Find the temperature u(x, t) for the boundary-value problem dx² = ди at' 0 < x < L, t> 0 u(0, t) = 0, u(L, t) = 0, t > 0 u(x, 0) = f(x), 0 < x < L when L = 1 and f(x) = 100 sin(67x). [Hint: Look closely at Un = X(x)T(t) = A„e-k(n²77²/L³)t sin(17x) u(x, t) = and u(x, 0) = f(x) = A sin(N7T x).]

Please help me solve this.

2. f is a differentiable function on the interval [0, 1] and g(x) = f(2x). The table below gives values of f '(x). What is the value of g '(0.1)? X f'(x) 1 2 3 -4 5 -4 0.1 0.2 0.3 0.4 0.5 2 Cannot be determined

1 Let f(y) -8 + In(y) Determine the derivative of f. f'(y) Determine the interval(s) on which f is differentiable. Report the solution using invterval notation. f is differentiable on