1.2
The Slope of a Tangent
A secant
A tangent
In the graph of f(x)=x2, the secant PQ passes through P(1, 1)
and Q(x, x2).
Q
The slope of secant is
P
mPQ
As point Q moves closer from the right toward P(1, 1), we have the following slopes of secant
Q
P
(1,
4.2 Critical Points, Local Max and Min
If
then,

f(x) has a local max/min value at

f(c) does not exist

x=c ,( f ' ( c )=0)
c is a critical number
(c, f(c) is a critical point
The First Derivative Test
Test for local maximum or local minimum point
Exa
4.3
Vertical and Horizontal Asymptotes
Vertical Asymptotes
a vertical line that a f(x) can never reaches
x
lim
x 0
0.1=10
1
=
1 2
x 0 (10 )
1
0.01=10
1
x2
lim
1
=
2 2
x 0 (10 )
2
lim
10
1
=
10 2
x 0 (10
)
0.0 01=10
lim
0.0 01=10100
lim
x 0
1000
0.0 01=10
4.4
Concavity and Points of Inflection
Concavity
 Concave upward

Concave downward
x=2, m1 =2
x=3, m2 =1
x=4, m3=0
x=5, m4=1
x=6, m 5=2
m=f ' ( x )
mis increasing f ' ( x ) is increasing
Let
If
g ( x ) =f ' ( x )
g (x )
Since,
g (x )
is increasing, then
7.6 The Cross Product of Two Vectors
The cross product a b of a and b in R3 is the vector that is
perpendicular to a and b , such that a , b and a b form a
righthand system.
b a=a b opposite direction
the cross product exists in R3 only
Example1 (P.401)
Worksheet: 7.3
Solve the following problems. Answer for each question is indicated inside cfw_.
1) WNBA basketball player Siddharth Swoopes scores on 0.89 of his free throws. What is the expected
waiting time before he misses the basket on a free throw? c
Worksheet: 2.3 Practice
Solve the following problems. Answer for each question is indicated inside cfw_.
1) A boy pushes a lawn mower (m=17.05 kg)
2) Block A is on a ramp inclined at 21. Block A is
starting from rest across a horizontal lawn by
connected
1.4
The Limit of a Function
lim f ( x )=L
x a
1)
2)
3)
from
The limit of f(x) as x approaches to a equals L.
The value of f(x) can be made arbitrarily close to L by choosing x sufficiently close to a
(but not equal to a)
f ( x ) exists if and only if the
7.5 Scalar and Vector Projections
Scalar Projection of a on b
ON =a cos
o
0 180 , b 0
o
a) 0 90
o
b) =90
c)
o
90 180
o
Calculate Scalar Projections
a) The scalar projection of a on b
a)
The scalar projection of b on a
b)
The scalar projection of a on b
Worksheet: 7.2 Practice
Solve the following problems. Answer for each question is indicated inside cfw_.
1) What is the expectation for a binomial distribution with p=0.96 and n=8? cfw_7.68
2) Suppose that 0.41 of the first batch of engines off a new prod
Chapter 8 Combining Functions
Chapter 8 Prerequisite Skills
Chapter 8 Prerequisite Skills
a)
Pattern A
Question 1 Page 414
Pattern B
Pattern C
b) A: linear increasing one step each time.
B: exponential increasing by a multiple of 2.
C: quadratic increasin
1.6
Continuity
The graph of f (x) has no breaks, jumps or gaps f (x) is
continuous.
The function f ( x) is continuous at
f (a) is defined at x=a
lim f ( x )=f (a)
x a
Removable Discontinuity
cfw_
f ( x )= 2 x 2,if x <1
2 x 2,if x >1
Jump Discontinuity
cfw
1.5 Properties of Limits
lim f ( x )=L
x a

As x approaches to a (from both sides of a), the value of f ( x) become closer and closer to
the number L.

In fact, there is no need to consider

The behavior of f ( x) near a (not at a) matters.
x=a , f (a)
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru Gugoiu
2010 Iulia & Teodoru G
Worksheet: 7.4
Solve the following problems. Answer for each question is indicated inside cfw_.
1) What is the expected number of men on a 5member jury randomly chosen from 13 women and 11
men? cfw_2.29
2) What is the expected number of women on a 8memb
Chapter 4
Curve Sketching
4.1 Increasing and Decreasing Functions
Increasing Function
Decreasing Function
Theorem for Increasing/Decreasing Functions

'
f ( x )> 0 f ( x) is increasing
f ' ( x )< 0 f ( x) is decreasing
Example 1(P. 166)
Use the derivativ
7.2 Velocity
Example 1 (P.365, Example 2)
A plane is heading due north with an air speed of 400 km/h when it is
blown off course by a wind of 100 km/h from the northeast. Determine
the
resultant ground velocity of the airplane.
air Speed
 the velocity of
7.3 The Dot Product of Two Geometric Vectors
a b=ab cos
0o 180o
a b is a scalar
1)
0o 90 o
cos> 0, a b > 0
2)
o
=90
o
cos 90 =0, a b=0
3)
90o 180o
cos< 0, a b < 0
Example1(P.373, example2)
a)
If a= 7 , calculate a a .
b)
Calculate i i
Properties of
7.4 The Dot Product of Algebraic Vectors
In R3, if
a =( a1 , a2 , a3 ) , b=( b1 ,b 2 , b3 )
then
a b=a1 b 1+ a2 b2 +a3 b3
1)
a1 b1 , a2 b2 ,a 3 b 3
2)
3)
,
are real numbers, a b is a scalar
a b= b a , because a1 b1=b1 a1 , a 2 b2 =b2 a2 , a3 b3=b 3 a 3
a
MHF4U Summative Task
Create a Desmos Roller Coaster meeting the following criteria
D: cfw_t R 0<t< 60 (Where t is the time in seconds)
R : cfw_h R0< h<500
(The current tallest roller coaster is 459 feet!)
Including polynomial functions of the form y