1. Newton’s binomial theorem continued
Newton’s binomial theorem is a powerful tool. Given a binomial (x + y)n . We
can more quickly expand it using the binomial coefficients.
Example 1.
(1) (x + y)3 = x3 + 3x2 y + 3xy 2 + y 3 .
(2) (x + y)5 = x5 + 5x4 y + 10x3 y 2 + 10x2 y 3 + 5xy 4 + y 5 .
(3) (x + 3)4 = x4 + 4x3 ⋅ 3 + 6x2 ⋅ 32 + 4x ⋅ 33 + 34 = x4 + 12x3 + 54x2 + 108x + 81.
If n is small enough and it is difficult to find the binomial coefficients, we can
use Pascal’s triangle for aid.
2. Functions
We have used the term “function” several times in this course. For example,
we have looked at area as a “function” from the plane to the real numbers. In
this section we defined functions in a more familiar way. One way to think of
a function is as a set of ordered pairs (x, y) such that for a given x, there is at
most one y such that (x, y) satisfies a given equation. For example, consider the
equation y = x2 . A function is the set of pairs (x, y) such that for every x there is
at most one y such that (x, y) satisfies y = x2 . So for whatever x we choose, there
is at most one y that could be its pair. Thus, we can say f (x) = x2 is a function.
√
Consider the equation y √
= x. Here, the function “doesn’t make sense” for all
√
−1
does
not
exist
for
us.
x only “makes√sense” for
values of x. For example,
√
x ≥ 0. However, f (x) = x can still be a function. If we assume that x always
yields the positive √
root, and not both the positive and negative, we that √
that for
every x√such that x makes sense, there is at most one y such that y = x. So,
f (x) = x is another function.
Now consider the function y 2 = x. This equation seems very similar to our first
example. Consider x = 4. There are two y values, y = 2, −2 such that y 2 = x. This
means that it is not a function.
Another way to think of functions is as algorithms. We can say a function is
rule that assigns exactly one real number to each number in a particular set of
real numbers. For example, f (x) = x√2 is a function from all real numbers to the
non-negative real numbers. f (x) = x is a function from the non-negative real
numbers to the non-negative real numbers.
1
2
A function of the form f (x) = mx + b with m and b real numbers, is called a
linear function. In the case that m = 0, we call the function a constant function.
Finally, a function of the form f (x) = ax2 + bx + c is called a quadratic function.
3. Graphs
Let f (x) be a function. The graph of the function f (x) is the set of points in
the cartesian plane (x, y) such that x and y satisfy y = f (x). So if f (x) = 3x + 5,
the graph is the set of points (x, y) such that y = 3x+5. We know that this is a line
with slope 3 and y-intercept 5. In fact, for all linear functions f (x) = mx + b, the
corresponding graph will be a line with slope m and y-intercept b. As you would
expect, the graph of a quadratic function will be a parabola, assuming a ≠ 0.
√
For example, f (x) = x2 , g(x) = x3 , and h(x) = 3√x are all functions that “make
sense,” or are defined on all real numbers. f (x) = x is a function that is defined
only on the non-negative real numbers. g(x) = x1 and h(x) = x12 are function that
are defined everywhere except for x = 0. See figure 5.25 in your text to see graphs
for these functions.
Finally, consider the graph of y 2 = x. It is shown in figure 5.26 in your text.
It looks like a sideways parabola. We’ve already determined that y 2 = x is not a
function, but the graph of it gives us a more clear picture of what is going on.
Consider all the graphs we’ve seen. Those that are not functions have two points
in the graph (x, y1 ) and (x, y2 ) for y1 ≠ y2 . Pictorially, if we can draw a vertical
line through the graph in such a way that it intersects more than one point, the
graph is not a graph of a function. This process is known as the vertical line test
and is quite helpful when we know what the graph for an equation looks like.
4. Tangent lines
Consider some curve in the cartesian plane and fix some point P on it. See
figure 5.10 in your text for reference. Let Q be some other point on the curve, and
let P Q be the line between them. Now imagine pushing the point Q along the
curve closer to P . As it gets closer and closer, the slope of the line P Q gets closer
to that of what is called the tangent line at P . Call the slope of the tangent line
at P , mP .
Say P = (x, y) and Q = (x′ , y ′ ). Let ∆x = x′ − x and ∆y = y ′ − y. So the slope of
the line P Q is
y ′ − y ∆y
=
.
x′ − x ∆x
3
Again, imagine we push Q along the curve close to P . This is the same as saying
we push x′ close to x, or that we push ∆x close to 0. Recall that as Q gets closer
∆y
to P the slope of P Q, ∆x
, gets close to the mP . In limit notation, we get
∆y
.
∆x→0 ∆x
mP = lim
The above definition is how Leibniz defined the notion of a tangent line. Notice
that he could not simply say “when ∆x = 0...” as this would give us ∆y = 0. So
∆y
0
∆x = 0 , which is not defined.