> # ASSIGNMENT #1, MATH 210, SECTION 202 # # QUESTION # 1. # # Consider the equation # x^4 -x -1=0. (1) # # (a) Find the exact solutions of (1) using the solve command. # (b) Use evalf to numerically evaluate these solutions. # (c) Use fsolve to numerically compute the solutions. Hand in the value of the largest # positive solution to 10 decimal places. # (d) How many of the solutions have absolute valueless than 1? # # QUESTION # 2. # # The following equation occurs in the theory of crystal growth in solutions: # # x^3 + exp(1/x)=c. (2) # # (a) Letting f(x)=x^3 + exp(1/x)=c, use Maple to compute f'(x), f''(x). # (b) Show that for a given value of c there are 0, 1 or 2 solutions of (2) with x>0. # (c) Find numerically the smallest value of c for which equation (2) has positive solutions. # Explain your reasoning. # (d) Find numerically the positive solutions of (2) when c=4. # (e) Discuss the accuracy of the solutions in (d). # # QUESTION #3. # # Consider the following equation for x<0: # # (sin(x))^2=exp(-x)*cos(x) (3) # # (a) Write the equation in a form more convenient for plotting for negative x. # (b) Find the first 3 negative solutions to 10 decimal place accuracy. # (c) Give an asymptotic formula for the n^th negative solution for large n.
