# 3.10 Assignment 1 (Sjerve)

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#                           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.
# (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.

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