Introduction Definition Motion Product Rule Quotient Rule

Introduction Definition Motion Product Rule Quotient Rule

Introduction • Calculus is the ability to calculate the rate of change, known as the derivative, of one quantity with respect to another. • Sir Isaac Newton (1642–1727) and Gottfried Leibnitz (1646–1716) discovered calculus in the seventeenth century. It is one of the new branches of mathematics.

Sir Isaac Newton • Newton is accepted as one of the greatest minds in the history of man. • He discovered much of the maths and physics we still use today. But he also spent time thinking about what happens after death and the possibility of eternal life. As a result he spent a large part of his life studying alchemy (a form of chemistry).

George Berkeley • There is an Irish connection to the story of calculus. • One mile outside Thomastown in County Kilkenny on a bend of the River Nore is Dysart Castle and Church. • Bishop George Berkeley was born there in 1685. He opposed the new maths of calculus on the grounds that the small increments used were infinitely small. If they were zero, the whole grounds on which calculus is based is flawed. • Berkeley’s opposition to calculus is still valid but it is ignored because calculus is so useful.

dx dt Change in x Change in t __________ __ = Definition Rule for simple Differentiation 1. Look at the number in front of the t 2. Multiply this number by the power of t 3. Reduce the power of t by 1

Multiply the power by the number in front and drop the power by one dxdt dxdt dxdt = = = Examples x = 5t2 + 3t x = 5t2 x = 3t 3 10t + 3 5(2)t1 = 10t

Multiply the power by the number in front and drop the power by one dxdt = Examples x = 6t4 – 2t3 + 4t2 + 3t– 2 6(4)t3 – 2(3)t2 + 4(2)t1 + 3 The differential of a constant is zero = 24t3 – 6t2 + 8t+ 3

Multiply the power by the number in front and drop the power by one dxdt = Examples x = 6t–2 6(–2)t–3 = – 12t–3

dx dt dv dt __ __ p Change in distance Time Change in velocity Time _______________ _______________ Motion q We will look at calculus from the point of view of the motion of a car moving to the right: x is the distance moved. Velocity (v) is defined as the = Acceleration (a) is defined as the =

Motion The manufacturers of a small rocket claim its motion is given by x = 3t2+ 2t. Therefore the distance x it will move in metres can be found by putting the time t into the formula. After three seconds it will move 3(3)2+ 2(3) = 33 metres. Using calculus we can find its velocity or acceleration at any time t.

dx dt dv dt __ __ Velocity is defined by = Acceleration is defined by = Motion Motion is defined by x = 3t2+ 2t 6t+ 2 6

dx dt dv dt du dt __ __ __ If x=uv then =u +v Product Rule

dxdt dxdt =udv+vdu = Differentiate (t2– 3)(2 – 3t3) with respect to t. u v Product Rule u=t2– 3 v= 2 – 3t3 du= 2t dv=–9t2 (t2– 3) (–9t2) + (2 – 3t3) (2t) =–9t4 – 6t4 + 27t2 + 4t =–15t4 + 27t2+ 4t

v –u u v dx dt du dt dv dt __ __ __ __ __________ If x= then = v2 Quotient Rule

2t+ 5t2+ 1 _____ Differentiate with respect to t. dxdt dxdt 2t+ 5 t2+ 1 _____ x = vdu–udv = v2 –2t2 –10t + 2 = (t2+1)2 u v u= 2t+ 5 v=t2 + 1 du= 2 dv= 2t Quotient Rule (t2+1) (2) –(2t+ 5) (2t) 2t2 + 2 – 4t2 – 10t = = (t2+1) 2 (t2+1)2

Introduction Definition Motion Product Rule Quotient Rule