivative you THEN substitute the x-value into the equation to find the gradient. Lets just start by finding the derivative though remembering the rule;

f(x)=ax

f'(x)=nax

Okay so for f(x)=4X find the derivative.

f(x)=4x

f'(x)=24x

Now that you've seen how to do it. Do this question;

Find the derivative of f(x)=8x The correct answer is f'(x)=24x Now you have a basic undersatnding of the basis of differential calculus.

WE will move on to the very basics of integral calculus. INtegral calculus is often used to find the area between the curve and the

x-axis. There is a symbol VERY often used in integral calculus is This sign means "sum" and is often said to be a big, streched

"s." It also means add the area. The "sum" symbol is also often written like this:

b

f(x)dx

a This means "find the area between the curve f(x) and the x-axis between the points (a,f(a)) and (b,f(b)) a b This formula will help you with the area fo polynomials: f(x) dx=ax

. n+1 +c It helps to remember that 1=1x Thats as far as we're going to go into it, because it gets VERY complicated after that.

But before we finish, Lets take a quick look back, to the history, to the roots of this tree that is calculus. Calculus was discovered by two people at the same time.

Liebniz, who was German, and Newton, who was English. Both discovered it around 1676, but did

not publish their methods immediately. Newton waited nearly twenty years to publish. They ended up hating each other, in one of the

most famous feuds in science history. Newton's notation of calculus lead to our writing of f'(x)

Leibniz however gave us the sum of symbol and dy

. dx On Newton's tomb in westminster abbey was written;

"If i have seen further than others it is becauseI have stood on the shoulders of giants." On Newton's tomb in Westminster Abbey was written;

"If I have seen further than others it is because I have stood on the shoulders of giants." This was a clear shot at Leibniz who was a short man. References

www.sosmath.com/calculus/calculus.html

www.calculus.org

en.wikipedia.org/wiki/Differential_calculu

sydney.edu.au/stuserv/documents/maths.../differentialcalculus.pdf

www.math.about.com

www.google.com.au/images

And the mathematics genius' that are Mrs Royce, Mr Middleton and Mrs Middleton By Katherine Middleton

and Owen Metcalfe