1. There are two functions that have some interesting uses in the real world, and they are the hyperbolic trig functions cosh(x) = tez 2 and sinh(x) = 2 The graph of the hyperbolic cosine is called a catenary, and models the shape of an object hanging by its own weight between two points. This can be found in nature in spiderwebs, vines, the rope at the movie theater, power lines, and the Gateway Arch in St. Louis (although it is inverted). Some suspension bridge cables have this shape, but the Golden Gate Bridge actually does not because the roadway is so heavy that it pulls the cables into a parabolic shape instead of a catenary. Let's explore the properties of these two functions. (a) Compute cosh(a) + sinh(z). (b) Compute (sinh(z)). (c) Compute & (cosh(I)). (d) There is also a hyperbolic version of tan(r) defined as tanh(r) = sinh(r)/ cosh(x). Compute & (tanh(z) (e) Show that hyperbolic sine satisfies the following double angle formula: sin(2r) = 2 sinh(r) cosh(r). (f) Does hyperbolic cosine also satisfy a double angle formula? (Do not just say yes or no; you should show your work and/or explain your reasoning.) (g) Let a E R be a fixed constant. Find all critical points of cosh(ar) and determine if they are local maxima, local minima, or neither. Does your answer depend on a