A SHORT INTRODUCTION TO CALCULUS.   1. The ideas of Calculus were discovered at the same time by NEWTON in England and LEIBNITZ in Germany in the 17th century. There was a lot of ill feeling between them because each one wanted to take the credit for discovering Calculus. 2. Newton had a particular interest in the orbits of planets and gravity. He “invented” calculus to help him study such topics. His theory was used extensively in putting the first men on the moon and his equations of motion clearly describe the paths of objects thrown through the air.   Calculus can be applied to many subjects: finding equations to model the growth of animals, plants or bacteria; finding maximum profits in economics; finding the least amount of material to make boxes and cylinders; all sorts of velocity and acceleration problems.

So far we have only dealt with the gradients of lines. B The very basic idea of calculus is how to find the changing steepness of curves. So far we have only dealt with the gradients of lines. B 3 Gradient of AB = 3 4 A . 4

P We will move this line until it just touches the curve at one point.

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the gradient of the curve at P is the gradient of the tangent at P This line is called a TANGENT to the curve at P. We say that: the gradient of the curve at P is the gradient of the tangent at P P

Q To ESTIMATE the gradient of the tangent at P we use a CHORD PQ. NOTE: Q is meant to be a point VERY close to P. The diagram is very much enlarged for clarity. P

Q The Gradient of PQ is an approximation to the gradient of the tangent at P P

Q Clearly, the gradient of chord PQ is greater than the gradient of the tangent. To improve the approximation, we could move the point Q closer to the point P P

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The gradient of PQ is getting closer and closer to the actual gradient of the tangent at P. We now do this process using ALGEBRA. Q P

Suppose the curve’s equation is y = x2 32 R S 3 3 + h If x = 3, the distance PR = 32 = 9 If x = 3 + h, the distance QS = (3 + h)2 NOTE: h is a very small distance such as 0.01

y = x2 The gradient of PQ = QT PT = (3 + h)2 – 32 h = 9 + 6h + h2 – 9 And when h reduces to 0 then the gradient is equal to 6 Now we consider the triangle PQT in order to find the gradient of the CHORD PQ Q (3 + h)2 - 32 P T h (3 + h)2 32 R S 3 3 + h The distance PT = h The distance QT = (3 + h)2 - 32

Now let us repeat this process to find the gradient at x = 4 Q P T (4 + h)2 42 R S 4 4+ h If x = 4, the distance PR = 42 = 16 If x = 4 + h, the distance QS = (4+ h)2

y = x2 4 4 + h The gradient of PQ = QT PT = (4 + h)2 – 42 h And when h reduces to 0 then the gradient is equal to 8 Q (4 + h)2 - 42 P T h (4 + h)2 42 R S 4 4 + h The distance PT = h Remember h is very small! The distance QT = (4 + h)2 - 42

Now there is definitely a pattern here! Perhaps a better way to see the pattern is to choose a general position “x” instead of specific values like x = 3 and x = 4.

Now let us repeat this process to find the gradient at a general position x Q P T (x + h)2 x2 R S x x+ h At position x, the distance PR = x2 Also at x+ h, the distance QS = (x+ h)2

y = x2 x x + h The gradient of PQ = QT PT = (x + h)2 – x2 h = x2 + 2xh + h2 – x2 = 2xh + h2 = h( 2x + h) = 2x + h And when h reduces to 0 then the gradient is equal to 2x Q (x + h)2 – x2 P T h (x + h)2 x2 R S x x + h The distance PT = h Remember h is very small! The distance QT = (x + h)2 – x2

y = x2 at any point x is yꞌ = 2x The main symbol we use for the gradient of a curve is yꞌ (pronounced “y dash”) We have just found that the gradient of the curve y = x2 at any point x is yꞌ = 2x This means that for the curve y = x2 : at x = 1, the gradient yꞌ = 2×1 =2 at x = 2, the gradient yꞌ = 2×2 =4 at x = 3, the gradient yꞌ = 2×3 =6 at x = 4, the gradient yꞌ = 2×4 =8 at x = 10, the gradient yꞌ = 2×10 =20 at x = -6, the gradient yꞌ = 2×(-6) = -12 at x = ½ , the gradient yꞌ = 2× ½ =1

We found a simple pattern for y = x2 but there is also a pattern for any power of x

We can use the same basic diagram and theory to find the gradients of curves such as: y = x2 y = x3 y = x4

Now let us repeat this process to find the gradient of the graph y = x3 at a general position x Q P T (x + h)3 x3 R S x x+ h At position x, the distance PR = x3 Also at x+ h, the distance QS = (x+ h)3

y = x3 x3 x x + h = (x + h)3 – x3 h Q (x + h)3 – x3 The gradient of PQ = QT PT = (x + h)3 – x3 h We need more room to work this out. P T h (x + h)3 x3 R S x x + h The distance PT = h Remember h is very small! The distance QT = (x + h)3 – x3

The gradient of PQ = QT PT = (x + h)3 – x3 h = x3 + 3x2h + 3xh2 + h3 – x3 = 3x2h + 3xh2 + h3 = h(3x2 + 3xh + h2) = 3x2 + 3xh + h2 = 3x2 when h reduces to zero

We will repeat this process for the curve y = x4 then the pattern will be obvious to everybody!

Now let us find the gradient of the graph y = x4 at a general position x Q P T (x + h)4 x4 R S x x+ h At position x, the distance PR = x4 Also at x+ h, the distance QS = (x+ h)4

y = x4 x4 x x + h = (x + h)4 – x4 h Q (x + h)4 – x4 The gradient of PQ = QT PT = (x + h)4 – x4 h We need more room to work this out. P T h (x + h)4 x4 R S x x + h The distance PT = h Remember h is very small! The distance QT = (x + h)4 – x4

The gradient of PQ = QT PT = (x + h)4 – x4 h = x4 + 4x3h + 6x2h2 + 4xh3 + h4 – x4 = 4x3h + 6x2h2 + 4xh3 + h4 = h(4x3 + 6x2h + 4xh2 + h3) = 4x3 + 6x2h + 4xh2 + h3 = 4x3 when h reduces to zero

CONCLUSION! If y = xn the gradient is yꞌ = n×x(n – 1) If y = x2 the gradient is yꞌ = 2x1 If y = x3 the gradient is yꞌ = 3x2 If y = x4 the gradient is yꞌ = 4x3 If y = x5 the gradient is yꞌ = 5x4 If y = x6 the gradient is yꞌ = 6x5 If y = xn the gradient is yꞌ = n×x(n – 1)

Generally if y = axn then yꞌ = nax(n – 1) We could repeat the theory to be absolutely sure, but I think we can easily accept the following: If y = 5x2 then yꞌ = 2×5x1 = 10x If y = 7x3 then yꞌ = 21x2 If y = 3x5 then yꞌ = 15x4 If y = 2x7 then yꞌ = 14x6 Generally if y = axn then yꞌ = nax(n – 1) Note: It is a good idea to think of this in words: “Multiply by the power and reduce the power by 1”.

SPECIAL NOTES: The equation y = 3x represents a line graph and we already know that its gradient is 3 so we could write yꞌ = 3. Interestingly this also fits the pattern: We “could” say y = 3x1 so yꞌ = 1×3 × x0 = 3 Similarly, the equation y = 4 represents a horizontal line and we already know that its gradient is zero.

The process of finding the gradient is called DIFFERENTIATION. When we have an equation with several terms such as: y = 3x5 + 6x4 + 2x3 + 5x2 + 7x + 9 …it is a good idea to treat each term as a separate bit and we just apply the general rule to each term in turn. If y = 3x5 + 6x4 + 2x3 + 5x2 + 7x + 9 then yꞌ = 15x4 + 24x3 + 6x2 + 10x + 7 + 0

Usually we would just write the following: Question. Differentiate the function y = x3 – 5x2 + 3x + 2 Answer: yꞌ = 3x2 – 10x + 3

Because of this, we go through the same process in a general form. In Year 13 we learn to differentiate many new functions which are not just powers of x, such as: y = 2x y = log(x) y = sin(x) y = cos(x) y = tan(x) and other new functions. Because of this, we go through the same process in a general form.

y = f(x) which stands for ANY expression using “x” Q f(x + h) – f(x) The gradient of PQ = QT PT = f(x + h) – f(x) h P T h f(x + h) f(x) R S x x + h The distance PT = h The distance QT = f(x + h) – f(x)