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- compute the length of the polar curve. The cardioid r = 1 − cos θ in Figure 16If Z1= −4+j8 , Z2= 8−j5 , Z3= 3−j7 and Z4= −8−j9 determine i) Z = (Z1*Z2*Z3*Z4) / (Z1−Z2+Z3−Z4) ii) ln(Z2) in both Polar and Rectangular forms.Sketch and find additional polar three points for (2,4pi/3). When r>0 and r<0. And theta is between -2pi to 2pi.
- Suppose that U is a solution to the Laplace equation in the disk Ω = {r ≤ 1} andthat U(1, θ) = 5 − sin2θ.(i) Without finding the solution to the equation, compute the value of U at theorigin – i.e. at r = 0.(ii) Without finding the solution to the equation, determine the location of themaxima and minima of U in Ω.(Hint: sin2θ =(1−cos 2θ)/2.)Identify the conic with th e given equation and give its equation in standard form 2xy + 2√2x - 1 = 0Calculate the length of the polar curve r=1-sinθ on the given interval (0,2π).
- Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.If P1 and P2 are two points on the conic whose angles θ1 and θ2 differ by 180° , show that the harmonic mean of their respectivve values of r is a constant ( you will need to look up what harmonic mean is).Find the points of horizontal tangency to the polar curve r = 2 csc + 3 .