Provide the answers to the following questions and very brief justification.

1) The function g(t) =a cos(at) + b exp(t) with free parameters a, b R with a, b > 0 is defined for all values t R. For which values of a and b is g(t) a 7-periodic function?

2) The function f(x, y) = g(x)h(y) where g(x) = 1 + 2x and h(y) = 3 + 2y. Write down the Taylor series of f(x, y) including all terms up to fifth order around the point (0, 0).

3) Consider an even function f(t) on t [1, 1] with f(t) > 0 for all t. The 2-periodic function g is defined for the interval [1, 1] as g(t) = t f(t) + 3. Determine all Fourier cosine coefficients ak for k 0 of g(t) (but not the coefficients b_k ).