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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. - PowerPoint PPT Presentation
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Introduction
Definition
Motion
Product Rule
Quotient Rule
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• 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.
IntroductionIntroduction
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• 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).
Sir Isaac NewtonSir Isaac Newton
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• 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.
George BerkeleyGeorge Berkeley
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DefinitionDefinition
Change in xChange in t
dxdt =__ __________
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
Rule for simple DifferentiationRule for simple Differentiation
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ExamplesExamplesMultiply the power by the number in front and drop the power by one
dx dt
= 5(2)t 1
= 10t
x = 5t2 x = 3t
dx dt
= 3dx dt
= 10t + 3
x = 5t2 + 3t
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ExamplesExamplesMultiply the power by the number in front and drop the power by one
dx dt
= 6(4)t 3
x = 6t 4 – 2t
3 + 4t 2 + 3t – 2
– 2(3)t 2 + 4(2)t
1 + 3
= 24t 3 – 6t
2 + 8t + 3
The differential of a constant is zero
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ExamplesExamplesMultiply the power by the number in front and drop the power by one
dx dt
= 6(– 2)t –3
x = 6t –2
= – 12t –3
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Change in distanceTime
_______________
MotionMotionp 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 =dxdt__
Change in velocityTime
_______________Acceleration (a) is defined as the =dvdt__
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MotionMotionThe manufacturers of a small rocket claim its motion is given by x = 3t
2 + 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.
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MotionMotionMotion is defined by x = 3t
2 + 2t
Velocity is defined by =dxdt__ 6t + 2
6 Acceleration is defined by =dvdt__
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Product RuleProduct Rule
If x = uv then = u + v dxdt__ dv
dt__ du
dt__
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u v
Product Rule u = t
2 – 3
du = 2t
v = 2 – 3t 3
dv = – 9t2
(t 2 – 3)(– 9t2) + (2 – 3t
3)(2t)
= – 9t 4 + 27t2 + 4t
= –15t 4
Differentiate (t 2 – 3)(2 – 3t
3) with respect to t.
= udv + vdudxdt
dxdt =
+ 27t 2 + 4t
– 6t4
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Quotient RuleQuotient Rule
__________If x = then =dxdt__
dudt__ dv
dt__
v – u u v__
v2
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+ 2 – (2t + 5) – 4t 22t2(t
2 + 1)
uv
Quotient Rule
(2) (2t)
(t 2 + 1)
u = 2t + 5
du = 2
v = t 2 + 1
dv = 2t vdu – udv
v 2
dxdt
=
dxdt
=(t
2 + 1)2=
(t 2 + 1)2
– 2t 2 – 10t + 2
=
2
– 10t
Differentiate with respect to t.2t + 5
t 2 + 1
_____
2t + 5 t
2 + 1 x = _____
Yes No
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