BRIEF QUIZ ANSWERS FOR CALCULUS II STUDENTS
In this document you will find brief answers to quiz questions.
If the quiz question is a homework problem, and the answer
is in the back of the book, then the brief answer that you
will see below will be only ``See the back of the book.''
Week 1
Week 2
- See the back of the book.
- See the back of the book.
- See the back of the book.
Week 3
- See the back of the book for (6.10.17). I mistyped
problem (6.9.19), so the answer in the back of the book is 0,
but the answer to this problem, as typed, is 2/(x*square_root(x^2-1)).
- See the back of the book.
- No, the equation tanh(x) = 1 cannot be solved.
The reason is that this equation is equivalent to
sinh(x)/cosh(x) = 1, or sinh(x) = cosh(x).
There is no solution to this equation since
(as we observed in class) sinh(x) < (1/2)*e^x < cosh(x) for
all values of x.
Week 4
- See the back of the book.
- See the back of the book.
- See the back of the book.
Week 5
Week 6
- See the back of the book.
- See the back of the book.
Week 7
- See the back of the book.
- We must find the centroid of the region in the
first quadrant cut out by a circle of radius 3 centered
at the origin. By symmetry, the x- and y-coordinates of the
centroid are equal, so we only need to calculate one of them.
The x-coordinate of the centroid is M_y/M where
M_y is the integral from 0 to 3 of the function
x*square_root(9-x^2) and M is the area of the quarter circle.
The integral for M_y
can be evaluated by the substitution u = 9-x^2; the answer
is 9. The area M of a quarter of a circle of radius 3 is
Pi*9/4. Thus M_y/M = 4/Pi. This means that the centroid is
(4/Pi,4/Pi).
Week 8
- See the back of the book.
- See the back of the book.
- See the back of the book.
Week 9
- See the back of the book.
- See the back of the book.
- See the back of the book.
Week 10
Spring Break
Week 11
Test 2.
Week 12
- Converges by comparison with 1/n^p, p = 3/2.
(See the back of the book.)
- Converges by ratio or root test.
(See the back of the book.)
- Diverges by comparison with harmonic series.
(See the back of the book.)
- Diverges by comparison with harmonic series.
(See the back of the book.)
- Converges. Since 3^(ln(n)) = (e^(ln(3)))^(ln(n)) =
(e^(ln(n)))^(ln(3)) = n^(ln(3)), it follows that this
series is none other than the sum of 1/n^p for
p = ln(3). This p is greater than 1, so the series converges.
Week 13
Week 14
Week 15
Last modified on Feb 5, 1998.