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 {an} = {(n2+n+2)/(2n2 + 5)} and the series an = (n2+n+2)/(2n2+5).
(a) Determine if the sequence {an} 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 an 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.

[Graphics:cardioid.gif]

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.


Dan Cohen
Fall 1999
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