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31
)1;6;2(
--=
)2;3;4(
--=
)2;2;4(
--=
162
0150
=-
0100
160
-
,
0
0150
=-
0100
Find the coordinates of the vectors:
162
--
234
=
=×× ADACAB
--
224
--
AB
AC
AD
--
Reading Questions
1. Find the volume of the parallelepiped determined by the vectors,
i−7j−5k, i−2j−6k,3i+2j+3k.
2. What does it mean geometrically if the product of three vectors
gives zero?
32
Section 2. Basic calculus
Topic 9. Limits
Introduction. This lecture is devoted to the discussion of the following concepts: «epsilon-delta definition» of limits, the rule of Lapetal,
uncertainties, some basic theorems of limits.
We have come to the «epsilon-delta definition» of limits. First, Socrates chooses E. He has to be shown that f (x) is within E of L, for every x
near a. Then somebody else (maybe Plato) replies with a number δ. That
gives the meaning of «near a.» Plato's goal is to get f(x) within E of L,
by keeping x within δ of a: if 0 < | x – a | < S then (f(x) – L) < E. The
input tolerance is δ (delta), the output tolerance is E. When Plato can find
a δ for every E, Socrates concedes that the limit is L.
.)(lim Axf
x=¥®
can graphically represent:
Fig.1 – graphic representation of limits
33
To the category uncertainties referred to the following relation
¥
0
¢
e
e
-
0
х
±=±
¹
12
8
+
-
x
x
0
¥
¥
0
;1;;0;;
¥-¥¥×¥
L” Hospital Theorem (rule of L” Hospital).
Example. Найти предел
lim
x
®
2
1
xx
+-
ln1
x
.
lim
xf
lim
=
®®
)(
xg
)(
As you can see, when you try to directly calculate the limit you get
indeterminate form
0
. Functions included in the numerator and denomi-
)(
xf
axax
¢
)(
xg
nator of the fraction meet the requirements of theorem of L” Hospital.
f¢(x) = 2x +
¢
xf
lim
x
®
)(
1
=
¢
xg
)(
1
; g¢(x) = e
1
x
2
+
x
x
e
x
;
312
+
=
;
=
ee
It should be noted that in this case the rule of L” Hospital is the only
one way of calculating the limits. Often in the specific example we can
use this and any other method
Basic theorems of limits.
Theorem 1.
lim
ax=®
, где С = const.
CC
The following theorem is valid under the assumption that the function
f(x) and g(x) have a finite limits if х®а.
Theorem 2.
Theorem 3.
Consequence.
Theorem 4.
lim
®
xf
ax
=
)(
xg
)(
Example. To find a limit
ax
®
®
lim
x
×=×
×=×
)(lim
xf
if
)(lim
xg
ax
2
2
2
®
)(lim)(lim xfCxfC
axax ®®
®xgax
86
+-
xx
.
)(lim)(lim))()((lim xgxfxgxf
axaxax ®®®
)(lim)(lim)]()([lim xgxfxgxf
axaxax ®®®
0)(lim
x2 – 6x + 8 = 0; x2 – 8x + 12 = 0;
34
D = 36 – 32 = 4; D = 64 – 48 = 16;
-
-
-
x
x
-
lim56(2)(3)
=-+=--=
()()
fttft
t
+D-
x1 = (6 + 2)/2 = 4; x1 = (8 + 4)/2 = 6;
x2 = (6 – 2)/2 = 2 ; x2 = (8 – 4)/2 = 2;
lim
)4)(2(
xx
xx
lim
=
xx
)6)(2(
--
4
x
22
®®
6
-
x
==
2
4
1
2
Example. To find a limit.
22
xxxx
+--++
lim
x
®
11
0
2
multiply the numerator and denominator
of the fraction on the adjoint expression:
lim
=
11
0
2
+×-
.
1
-=
)11(1
22
xxxx
-+-++
+-+++-
lim
=
22
0
xx
®®
)11)(1(
xxxxxx
2
x
22
)11)(1(
xxxxxx
+-+++-
Example. To find a limit.
2
56
xx
-+
2
3
x
®
lim
===
x
®
9
x
-
(2)(3)321
xx
---
3
(3)(3)336
xx
-++
2
xxxx
{ }
Reading Questions.
1. What is the limit of the sum of two functions?
2. What is the limit constants?
Topic 10. Derivative
and differential of a function
Introduction. This lecture is devoted to consideration of the basic
rules of differentiation, the derivative of the derivative and using derivatives in curve tracing.
The definition of the derivative.
At time t, the derivative f'(t)or df /dt or v(t) is
=
¢
()lim
ft
=
0
t
D®
D
.
35
We use symbols f ' (f prime of t) and
df
dt
tt
+D
dff
dtt
D=D
f
t
D
D
df
dt
2
ftt
=
ftD
D
()()
fttft
t
+D-
tt
t
=+D
f
t
D
D
¢
for the derivative. The ratio
on the right is the average velocity over a short time Δt. The derivative,
on the left side, is its limit as the step Δt (delta t) approaches zero.
Go slowly and look at each piece. The distance at time
is
f (t + Δt). The distance at time t is f(t). Subtraction gives the change in
distance, between those times. We often write Δt for this difference:
Δf = f (t + Δt) – f (t).
The average velocity is the ratio Δf / Δt (delta f over delta t) – change
in distance divided by change in time. The limit of the average velocity
is the derivative, if this limit exists:
lim
D®
This is the neat notation that Leibniz invented that:
Look at the calculation for
222
ttttt
=
D
+D+D-
=
()
2()
:
D
.
0
t
2
approaches
.
.
Important point: Those steps are taken before Δt goes to zero. If we
set Δt = 0 too soon, we learn nothing. The ratio
becomes 0/0 (which
is meaningless). The numbers Δf and Δt must approach zero together,
not separately. Here their ratio is 2t + Δt t, the average speed.
Then and only then can we approach Δt =0. The limit is the derivative 2t.
The basic rules of differentiation. We denote f(x) = u, g(x) = v –
functions which are differentiable at the point х.
1) Sum Rule. When we add functions, we add their derivatives.
(u ± v)¢ = u¢ ± v¢
Example; The derivative of x + sin x is 1 + cos x.
2) Product Rule. (u×v)¢ = u×v¢ + u¢×v
The derivative of uvw is uvw' + uv'w + u'vw – one derivative at a
time.
¢
¢
3) Quotient Rule.
u
ö
æ
=
÷
ç
v
ø
è
uvvu
-
2
v
, если v ¹ 0
36
Derivatives of basic elementary functions.
¢
¢
(
)
¢
x
cos
¢
x
sin
(
)
¢
(
)
¢
x
11x
a
x
ln
11x
+
¢×¢=¢
xuuyx
D
D
D
xuuyx
D
D
D
dxdududydx
¢=¢
1) С¢ = 0; 9)
2) (x
3)
æ
4)
ç
è
5)
6)
7)
8)
( )
( )
m
)¢ = mx
ö
÷
x
ø
x
ln =
log =
m-1
; 10)
1
=
¢
x
a
11)
xx2
11
12)
-=
2
x
xx
13)
ee =
xx
14)
aaa
ln=
1
15)
1
¢
16)
( )
( )
¢
( )
tgx
( )
ctgx
( )
arcsinxx
( )
arccosxx
( )
arctgx+=
( )
arcctgx
=
¢
¢
xx cossin =
xx sincos -=
1
2
1
-=
¢
¢
¢
2
1
=
-=
-=
2
1
-
1
2
1
-
2
2
Theorem. If y = f(u);; u = g(x), and the value function u is included
in the domain of the function f then uufy
)( .
The proof
y
=
D
D
y
=
D
×
D
×
limlimlim
D
xux
®D®D®D 000
D
( Dx®0, Du®0, because. u = g(x) – a continuous function)
dy
then
.
×=
)(
For example,
( )
)(ln
xf
xf
)(
xf
Power Rule . This section is full of rules, and I hope you will allow
one more. It goes beyond xn to (u(x))n. A power of x changes to a power
of u(x) – as in (sin x)6, or (tan x)7 or (x2 + 1)8. The derivative contains
n-1
nu
(copying nu
n-1
), but there is an extra factor du/dx. Watch that factor
in 6(sin x)5 cos x and 7(tan x)6 ~ sec2 x and 8(x2 + 1)7 (2x):
37
Example.
()5
ftt
=
5,0
dtdt
==
()
dyfxdx
¢
=
()dyfx
dx
¢
df
dx
df
dx
22222222
222
)2()1)(22(
¢
=
y
x
2
x
=
x
24
)1(2
++
xxxe
22
)1(
+
-++
22
)1(
+
exxxxexxe
=
22
)1(
+
x
3353
xxxxxxxx
22222
-+++
exexxeexex
The second derivatives.
We now introduce the derivative of the derivative. That is the second
derivative of the original function. It tells how fast the slope is changing,
not how fast y itself is changing. The second derivative is the "rate of
change of the velocity." A straight line has constant slope (constant velocity), so its second derivative is zero:
dfdf
has
2
2
A short form for the second derivative is f" or y". (This is pronounced
f double prime or y double prime). Example:
The second derivative of y = x3 is y" = 6x.
The differential
is consistent with the derivative
=
=
.
Topic 11. Using derivatives in curve tracing
Introduction. This lecture is devoted to consideration of finding the
asymptotes and creating a graph of functions
Our goal is to learn about f(x) from
questions. If
is positive, what does that say about f? If the slope is
negative, how is that reflected in the function? Then the third question is
the critical one: How do you identify a maximum or minimum? Normal
answer: The slope is zero.
To define increasing and decreasing, look at any two points x < X.
"Increasing" requires f(x) < f(X). "Decreasing" requires f(x) > f(X). A
positive slope does not mean a positive function. The function itself can
be positive or negative.
. We begin with two quick
38
Theorem. If
df
dx
df
dx
()2
fxxx
=-
df
dx
df
dx
df
dx
34
()43
fxxx
23
1212
xx
23
1212
xx
dy
dx
2
2
dy
dx
2
fxx
=
()
fx
¢
()
fx
¢¢
> 0 then f(x) is increasing. If
< 0 then f(x) is de-
creasing.
Example.
2
has slope 2x -2. This slope is positive when
x > 1 and negative when x < 1. The function increases after x = 1 and
decreases before x = 1.
Which x makes f(x) as large as possible? Where is the smallest f(x)?
Without calculus we are reduced to computing values of f(x) and com-
paring. With calculus, the information is in
.
Theorem. Local Maximum or Minimum.
Suppose the maximum or minimum occurs at a point x inside an in-
terval where f(x) and
are defined. Then f(x) = 0.
The word "local" allows the possibility that in other intervals, f(x) goes
higher or lower. We only look near x, and we use the definition of
Example.
=-
has slope
. That derivative is ze-
-
.
ro when x2 equals x3, at the two points x = 0 and x = 1. To decide between minimum and maximum (local or absolute), the first step is to
evaluate f(x) at these stationary points.
We find f(0) = 0 and f(1) = 1.
Now look at large x. The function goes down to – co in both directions. (You can mentally substitute x = 1000 and x = -1000). Conclusion
f(x) = 1 is an absolute maximum. f(x) = 0 is not a maximum or minimum
(local or absolute). We have to recognize this exceptional possibility,
that a curve (or a car) can pause for an instant (f' = 0) and continue in the
same direction. The reason is the "double zero" in
-
, from its
double factor x2.
When f '(x) is positive, f(x) is increasing. When
is negative, y(x) is
decreasing. That is clear, but what about the second derivative? From
looking at the curve, can you decide the sign of f''(x) or
swer is yes.
y" > 0 means that y' increases so y bends upward (concave up)
y" < 0 means that y' decreases so y bends down (concave down).
()
has
= 2x and
? The an-
= 2 (this parabola bends up)
39
An increasing population means
()
fx
¢
()
fx
¢¢
()
fx
¢
()
fx
¢¢
()
fx
¢
()
fx
¢¢
x
®+¥
x
®-¥
()
fx
®+¥
()
fx
®-¥
x
[
]
x
limlimlim22
==-=
means
> 0. Those are different.
> 0. An increasing growth rate
Theorem. When
When
= 0 and
= 0 and
> 0, there is a local minimum at x.
< 0, there is a local maximum at x.
A formula is now given for f(x). The problem is to create the graph.
Our job is to apply calculus. We extract information from f'(x) and
f''(x) as well as f(x). Small movements in the graph may go unnoticed,
but the important properties come through. Here are the main tests:
1. The sign of f(x) (above or below axis: f(x) = 0 at crossing point)
2. The sign of f(x) (increasing or decreasing: f'(x) = 0 at stationary
point)
3. The sign of f''(x) (concave up or down: f''(x) = 0 at injection point)
4. The behavior of f(x) as
5. The points at which
and
or
6. Even or odd? Periodic?
The straight line y = kx + b is a sloping asymptote..
xf
k
lim
=
x
)(
¥®
x-=¥®
,
kxxfb
)(lim
Creating a graph of a function has several stages. Consider the case.
Example. Find the asymptotes and create a graph of functions
2
xx
12
-+
y
=
.
1) Vertical asymptotes: y®+¥ x®0-0: y®-¥ x®0+0, so, х = 0 is a
vertical asymptote.
2) a sloping asymptote.:
2
lim
=
k
lim(())lim
bfxxx
=-=-=
xx
®¥®¥
22
æö
21211
xxxx
+---
ç÷ç÷ç÷
xxx
®¥®¥®¥
èø
12
-+
xx
x
æ
1lim
ç
¥®¥®
xx
è
2
æö
xx
+-
ç÷
èø
æöæö
xxx
èøèø
12
ö
1
=
-+=
÷
22
x
x
ø
21
x
Thus, line у = х + 2 is a sloping asymptote..
Draw the graph of the function (Figure 2):
40
4
6
99x
-
9
-
x
x
6
2
2
dy
dx
df
dx
2
-3 -2 -1 1 2 3
-2
Figure 2
Fig.2 – graphic representation of the function.
Example. Find the asymptotes and graph functions
x
y
=
2
Direct x = 3 and x = -3 are the vertical asymptotes of the curve.
Find a sloping asymptote.:
9
9
x
lim
=
b
=
2
9
-
x
lim
xx
x
¥®¥®
=
9
1
-
2
=
k
0
lim
x
9
¥®
0
=
2
y = 0 there is a horizontal asymptote.
4
2
.
-7.5 -5 -2.5 2.5 5 7.5
-2
-4
-6
Figure 3
Fig.3 – graphic representation of the function.
Reading questions
1. What is the derivative of constants?
2. From looking at the curve, can you decide the sign of f''(x) or
3. If
is positive, what does that say about f?
?
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