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} = {(n²+n+2)/(2n² + 5)} and the series a_n = (n²+n+2)/(2n²+5).
(a) Determine if the sequence {a_n} converges or diverges. If the sequence converges, find the limit.
The sequence converges to 1/2.
(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 2²ⁿ/5ⁿ.
(a) Write out the fourth partial sum, s₄, of this series.
s₄ = 2²/5 - 2⁴/5² + 2⁶/5³ - 2⁸/5⁴
(b) Determine if this series converges or diverges. If the series converges, find its sum.
This is a geometric series with first term a = 4/5 and ratio r = -4/5.
The series converges to a/(1-r) = 4/9.

QUIZ 5

Find the radius of convergence and interval of convergence of the power series (x-4)ⁿ / n 5ⁿ
The radius of convergence is R = 5, and the interval of convergence is [-1,9).

QUIZ 6

Find the Taylor series of the function f(x) = 1/x² centered at a = 1.
The Taylor series is (-1)ⁿ(n+1)(x-1)ⁿ = 1 - 2(x-1) + 3(x-1)² - 4(x-1)³ + ...

QUIZ 7

Sketch the graph of the curve r = 1-sinq in polar coordinates.

Find the slope of the line tangent to the curve r = 1-sinq at q = p.
The slope of the tangent line is equal to 1.

QUIZ 8

1. Find an equation of the ellipse with foci (±4,0), and vertices (±5,0).
An equation for this ellipse is x²/25 + y²/9 = 1.

2. Show that x²+y²+z²+2x-6z+5=0 is the equation of a sphere, and find its center and raduis.
The center is (-1,0,3) and the radius is equal to the square root of 5.