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} = {(3n2+7)/(n2 + 3n+1)} and the series an = (3n2+7)/(n2 + 3n+1).
(a) Determine if the sequence {an} 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 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 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.


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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