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- The Celsius temperaturein a region in space is given by T(x, y, z) = 2x2 - xyz. Aparticle is moving in this region and its position at time t is givenby x = 2t2, y = 3t, z = -t2, where time is measured in secondsand distance in meters. How fast is the temperature experienced by the particle changing in degrees Celsius per meter when the particle is at the point P(8, 6, -4)?x = 2(3y)^0.5, x = 0, y = 9, about the y-axix. Sketch the region.(a) Find the overlapping area of two equations also give its geometrical representation: u^2+v^2=n, u^2+(v-√n)^2=1 Where n is your arid number for example if your arid number is 19-arid-1234 then n=1234. (b) Find the points that touch the x axis of curve v=u^2+pu+(n+1). Where n is your arid number for example if your arid number is 19-arid-1234 then n=1234. Also find the equation of lines at that points give geometrical representation
- A space curve Let w = x2e2y cos 3z. Find the value of dw/ dt at the point (1, ln 2, 0) on the curve x = cos t, y = ln (t + 2), z = t.StokesTheorem.Evaluate∫ F·dr,whereF=arctanx/yi+ln√x2+y2j+k and C is the boundary of the triangle with vertices (0, 0, 0), (1, 1, 1), and (0, 0, 2).Perform the given problem solving on Finite Divided Difference or Euler's method. A storage tank contains a liquid at depth y where y = 0 when the tank is half full. Liquid is withdrawn at a constant flow rate Q to meet demands. The contents are resupplied at a sinusoidal rate 3Qsin^2(t)
- Let S be the solid of revolution obtained by revolving about the x-axis the bounded region R enclosed by the curve y=e−2x and the lines x=−1, x=1 and y=0. We compute the volume of S using the disk method. a) Let u be a real number in the interval −1≤x≤1. The section x=u of S is a disk. What is the radius and area of the disk? Radius: Area: b) The volume of S is given by the integral (b to a) ∫f(x)dx, where: a= b= and f(x)= c) Find the volume of S. Give your answer with an accuracy of four decimal places. Volume:A flat circular plate has the shape of the region x2 + y2<= 1. The plate, including the boundary where x2 + y2 = 1, is heated so that the temperature at the point (x, y) is T(x, y) = x2 + 2y2 - x. Find the temperatures at the hottest and coldest points on the plate.Sketch the region enclosed by the given curves. Decide whether to integrate with respect to x or y. Draw a typical approximating rectangle. x = 1 − y2, x = y2 − 1 The x y-coordinate plane is given. There are 2 curves, a shaded region, and an approximating rectangle on the graph. The first curve enters the window in the third quadrant, goes up and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the second curve, changes direction at the point (0, 0.5), goes down and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the second curve, and exits the window in the fourth quadrant. The second curve enters the window in the second quadrant, goes down and right becoming less steep, crosses the x-axis at approximately x = −0.71 crossing the first curve, changes direction at the point (0, −0.5), goes up and right becoming more steep, crosses the x-axis at approximately x = 0.71 crossing the first curve, and exits the window…
- Heat flux Suppose a solid object in ℝ3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is∇ ⋅ F = -k∇⋅ ∇T = -k∇2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions.Heat flux Suppose a solid object in ℝ3 has a temperature distribution given by T(x, y, z). The heat flow vector field in the object is F = -k∇T, where the conductivity k > 0 is a property of the material. Note that the heat flow vector points in the direction opposite to that of the gradient, which is the direction of greatest temperature decrease. The divergence of the heat flow vector is∇ ⋅ F = -k∇⋅ ∇T = -k∇2T (the Laplacian of T). Compute the heat flow vector field and its divergence for the following temperature distributions. T(x, y, z) = 100e-x2 + y2 + z2Walking on a surface Consider the following surfaces specified in the form z = ƒ(x, y) and the oriented curve C in the xy-plane. a.In each case, find z’(t). b.Imagine that you are walking on the surface directly above the curve C in the direction of positive orientation. Find the values of t for which you are walking uphill (that is, z is increasing). z = 4x2 - y2 + 1, C: x = cos t, y = sin t; 0 ≤ t ≤ 2π