3.2
Use Parallel Lines and
Transversals
Goal
Your Notes
p Use angles formed by parallel lines and
transversals.
POSTULATE 15 CORRESPONDING ANGLES
POSTULATE
t
Example 1
p
2
If two parallel lines are cut
by a transversal, then the pairs
of corresponding ang
Evaluating limits when x .
x
= 1/2.
+11
x+2
= 1.
x x 2
6. Show lim
3x2 + 2x 5
= 3/5.
x 5x2 + 3x + 1
7. Show lim 2x = 0.
1. Show lim
2. Show lim
x2 7x + 11
= 1/3.
x
3x2 + 10
3. Show lim
2x3 5x + 7
= 2/7.
x 7x3 + 2x2 6
x
4x2
x
8. Show lim
t+1
= 0.
+1
9. Sh
Evaluating limits when x a.
1. Show lim (6x2 4x + 3) = 5.
x1
x2 49
= 14.
x7 x 7
2. Show lim
x2 6x + 8
= 2.
x2
x2
x5/2 a5/2
= (5/2)a3/2 .
xa
xa
11. Show lim
(x + 2)5/3 (a + 2)5/3
= (5/3)(a + 2)2/3 .
xa
xa
12. Show lim
3. Show lim
x3 64
= 6.
x4 x2 16
13. Sh
APCalculusAB
LimitsReview(Sections1.21.5,2.1A,&3.5)
Name_
1.Usethegraphoffattherighttoevaluatethefollowing:
lim f (x)
a)
c)
x 1
lim f ( x)
d)
x 6
lim f ( x)
f)
x 6
e)
lim f ( x )
b)
x 0
x 5
h)
lim f ( x)
lim f ( x)
x 5
lim f (x)
g)
f(5)
i)
Atwhatxvalue(
The Conjugate Method/Trig Limits
Recall from Precalc that the conjugate of a+bi is_
The conjugate of x 2 is: _
The conjugate of x 7 is: _
The conjugate of 5 y is: _
(what happens if you multiply these two together?)
How can the conjugate help us?
Try to e
Lesson 3 : Solving for Limits by the conjugate
You need to multiply by the conjugate whenever you see a
f ( x)
For example:
If you see
If you see
3x 5 , we need to multiply this by its conjugate,
3x 5
3x 5 7 , we need to multiply this by its conjugate,
3x
lim f ( x) L
Read: The limit as x approaches c of f
of x is equal to L
x c
Direct Substitution
Factor and reduce
Rationalization Technique
The Squeeze Theorem
sin x
1 cos x
lim
1 , lim
0
x 0
x
0
x
x
Note: c is an x value, and L is a y
value.
Create a tab
Difference Quotients
1. Find the difference quotient of the function, f, given by f (x ) .5x 2 if h 2 , i.e. find
f (x 2) f (x )
.
2
H (-6,8)
D (6,18)
18
16
14
12
10
I (-4,8)
C (4,8)
8
6
4
J (-2,2)
2
B (2,2)
2. Now evaluate this difference quotient at x 2
Difference Quotient Problems
1. f(x) = 2x3 + x
[2(x h) 3 (x h)] - [2x 3 x]
[2x 3 6 x 2 h 6 xh 2 2h 3 x h] - [2x 3 x]
=
h
h
2
2
2
2
3
h(6 x 6 xh 2h 1)
6 x h 6 xh 2h h
=
= 6 x 2 6 xh 2h 2 1
h
h
DQ =
2. f(x) =
x
x 1
xh
x
x h x 1 x x h 1
DQ = x h 1 x 1 = x h
Calculus HW from 2.4
Compute and simplify the difference quotient (slope of the tangent line)
lim f ( x h) f ( x)
h0
h
1)
2)
3)
4)
f ( x) 6 x 9
f ( x ) 5 x 2 3x 3
1
f ( x)
3x
f ( x) 4 x 3 6
y 7x 8
3x
6) y
4 2x
7) Find the derivative for the f(x) and the
1.3 Average Rate of Change
Name _
Precalculus H
If (x 1 , y 1) and ( x 2 , y 2) are two ordered pairs of a function, then the average rate of change of the function
y2 y1
as x varies from x 1 to x 2 is defined to be
.
x2 x1
Example 1:
Find the average rat
Average Rate of Change and the Difference Quotient
-Two Ways to show the same thing
1. If f(10) =26 and f(14) = 36,
find the average rate of change of f
from x=10 to x=14. (show all steps)
= = 2.5
10 What is x? 4_
What is y? _
2. Now find the average rate
Name _
Date _
Chapter 1.3
Evaluating Limits Analytically
Let b and c be real numbers and let n be a positive integer.
b b
x c
x n c n
1. lim
2. lim
3. lim
x c
x c
x c
The above functions are nice, smooth, and what we call continuous at
c. Because of their
3aeafafa096cd0855859df80894fdbd968ad03bc.doc
What type of discontinuity is at
f ( x)
x 5
in f(x) below?
x 2 16
x 2 x 20
A) Asymptotic (infinite) Discontinuity
B) Removable (point) Discontinuity
C) Jump Discontinuity
D) The function is Continuous
What typ
3.6
Prove Theorems About
Perpendicular Lines
Goal
Your Notes
p Find the distance between a point and a line.
VOCABULARY
Distance from a point to a line
THEOREM 3.8
If two lines intersect to form a linear
pair of congruent angles, then the lines
are
.
If 1
3.5
Write and Graph Equations
of Lines
p Find equations of lines.
Goal
Your Notes
VOCABULARY
Slope-intercept form
Standard form
Write an equation of a line from a graph
Example 1
Write an equation of the line in slope-intercept form.
y
(0, 3)
1
x
1
(2, 21
3.4
Find and Use Slopes of Lines
Goal
Your Notes
p Find and compare slopes of lines.
VOCABULARY
Slope
SLOPE OF LINES IN THE COORDINATE PLANE
Negative slope:
as in line j
from left to right,
Positive slope:
as in line k
from left to right,
rise
m5
run
y 2
Evaluating limits when x 0.
1. Show lim (x2 2)2 + 6 = 10.
10. Show lim
x0
2. Show lim
x0
x0
5x
= 5.
x
p
2x
= 2 (a).
a+x ax
11. Show lim
1+x1
= 1/2.
x
12. Show lim
x
= 2.
1+x1
x0
17x
= 17/2.
x0 2x
3. Show lim
4. Show lim
x0
5. Show lim
x0
x0
317x
= 317/4