MATH 1552 QUIZZES FALL 1999

QUIZ 1

Evaluate the integral   (x+sin² x) sin x dx

(x+sin² x) sin x dx = -x cos x + sin x - cos x +(1/3)cos³x + C

QUIZ 2

Evaluate the integral   x²/(x²-4) dx

x²/(x²-4) dx = x + ln|x-2| - ln|x+2| + C

QUIZ 3

Find the solution of the differential equation y' = (xy+3x)/(x²+5) that satisfies the initial condition y(2)=3.

y = 2(x²+5)^1/2 - 3

QUIZ 4

1. Consider the sequence {a_n} = {(3n²+7)/(n² + 3n+1)} and the series a_n = (3n²+7)/(n² + 3n+1).
(a) Determine if the sequence {a_n} converges or diverges. If the sequence converges, find the limit.
The sequence converges to 3.
(b) Can you use your answer to part (a) to determine if the series a_n converges or diverges? Explain.
Since the terms of the series do not tend to zero as n approaches infinity, the series diverges by the n-th term test.
2. Consider the series (-1)^n-1 5ⁿ/3²ⁿ.
(a) Write out the fourth partial sum, s₄, of this series.
s₄ = 5/3² - 5²/3⁴ + 5³/3⁶ - 5⁴/3⁸
(b) Determine if this series converges or diverges. If the series converges, find its sum.
This is a geometric series with first term a = 5/9 and ratio r = -5/9.
The series converges to a/(1-r) = 5/14.