Compute the following Laplace transforms by hand using the definition of a Laplace trans- form (not using a table, although you can check your answers with a table): (a) Delta function: f(t) = 8(t) Note: the Laplace transform integral is from just before 0 (what you called 0 in calculus) to infinity. So, integrate from 0 to ∞. (b) Heaviside function: f(x) = {1; t≤0 1, 20 (c) Time-shifted delta function: f(t) = 8(t-a).

Please assist the following questions in the image with full handwritten working

et △ABC be a triangle in S^2 (the two-dimensional sphere). Let the dual point C'∈S^2 of C be defined by the following three conditions:(i) d(C' , A)=π/2(ii) d(C' , B)=π/2(iii) d(C' , C)≤π/2A' and B' are defined analogously. Thus we get a dual triangle △A' B' C'. More precisely, △ABC is a non-degenerate triangle, andit follows (you may assume) that △A' B' C' is too.(question) Are there triangles △ABC in S^2 identical to their own dual △A'B'C'?   I would be very thankful if you could provide some explanation with the steps, thank you in advance.

b) Using the table of standard transforms, find the Laplace transforms of the following functions: (i) 6e-²t cosh(5t) (ii) t² e ³t