In a study of frost penetration it was found that the temperature T at time t (measured in days) at a depth x (measured in feet) can be modeled by the function

T(x, t) = T₀ + T^1e2−λx sin(ωt − λx)

where ω = 2π/365 and λ is a positive constant.

(a) Find ∂T/∂x. What is its physical significance?

(b) Find ∂T/∂t. What is its physical significance?

(c) Show that T satisfies the heat equation T_t = kT_xx for a certain constant k.

(d) Graph T(x, t) for λ = 0.2, T₀ = 0, and T₁ = 10.

(e) What is the physical significance of the term −λx in the expression sin(ωt − λx)?