history of mathematics - National STEM Centre

: = vtcoscot, y = vt sin cot; r = vO
3 a (vcot 2 cos(cot-^n),vcot 2 sm(cot-^n)} or
vox sin (at, -van cos ox
)
u b r- Gradient A- t T>T PT = —————————— -at cos cot -sin cot ; from the
cot sin cot - cos cot
_ . . . , .. . dy I Ax
Cartesian parametric equation the gradient is -^- / — .
dt I At
which leads to the same result.
. „ ...... , . . , OT rco
4 From the similar right-angled triangles —— = — .
r v
Using r = vt leads to the result. The length of the arc PK
is also rcot .
Activity 9.3, page 115
b z-z' = e 2 + e(2b-x)
c If the coefficient, (2b - x), of e was negative, then
you could find a small positive value of e such that
z < z' , and z is not the maximum. Similarly, when the
coefficient of e is positive. Therefore the coefficient of e
is zero.
2 a z' = a2 X + e-X + e 3
b
c X 2 = \a 2 , giving the two solutions X = -"^ a - You
need to check which gives the maximum by another
method.
Activity 9.4, page 116
b x + e, a
2 4 - 2s) + se 2 (lx 2 - s 2 ) + X 2e3 }
3 a 5 = \x
c Differentiate to find that — = — ~- , and then use the
ck 3y 2
equation of the curve to show that you can write this as — .
s
Activity 9.5, page 116
.
1 a \ x + e,
CO
=a(
14 Answers
c e(3x 2s 2 - 2xs 3 ) + e 2 (lx 2s -s 3 ) + x 2 e 3 = 0
d (3* V - 2xs 3 ) + e(3x 2s - s 3 ) + x 2e 2 = 0
2 ^ = |jc
Activity 9.7, page 118
1 w, omn. w, xw , ult. xxomn. w, omn.omn. w
Activity 9.8, page 119
Activity 10.1, page 124
1 In a practical way negative numbers arose naturally in
accounting and financial systems. This is the way that they
were first used by the Chinese about 2000 years ago. A
positive number might have been used to represent money
received while a negative number represented the opposite,
that is money spent.
2 a A consistent interpretation of addition is as a
movement to the right of the number line through a certain
distance.
+3
H——\——\——\——\——\——I—
-1012345
2 + 3 = 5
Thus you can justify that the result of adding 3 to 2 gives 5.
This model gives you the same result if you started with 3
and added 2 to it. Your model is therefore consistent with
the rules of addition as it should be.
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