a)
b)
c)
For , .
If , then clearly .
Therefore, .
From b) . Since ,
Replacing with , we get with and so
Giving, as required,
d)
From b),
.
From c),
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
yoyoyo
there was a nice question i found a while back and thought i would post it here:
given that e^x can be written as a sum of an odd and even function, find the two functions.
Last edited by mrbunton; 20 Nov 2018 at 11:20 AM.
Last edited by fan96; 20 Nov 2018 at 12:02 PM.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
its Hyperbolic function of sin and cos
so e^x = sinhx+coshx
Its legit like deriving cosx and sinx in complex field
The taylor expansion is just an approximate for e^x for values less than 1 or close to 0 but can get quite close. You would just factorise the even and odd parts of the function. However, that does not mean you cant use it, I tried it with Maclaurin series and it seems to look solid.
Last edited by HeroWise; 20 Nov 2018 at 2:05 PM.
Yeah it clicked in when i was thinking of expressing these interms of Complex version of cos and sin and basically simultaneous
More qtns
The equation of the tangent at the hyperbola is given by
Because the shortest distance between a point and a line is the perpendicular distance, is the intersection between the tangent line and the line perpendicular to the tangent which also passes through the origin.
That line is
Solving these two equations gives
Now,
Hence the locus of is given by the equation .
And similarly,
So,
Last edited by fan96; 1 Jan 2019 at 3:35 PM.
HSC 2018: English Adv. [80] • Maths Ext. 1 [98] • Maths Ext. 2 [95] • Chemistry [87] • Software Design [95]
ATAR: 97.40 | Uni Course: Advanced Mathematics (Hons) / Engineering (Hons) at UNSW
Last edited by stupid_girl; 17 Feb 2019 at 12:16 AM.
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