AN ELECTROMAGNETIC ROCKET HYPER-LIGHT STELLAR DRIVE
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- leçon - matière potentielle : inside the circle
- light speeds
- sinh sinh
- charge density
- ring
- magnetic field
- particle
- electric field
- velocity
- electron
Solutions Sheet 3, DifferentiationQuestions 19-28v2
19) Go back to the definition and forx6= 0 consider
sinx f(x)−f(0)−1 sinx−xsinx−x x = = =x . 2 3 x−0xx x Thus, by the Product Rule, f(x)−f(0) sinx−xsinx−x lim = limx= limxlim 3 3 x→0x→0x→0x→0 x−0x x 1 = 0× −= 0. 6
Since the limit exists the functionfis differentiable atx= 0 with derivative 0.
20)Chain or Composite Rule Ifgis differentiable ataandfis differentiable atg(a)thenf◦gis differentiable ataand
0 0 0 (f◦g) (a) =f(g(a))g(a).
i) By the Composite Rule, and an earlier Question, d1 cosy1 −1 tanh (siny) = cosy= =. 2 2 dy1−(siny) cosycosy
This is valid as long as|siny|<1,which is certainly true fory∈(−π/2, π/2).
ii) By the Composite Rule, and an earlier Question, dcos1 1 y1 −1 sinh (tany) =q×= =. 2 2 dy2cosycosycosy 1 + (tany)
True as long as cosy6= 0 and so certainly true fory∈(−π/2, π/2).
iii) Recall from an earlier Question 12 that
d1 −1 coshy=p dy 2 y−1
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