Hello! would you be able to provide a detailed explanation of these questions and their answers with each step? Also, feel free to write it out if it is easier... Thank you!

1) a) (Ex. 3.1.12 (b), 3pts) Show that T(..'c, t) = e'\"(cos(:z:) + cos(y)) satises the two dimensional heat equation + at' 16 82T 82T 8T 6:32 Byz b) (Ex. 3.1.29 (a), 3pts) Let f, g be (Eafunctions of one variable and set q(3:,t) = f(:t: t) + g(:z: + t). Show that gt) satises the wave equation 82gb/6t2 = 6295/6322. 2) Determine the second order Taylor expansion of the following functions f about the point (0, 0). Using a calculator, compute the value f (0.1, 0.1) and compare it to the value of the second order Taylor polynomial at (0.1, 0.1). a) (Ex. 3.2.4, 4pts) f(m,y) = 1/(32 + y2 + 1). b) (Ex. 3.2.6, 4pts) f(:r,y) = 3"\"2'3'2 cos(:r:y)