Chemistry 3730 Fall 2000 Assignment 1

1.

(a)

Evaluate $$\int_0^\infty e^{-ax^8}\,dx$$. [2 marks]

Maple hints: Maple will need to know that a>0. The exponential function is denoted exp() in Maple.

Note: The solution will involve the gamma function. Maple often uses functions with which you may not be familiar to express answers to various computations. Don't let this bother you.

(b)

Obtain a floating-point value for the integral when a=5. [2 marks]

Maple hint: Since you will have made an assumption on a, you can use assign() or subs(), but not := with this variable.

2.

Find the only real solution to the equation x³-2x²+x-1=0. Give your answer in floating-point format. [3 marks]

3.

Find all maxima and minima of the function $f(x) = \exp(-p(x))$ where p(x) = 20x⁴ + 3x³ - 13x² - x + 1. Verify whether you have a maximum or minimum using the second-derivative test. [10 marks]

Hint: It helps to know how many maxima and minima there are so plot f(x) first.