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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 valuetheorem 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 usingmean value theorem the officer was able to conclude that the driver exceeded the speed limit.
The Continuity of ClimatologyA schematic of Earth’s equatorial circle (shown in blue), wherePAandPBare antipodal points(shown in red). The temperatures are shown at each point,T(θ) andT(θ+π), respectively, whereθis the angle from the origin, which has a temperatureT(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)