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Continuity & Differentiability.

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Added on  2019-10-12

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Continuity & DifferentiabilityWarm Up: Using Calculus to issue speeding finesIf we letp(t) be the position function for the carand that the car passed through the highway on-ramp att = 0hrs. Thenp(0) = 0andp(10/60) = 25. Assumingpis differentiable, the mean value theorem says that there is ac∈(0,10/60)such thatv(c)=p'(c)=p(10/60)−p(0)1060−0¿25−01060−0=150km/hrIn other words, at some time during the drive, the car was traveling 150 km/hr. Hence by using mean value theorem the officer was able to conclude that the driver exceeded the speed limit.
Continuity & Differentiability._1
The Continuity of ClimatologyA schematic of Earth’s equatorial circle (shown in blue), where PA and PB are antipodal points (shown in red). The temperatures are shown at each point, T (θ) and T(θ + π), respectively, whereθ is the angle from the origin, which has a temperature T (0).Now, we are given that T is continuous inθon[0,2π], and we see that T is2πperiodic.Let∆: [0,2π] →Rto be the antipodal difference in temperature, i.e.∆(θ)= T(θ+π) − T(θ)Then ∆is also continuous on[0,2π], and we have that:∆ (0) = t(π) − t(0)and∆ (π) = t(2π) − t(π)So as t is2πperiodic, we get that∆ (0) = −∆ (π)
Continuity & Differentiability._2

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