Homework 11
Problem 1.
1) at 11:24 pm - thats about the time when the population stopped increasing
2) around 20 million - thats by eyeballing the yvalue when x is about 11:24 pm
3) at 3:36 am - thats when the population starts increasing drastically agai
Homework 5
Problem 1a.
We know that
8x6 = (2x2 )3
and
27
3
= ( )3
8
2
We apply the formula for dierence of two cubes:
27
3
9
= (2x2 )(4x4 + 3x2 + )
8
2
4
We can still factor the rst factor in the above, so we have:
3
9
3
3
9
2
4
2
f1 (x) = 2(x )(4x + 3x +
Homework 8
Problem 1.
One should observe that (0.25,0.5) lies on the given line 2x + y = 1. Thus,
the reection is itself, i.e. (0.25,0.5).
Problem 2. We should start by drawing the line L through our point (0, 3)
that is perpendicular to our line 2x + y =
Math W1003, Fall 2013
Lecture 21
Last time, we talked about exponential functions, and dened what we
mean by expressions like:
exp2 (x) = 2x
But quite often, we may be faced with equations like: what is the exponent x
such that 2x = 5? The answer, which w
Math W1003, Fall 2013
Lecture 23
Our last topic will be sequences, which will probably also be the rst
topic youll study in calculus. A sequence is a bunch of numbers placed in
a certain order:
2, , e2 3, 7
The above sequence has four elements, which can
Homework 14
Problem 1a.
If we shift the graph of sin a by units to the left, we get back the graph of
2
cos a:
sin a +
= cos a
2
If instead we shift the graph of cos a by units to the right, we get sin a
2
instead of sin a:
cos a +
= sin a
2
Problem 1b.
I
Math W1003, Fall 2013
Practice Midterm Exam
2 hours, 4 problems, 100 points. No calculators or notes/books, but
you are welcome (though not required) to use graph paper/ruler.
Explain all steps in your reasoning, or you may lose up to 75% of
points. Dont
Math W1003, Fall 2013
Practice Final 1
3 hours, 6 problems, 150 points. No calculators or notes/books, but
you are welcome (though not required) to use graph paper/ruler.
Explain all steps in your reasoning, or you may lose up to 75% of
points. Dont forge
Homework 1
Problem 1a.
am
1
am
= n = amn
an
a
Problem 1b.
5 (x4 y 3 z 2 )2 (3 z 1 x5 y 3 )1
=
5 (x4 y 3 z 2 )2
5 x8 y 6 z 4
5
5
=
= (x85 y 6(3) z 4(1) ) = x3 y 9 z 5
3 z 1 x5 y 3
3 x5 y 3 z 1
3
3
Problem 1c.
(
a2n
2a3 n
a2n
bn
a2n
a2n3n
1
)(
) =( n )( n
Homework 6
Problem 1.
Problem 2.
4x 3y + 12 = 0 y = 0, 4x = 12, x = 3 x = 0, 3y = 12, y = 4
1
Problem 3.
By examining the graph in problem 2, you can eyeball points that are 5 apart,
for example:
(3, 0) (0, 4)
The way we guess that these are distance 5 ap
Homework 19
Problem 1.
This is an even function because it is symmetric across the yaxis: f (x) =
f (x).
This is an odd function because it is symmetric across the origin: f (x) =
f (x).
Problem 2.
We have:
f (x) = x3 + x2
Since this is neither equal to f
Homework 12
Problem 1
= 30 :
= 45 :
= 60 :
x = x cos(30 ) y sin(30 ) = x 3 y
2 2
y = x sin(30 ) + y cos(30 ) = x + y 3
2
2
x = x cos(45 ) y sin(45 ) = x 2 y 2
2
2
y = x sin(45 ) + y cos(45 ) = x 2 + y 2
2
2
x = x cos(60 ) y sin(60 ) = x y 3
2 2
y = x s
Homework 15
Problem 1a.
5
)
4
x = r cos , y = r sin
(2,
hence:
x = 2 cos
5
4
=2
y = 2 sin
5
4
=2
2
2
2
2
= 2
= 2
So the point has Cartesian coordinates ( 2, 2)
1
Problem 1b.
5
)
3
x = r cos , y = r sin
(6,
hence:
x = 6 cos
y = 6 sin
5
3
5
3
1
2
=6
=6
Homework 18
Problem 1a.
arcsin(1) =
2
2
= 1
5
6
=
4
= 1
sin
because
Problem 1b.
arccos
3
2
=
5
6
because
cos
3
2
Problem 1c.
4
because
tan
2
3
because
cot
arctan(1) =
Problem 1d.
1
arccot
3
=
2
3
1
=
3
In all the above problems, one needs to also e
Math W1003, Fall 2013
Practice Final 2
3 hours, 6 problems, 150 points. No calculators or notes/books, but
you are welcome (though not required) to use graph paper/ruler.
Explain all steps in your reasoning, or you may lose up to 75% of
points. Dont forge
General ideas:
periodicity:
sin( + 2) = sin
cos( + 2) = cos
(sin )2 +(cos )2 = 1
Pythagorean theorem (PT):
defs of tan and cot:
application of PT:
tan =
1+(tan )2 =
sin
cos
1
(cos )2
cot =
cos
sin
1+(cot )2 =
1
(sin )2
Angle sum/dierence formulas:
Math W1003, Fall 2013
Lecture 22
Logarithms appear whenever we need to compute exponents. For example, a computer scientist may need to gure out how many bits of information
(zeroes and ones) he needs in order to encode a letter. A sequence of n bits
can
Math W1003, Fall 2013
Lecture 7
In our last class we discussed lines in the coordinate plane. The equation of
a line is:
ax + by + c = 0
(1)
for some real constants a, b, c. This should be read as:
the line cut out by the above equation is the set of poin
Math W1003, Fall 2013
Lecture 8
We will start this class by doing a bonus topic to our previous lecture.
Namely, we gave formulas for the reection in certain special lines (the axes
x = 0 and y = 0, and the diagonal x = y). These formulas were all in the
Math W1003, Fall 2013
Lecture 10
Last class was mostly about getting qualitative information from the graphs
of functions. For example, take the function:
f : R R
whose graph looks like the following:
We can infer the following from the graph:
The functi
Math W1003, Fall 2013
Lecture 9
We will now begin a two lecture mini-series about graph of functions, one
of the most important applications of coordinate geometry. You can see
graphs anywhere, from presentations in a board room to gathering data in
the n
Math W1003, Fall 2013
Lecture 6
Several lectures ago, we learned how to use numbers to represent points on
a line. This was our rst taste of coordinate geometry (coordinate is the
name we give to the number that represents a point) and today we will take
Math W1003, Fall 2013
Lecture 3
Today well be looking at functions more in depth. Remember that a function
is an assignment between two sets: a way to attach to each element of the
rst set a single element of the second set. For example, think about your