Answer Happy

Please answer 4 and 5
Suppose we have two 1 metre long rods. Each rod has a temperature of 0° on its left end with the temperature increasing linearly to 10º at its right end. The rods are placed end-to-end so that the left end of one rod touches the right end of the other rod, thus giving us (effectively) a 2 metre long rod with a discontinuous starting temperature. Question 2: (1 Point) Let f() be the instantaneous) temperature of the joined rod immediately when the two smaller rods are joined. (So 0 <r<2). f(x) will be a piece-wise function. What is this function? if some condition is met Note: piece-wise functions look like f(1) = { something somethingelse if some other condition is met Hint: Have Desmos (or similar software) draw a picture of your answer and make sure it agrees with the description in the box. Maybe hand-draw a sketch of the smaller rods and the combined rod first. Question 3: (1 Point) Extend f(t) from question 2 to make an even function named f* (*). What is this extended function? Hint: Use Desmos to visually check your working. Question 4: (3 Points) Calculate the Fourier cosine series for f(c). Question 5: (3 Points) Let u(x, t) be the temperature on the rod (i.e., the 2m long rod obtained by joining the two lm rods together) at the point r and time t. So 0 <r< 2 and t > 0. Time t = 0 is the starting point immediately after the two smaller rods are joined. Solve the Heat Equation for the combined rod where assuming that the ends are insulated (i.e., that du/ar = 0 when r = 0 and 1 = 2), and that the constant of proportionality is K= 4. Note: you have all the information required to work out the initial condition u(x,0). You will need to use that initial condition.