2 Unit Maths
Notes about these Notes
I did not have to pay attention in my 2 unit maths classes because I did extension 2 so these note are very incomplete. They may also be wrong. This is from the NSW HSC course that existed in 2008 so may not be completely relevant to the new course.Title: 16 October 2007 2U Maths
Date: 16 October 2007 6:59 PM
Category: School
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Logarithms
Differentiations of logarithms. Particularly natural logs.
Title: 17 October 2007 2U Maths
Date: 17 October 2007 7:05 PM
Category: School
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Generally the derivative of ax = ln a . ax
*later*
INTEGRATION
The reverse of differentiation. To find the area under a curve, you use this.
You have a function. The derivative of this function's area function is the original function.
∆x was involved.
There was that scrolly thing: ∫
∫ƒ(x)dx
This is how you write an integration statement. The f(x) is the function that you are integrating.
Title: 18 October 2007 2U Maths
Date: 18 October 2007 8:43 PM
Category: School
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HOW to integrate.
4
∫ f(x) dx
1
The number on the bottom and top is the range.
When given a curve and boundaries, change the function so that it is the primitive, though opposite of derivative.
Put it in brackets.
Insert the bottom number into the primitive and take it away from the bottom number in the primitive function.
Title: 19 October 2007 2U Maths
Date: 19 October 2007 11:18 AM
Category: School
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The Integral of (ax + b)n
∫(ax +b)n dx
= (ax + b)n+1 +C
a(n + 1)
(C is a constant)
Title: 22 October 2007 2U Maths
Date: 22 October 2007 8:25 PM
Category: School
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Areas between 2 curves: slightly more complicated than between the x axis, think more and you will get it.
Take the lower curve away from the higher curve.
Title: 24 October 2007 2U Maths
Date: 24 October 2007 7:38 PM
Category: School
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Odd/Even Functions
Show that f(x) = x3 - x
If you have an odd function and you are asked to find something like -3 to 3, the net area is zero.
If you have an even function and are asked to find something like -3 to 3, the net area is 2x 0 to 3
Find a value of m such that
m
ʃ x3 dx = 80
1
......
F(m) - F(1) = 80 etc etc
Approximate methods of Integration
1. Simpson's Rule
Theory: Given a function y=f(x) the value of
b
ʃ f(x) dx
a
can be found by
A= h/3(df + dL + 4dm)
H is the distance between slices. df is the first value of f(x) L the last and m the middle.
For multiple slices:
A = h/3 [(sum of first and last f(x) values) + 4(every odd f(x) values) + 2(even f(x) values)]
*The odd and even values are actually just the 1st, 3rd 5th... and 2nd, 4th, 6th... number you get out of the f(x)
2. Trapeziodal Rule
Like Simpson's Rule but divided into inaccurate trapeziums.
A = h/2[(1st and last f(x) value sums) +2(sum of all other f(x) values)]
Title: 29 October 2007 2U Maths
Date: 29 October 2007 8:40 PM
Category: School
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Volumes of Solids of Revolutions
Volume of a disk is πy2∆x
x=b
lim ∑ πy2∆x
∆x → 0 x=a
b
= π∫ y2 dx
a
Area aginst the y axis
rearrange the function to make y the subject.
then integrate
b
∫ y dy
a
Do this with volume too. Remember to square and be careful which sign to use with square roots.
Title: 31 October 2007 2U Maths
Date: 31 October 2007 8:55 PM
Category: School
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Now would be a good time to revise on Integrals. You have an exam next week.
Title: 5 November 2007 2U Maths
Date: 5 November 2007 6:31 PM
Category: School
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Number patters (Series)
0,2,4,6,8
a is the first term. a = 0
d is the difference between terms d = 2
T = a + (n-1)d
Sn = (n/2)(a + a+ (n-1)d)
Title: 8 November 2007 2U Maths
Date: 8 November 2007 9:02 PM
Category: School
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Seriously. You need to remember this. (For arithmetic series)
For term n:
Tn = a + (n-1)d
For sum of numbers to term n:
Sn = (n/2)(2a + (n-1)d)
Title: 12 November 2007 2U Maths
Date: 12 November 2007 6:51 PM
Category: School
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For geometric series
Tn = arn-1
Where a is the first, r is the ratio between 2 terms (found by dividing?) and n is the term you are finding.
Sn = a(1-rn)
1-r
Title: 13 November 2007 2U Mahts
Date: 13 November 2007 6:54 PM
Category: School
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Compound Intrest
A=P(1+r/100)n
Sn = a(1-rn)
1-r
EXT QUESTION
How much to deposit to get $400000 exactly?
An = a R(1-Rn)
1-R
Title: 2 February 2008 2U Maths
Date: 2 February 2008 6:38 PM
Category: School
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INTEGRATION!
The trapezoidal rule is either:
1/2(b-a)(f(a) + f(b)) (you add multiple instances of this equation depending on how many trapezoids you have)
h/2((y0 + yn) + 2 (y1 +y2 + y3 + ... + yn-1) (where n is the number of trapezoids and h is the width of each trapezoid)
Title: 4 March 2008 2U Maths
Date: 4 March 2008 6:32 PM
Category: School
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This was learnt in Ext 1 Maths:
Locus definition of a parabola:
The locus of a point whose distance from a fixed point (the focus) is the same as the distance from a fixed line.
The line is called the directrix.
For a parabola with vertex (h,k) the equation is
(x-h)2 = 4a(y-k)