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Sept lie-20, 203
Announcements
Tmorials Start this week! An x n ntial f nction is a function of the form
Also, the Math Help Centre opens this week! (51.3.:th u
Topi s: on gm (so) f(x)=ax*
A
where a is a positive real number‘called the
base a
Maximum and Minimum Values Maximum and Minimum Values
f (c) is a global (absolute) maximum of f if f (c) is a global (absolute! minimum of f if
f(c) 2f(x) for all x in the domain off. f(c)sf(x) for all x_i_n the domain off.
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Announcements Optimization Problems
'JplCS: - Optimization is the process of making
- section 4.3 (what derivatives tell us about f) something as fully effective as possible,
' seem“ 4'5 (Curve SketChlngl subject to a set of constraints.
_
Power Functions Rational Functions
Some specnal cases: rational function
1
X
PU)
Q06)
where Pand Q are polynomials and Q(x);=0.
a=-2: f(x) = x"2 =
f(x)=
Shape: hyperbola
Domain: gig—[R1 R
Asymptotes: x :0; -. Q
Rangezyéiﬂl VI: 4:,
Examples:
1 x+2
f(x
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U/_. M u‘ Vectors in 3 Coordinate System Vectors in a Coordinate System
We represent vectors i
Announcements
.opics:
- 4.1 (maximum and minimum values).
- Section 4.3 (what derivatives tell us about f).
— section 4.5 (curve sketching]
* Read these sections in your textbook!
Work On:
- Exercises in the textbook corresponding to the above
sections as
Chain Rule
More Examples:
Differentiate the following.
(a) f(x)=e'2fcos4x
Starts with The PI‘COUCT
rule The you use Chum
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(e '3‘) (anew x) 1+ (e “)(-.swxx~>
1 “'quh [(coslel “r (Q‘sinLlKH
(b) g(x) = (Jr2 + l)3(x2 + 2)6
the: J r.- Announcements
For al ebraic vectors in R2: TOPICS:
If a =(a1 a ) and b=<b b) then — sections 12.2 (vectors), 12.3 (dot/scalar product), and
' ' ' ' 12.4 (cross/vector product)
* Read these sections in your textbook!
a-b=alb1 +azb2e
For algebrai
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If f(x) =k, where k is a constant, then f'(x)=0.
Example:
I y m=0
f(x)= 3 [J
f'(x)=0
———l———~ x
l
l
l
l
l
C(JSTF‘HTE :‘~"ttl.‘!'li'féii€ LEE
1».-
Let C be a constant.
Then i , =
dxtc f(x)] c dxm
Slope of a linear l:i.1i‘iCijC5i”;
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Slope = change in output
m
m: f(x7_)—f(xl)
change in input
= 2—)]!-
x2 — xr.
x.—x,
T
y
Secant Lines and Tangent lines
A secant line is a line that intersects two points on a
curve.
Ar
Steps:
1
Announcements NDTETAKERS NEEDED
Topics: * 1 .y, * . 7 . . 7
- Secﬁbn 2.2 (limnofa funnier” Oli me Thai Are you in class regularly Whats m it for you.
. 23 I. . l k. ,Th h , 3d DO VOthakeleglble “NBS? - Avaluable volunteer
- Egmon 'm't aws [s [p e eorem a
Functions can be described in 4 ways:
' Numerically (table of values)
° Geometrically (graph)
‘ Algebraically (explicit formula)
- Verbally (description in words)
Modeling Exercise:
Give an interpretation of the following graph.
What physical phenomena co
The Second Derivative
Example:
Find the intervals of concavity and the inﬂection
points for the following functions.
(a) f(x) = Jr3 + 6.r2 — 15x 3
i‘
x
/ « w
= } Cw VAC. K
\
x + 5 \
(c) go) = xe” "
Second Derivative Test
(also used to find Local Maxima
Academic dishonesty:
You are expected to exhibit honesty and use ethical behaviour in all aspects of the learning process.
Academic credentials you earn are rooted in principles of honesty and academic integrity.
Academic dishonesty is to knowingly act or
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