WEBVTT
Kind: captions
Language: en

00:00:01.320 --> 00:00:04.160
>> Welcome to the Cypress
College Math Review

00:00:04.380 --> 00:00:05.660
on Difference Quotients.

00:00:06.920 --> 00:00:09.920
Here's the graph of
a function, y = f(x).

00:00:10.446 --> 00:00:13.626
A secant line is a line
through two points on a curve.

00:00:14.286 --> 00:00:15.676
Here's our two points.

00:00:16.756 --> 00:00:20.896
Delta x is a change in x.
So we have our first point

00:00:21.166 --> 00:00:25.346
with coordinates x, f(x)
and our second point

00:00:25.386 --> 00:00:29.216
with coordinates
x + delta x and f

00:00:29.216 --> 00:00:34.006
of the quantity x + delta
x. We want to find the slope

00:00:35.066 --> 00:00:37.716
of the line through
those two points.

00:00:38.666 --> 00:00:40.056
Here's the formula for slope.

00:00:40.800 --> 00:00:42.180
Now let's apply it.

00:00:46.560 --> 00:00:55.460
So the slope will be rise
over run, so the change

00:00:55.826 --> 00:01:00.366
in the y coordinates over the
change in the x coordinates,

00:01:04.916 --> 00:01:08.696
which gives us the formula
for the difference quotient.

00:01:09.666 --> 00:01:14.046
So the difference quotient is
the slope of the secant line.

00:01:18.886 --> 00:01:23.436
A lot of instructors, instead
of using delta x, use h,

00:01:24.036 --> 00:01:26.616
and that's what we'll
use in the problems.

00:01:27.196 --> 00:01:29.836
So instead of using
delta x, we will use h,

00:01:30.676 --> 00:01:32.816
so this is the formula
that we will use

00:01:32.816 --> 00:01:35.556
for the difference
quotients in the problems.

00:01:42.856 --> 00:01:45.226
Let's find the difference
quotient for this function.

00:01:46.486 --> 00:01:48.626
Here's our formula we're
going to need to use.

00:01:49.646 --> 00:01:55.696
Let's first find f of the
quantity x + h. So we're going

00:01:55.696 --> 00:02:01.446
to replace the variable
with x + h. We distribute

00:02:02.416 --> 00:02:10.456
and we get 5 x + 5 h. Now we
want our difference quotient.

00:02:16.056 --> 00:02:20.306
So we have f of the
quantity x + h

00:02:21.876 --> 00:02:28.306
minus f(x) all over h. This
is a really good idea.

00:02:29.016 --> 00:02:32.736
Brackets minus brackets
all over h. What do you put

00:02:32.736 --> 00:02:34.176
in the first brackets?

00:02:34.476 --> 00:02:37.326
Well, f of the quantity x + h,

00:02:37.866 --> 00:02:43.176
so that would be 5 x +
5 h - 7 in this case.

00:02:43.936 --> 00:02:45.466
What goes in the
second brackets?

00:02:45.466 --> 00:02:46.516
The original function.

00:02:47.806 --> 00:02:49.536
Now get rid of the brackets.

00:02:51.716 --> 00:02:56.996
So 5 x + 5 h - 7,
distribute the negative,

00:02:57.516 --> 00:03:03.636
so we have negative
5 x + 7 all over h,

00:03:06.226 --> 00:03:11.426
then we have positive 5 x - 5
x, negative 7 and positive 7,

00:03:12.776 --> 00:03:18.626
so we get 5 h over
h. The h's cancel,

00:03:18.996 --> 00:03:21.756
so we get 5 with
the restriction,

00:03:21.756 --> 00:03:24.036
of course, that h cannot be 0.

00:03:33.596 --> 00:03:35.566
Let's find the difference
quotient

00:03:35.566 --> 00:03:37.586
for this quadratic function.

00:03:42.306 --> 00:03:47.256
Every time you see an x, you're
going to replace it with x + h.

00:03:53.116 --> 00:03:55.226
And now we simplify
the right-hand side.

00:03:55.916 --> 00:03:59.506
Watch out, exponents
before multiplication,

00:03:59.646 --> 00:04:02.016
so we need to square
out this binomial.

00:04:02.436 --> 00:04:04.346
You cannot distribute the 3.

00:04:05.066 --> 00:04:07.406
Exponents come before
multiplication.

00:04:09.646 --> 00:04:11.406
So we square out the binomial.

00:04:12.016 --> 00:04:14.586
What do you get when you
square out a binomial?

00:04:14.586 --> 00:04:16.976
When you square out
a + b, you get what?

00:04:18.336 --> 00:04:21.496
A squared + 2 a b
+ b squared, right?

00:04:21.666 --> 00:04:29.316
So in this case, that would
be x squared + 2 x h or h x,

00:04:29.406 --> 00:04:32.526
either one, + h squared.

00:04:34.026 --> 00:04:41.706
Now I distribute the negative
2, so negative 2 x - 2 h + 3.

00:04:42.426 --> 00:04:43.736
Now we distribute the 3.

00:04:51.116 --> 00:04:53.096
And any time you change
what you're doing,

00:04:53.096 --> 00:04:55.716
you need to change your
label on the left-hand side.

00:04:56.076 --> 00:04:57.366
Now we're doing something else.

00:04:57.366 --> 00:04:58.506
We have to change our label.

00:04:59.116 --> 00:05:01.446
Now we're doing the whole
difference quotient,

00:05:01.986 --> 00:05:04.666
so we have to change our
label on the left-hand side.

00:05:05.376 --> 00:05:06.826
So we wrote down our label.

00:05:06.896 --> 00:05:08.476
Our new label is
the whole formula

00:05:08.476 --> 00:05:09.736
for the difference quotient.

00:05:10.266 --> 00:05:11.616
Brackets minus brackets.

00:05:11.616 --> 00:05:17.316
So we have the brackets
minus brackets all

00:05:17.316 --> 00:05:20.866
over h. What do we put
in the first brackets?

00:05:22.260 --> 00:05:27.780
f of the quantity x + h,
what we were just working on.

00:05:35.640 --> 00:05:37.900
What do we put in
the second brackets?

00:05:38.240 --> 00:05:39.280
The original function.

00:05:44.220 --> 00:05:46.020
Now we get rid of the brackets.

00:06:01.186 --> 00:06:02.466
So we distribute the negative.

00:06:07.476 --> 00:06:10.146
So we have a positive
3 x squared

00:06:10.146 --> 00:06:11.956
and a negative 3 x squared.

00:06:12.376 --> 00:06:13.656
We have a negative 2 x

00:06:13.656 --> 00:06:17.016
and a positive 2 x. We have a
positive 3 and a negative 3.

00:06:18.196 --> 00:06:22.286
Now we can factor out an h
from every term that's left

00:06:22.976 --> 00:06:34.546
and we're left with 6 x + 3 h
- 2 all over h. The h's cancel,

00:06:35.336 --> 00:06:39.466
so we're left with 6 x + 3 h

00:06:39.800 --> 00:06:44.400
minus 2 with the restriction
that h cannot be 0.

00:06:54.660 --> 00:06:56.516
So in this case, f

00:06:56.516 --> 00:07:00.886
of the quantity x + h
really can't be simplified.

00:07:01.736 --> 00:07:03.336
So here we go with the
difference quotient.

00:07:12.106 --> 00:07:15.436
So brackets minus
brackets, and in this case,

00:07:15.436 --> 00:07:17.106
the brackets don't
serve any purpose.

00:07:28.156 --> 00:07:29.496
All right, there's
two different ways

00:07:29.496 --> 00:07:32.756
to simplify this complex
fraction, and I'm going

00:07:32.756 --> 00:07:34.896
to do the problem twice so you
can see the two different ways.

00:07:35.596 --> 00:07:37.946
The first way is to
multiply it by the number 1

00:07:38.196 --> 00:07:39.966
which eliminates the
complex fraction.

00:07:40.666 --> 00:07:43.866
So I determine the LCD
of these fractions,

00:07:44.126 --> 00:07:47.976
and the least common denominator
would be x times the quantity x

00:07:48.016 --> 00:07:51.606
+ h. So I multiply the numerator

00:07:53.316 --> 00:07:55.286
and the denominator
by that expression.

00:07:55.286 --> 00:07:58.026
Of course, you can only multiply
an expression by the number one,

00:07:58.226 --> 00:08:01.866
and that's equal to
number one, distribute this

00:08:02.456 --> 00:08:06.226
to the first term in the
numerator, which gives me 3 x,

00:08:06.880 --> 00:08:10.340
distribute it to the
second term in the numerator

00:08:10.440 --> 00:08:13.920
and it gives me 3 times
the quantity x + h.

00:08:15.280 --> 00:08:21.336
In the denominator, it's
h times x times x + h,

00:08:21.336 --> 00:08:23.196
and it's all factored
and you want it factored,

00:08:23.196 --> 00:08:24.226
so we leave it like that.

00:08:25.596 --> 00:08:40.756
In the numerator, we distribute
the negative 3, 3 x - 3 x,

00:08:43.036 --> 00:08:54.436
so we have negative 3 h over h
x times x + h. The h's cancel,

00:08:55.556 --> 00:08:58.596
so we end up with negative 3

00:08:59.106 --> 00:09:03.446
over x times the quantity
x + h with, again,

00:09:03.446 --> 00:09:06.146
the restriction that
h cannot be 0.

00:09:14.236 --> 00:09:15.926
Referring back to
the same problem,

00:09:17.896 --> 00:09:19.976
to add or subtract fractions,

00:09:20.246 --> 00:09:22.516
where the least common
denominator is simply the

00:09:22.586 --> 00:09:28.016
product of the denominators,
you just cross-multiply, yes,

00:09:28.346 --> 00:09:33.316
a times d + b times c
over b times d, yes,

00:09:33.316 --> 00:09:34.856
to add fractions, okay?

00:09:35.396 --> 00:09:36.456
So we'll do that here.

00:09:37.346 --> 00:09:44.716
3 times x minus, in this case,
3 times the quantity x + h all

00:09:44.716 --> 00:09:49.876
over x times the
quantity x + h, yes.

00:09:50.126 --> 00:09:54.086
Now, I'm dividing by h. I
really don't want to do this all

00:09:54.086 --> 00:09:57.096
over h. That gives,
again, a complex fraction.

00:09:57.726 --> 00:10:00.806
So to divide by h,
that's the same thing

00:10:00.806 --> 00:10:03.796
as multiply by 1 over h, yes?

00:10:05.456 --> 00:10:07.136
Well, why write it like that?

00:10:08.086 --> 00:10:11.536
Multiply it by 1 over h.
Why don't I simply write it

00:10:11.536 --> 00:10:13.456
as h times it, yeah?

00:10:13.756 --> 00:10:15.756
Multiply h times
the denominator.

00:10:17.196 --> 00:10:22.746
Good. And then we do the
same thing we did earlier,

00:10:23.146 --> 00:10:31.686
since we've already done this
problem, and we get negative 3 h

00:10:32.276 --> 00:10:37.176
over h x times the quantity
x + h. The h's cancel

00:10:37.876 --> 00:10:42.876
and we get negative 3 over
x times the quantity x + h

00:10:42.876 --> 00:10:45.746
with the restriction
that h cannot be 0.

00:10:51.826 --> 00:10:53.826
Here are some problems
to try on your own.

00:10:54.806 --> 00:10:59.966
Pause the video, try these
problems, then restart the video

00:11:00.116 --> 00:11:01.346
to check your answers.

00:11:08.356 --> 00:11:09.976
Here are the answers
to the problems.