Math 1431
LAB session 1
Daria Kurzanova
[email protected]
•
We have Labs TTh 7-8.30 pm
•
I have CASA hours MW 1-3 pm
•
You are supposed to attend every Lab
•
During the Lab log in to your CASA account and find
Attendance Lab # in EMCF tab (have it open during the Lab)
•
If you cannot attend
the Lab you should watch the video
posted on the website and complete
Alternate Attendance Lab #
•
Alternate Attendance Lab is due either Thursday 6 pm or
Sunday 6 pm
1
Daria Kurzanova
[email protected]
[email protected]
Review
1
Review
equations
Quadratic
equations
1 Quadratic
Review
Quadratic
equations
1 Consider
Review
quadraticequation
equation x2 + bx
+c=0
Consider
quadratic
Quadratic
equations
Vista’s theorem
Quadratic equations
Consider quadratic equation x2 + bx + c = 0
Vieta’s
x1 + x2 quadratic
= b
Consider
equation x2 + bx Vista’s
+ c = 0theorem
theorem
x1 x2 =
c and then
we can
x2 + bxx + c = (x
Consider
quadratic
equation
x2 factor
+ bx +quadratic
Vista’s
theorem
xc1 =
+ 0x2 = equation
b
Vista’s
x1x+1 )(x
x2theorem
= x2b)
x1 x2 = c
x1x+
1 xx
2 2==c b
and then we can factor quadratic equation x2 + bxx + c = (x
x1and
x2and
=then
cthenwe
wecan
can factor
factor quadratic
quadraticequation
equation x2 + bx + c = (x x1 )(x x2 )
Daria
Kurzanova
and then we can factor quadratic equation x2 + bx + c = (x x1 )(x x2 )
How to write a thesis
ExampleMarch,2015
1:
x2
5x + 6 = 0
8
2
>
<Ax + B if x < 2
f (x) = 3.5x
if 2  x
>
:
Bx + 3A if x > 6
How6 to write a thesis
f (x) = 3x2
Factorization
2
2
a
b = (a
x2 + bx + c = (x
8x
10
Daria Kurzanova
b)(a + b)
)(x
March,2015
1
)
1
1
1
8
2
>
Ax
+ B if x < 2
<
f (x) = 3.5x
if 2  x
>
:
Bx + 3A if x > 6
Formula to remember
2
f (x) = 3x
a2
1
b2 = (a
8x
10
b)(a + b)
6
f (x) = 3x2
a2
b2 = (a
8x
b)(a + b)
x2 + bx + c = (x
Example 2:
lim (3x
x!5
Question #
1
10
Quiz 1 types problems
)(x
7)
)
+6=0
ew
Example 3:
x
lim
x!4 20
4
5x
equations
dratic equation x2 + bx + c = 0
rem
can factor quadratic equation x2 + bx + c = (x
6=0
1
x
lim
x!4 20
Example 4:
lim
x!3
x
x2
1
4
5x
10
5x + 6
x1 )(x
x2 )
Example 5:
lim
x!3
2x2
x2
18
6x + 9
1
Example 6: Evaluate the limit at x=10
8
>
>
3x
>
<
f (x) = 50
>
>
>
:6x
if x < 10
if x = 10
30 if x > 10
a
2
2
b = (a
b)(a + b)
+ bx + c = (x
Question
lim (3x #
x!5
)(x
f (x) =
)
7)
x 2
x!2 4x
8
lim
>
:
3.5x
if 2  x
Bx + 3A if x > 6
f (x) = 3x2
a2
b2 = (a
8x
lim (3x
10
b)(a + b)
x2 + bx + c = (x
x!5
6
)(x
)
7)
x 2
x!2 4x
8
lim
A)
1
4
B) doesn’t exist
C)
1
8
D) 0
1
1
E) 2
F)
3
4
>
>
3x
if x < 10
>
<
) = 50
if x = 10
>
>
>
:6x
30 7: if x > 10
Example
x
x!4 |x
4
4|
lim
8
>
>
3x
>
<
= 50
>
>
>
:6x
lim+
x!4
8
>
>
3x
>
<
= 50
>
>
>
:6x
lim
x!4
if x < 10
if x = 10
30 if x > 10
x
|x
4
4|
if x < 10
if x = 10
30 if x > 10
x
|x
2
4
4|
Quiz 2 types problems
C)
8
D) 0
E) 2
A)
| < Example
" whenever
8: " |x
1
4
lim f (x) = L
x!c
F)
x 2
3
lim
B) doesn’t exist C)
D) 0 E)
2
F)
x!2 4x
8 4
8
c| definition
<
3
1
3
of
a limit
2 B) 1
A)
C)
D)
0 F) doesn’t exist "
>
50
16
Ax + B
if 1x80< 2 1 10 E)
<
1
1
exist B) 0 C) 18 D) 4 E) 16
F) 15
f (x) =8" >3.5x
if 2  x 6
> 0 9 s.t. |f (x) L| < " whenever |x c| <
4
1
8
:
Bx + 3A if x > 6
lim f (x) = L
x!c
2
f (x)Find
= 3x
8x 10
the largest
that works for " = 0.7
a2
b2 = (a
b)(a + b)
x2 + bx + c = (x
lim (3x
x!5
)(x
7)
)
8
>
x 2
>
3x
if x < 10
>
lim
<
x!2 4x
8
x) = 50 1
if x = 10
1
1
>
A)>
B)
doesn’t
exist
C)
D) 0 E) 2 F)
>
4
8
:6x 30 if x > 10
"
1
8" > 09 8s.t. |f (x) L| < " whenever |x c| <
x >4
lim
>
3x
if x < 10
>
<
x!4 |x
4|
lim f (x) = L
f (x) =
50
if x = 10 x!c
>
works for ✏ = >
0.7
>
Find the :
largest
works
6x 30 that
if x >
10 for " = 0.7
x
lim 7x = 21
lim
x!3
rgest
x!4
|x
4
4|
that works for ✏ = 0.7
lim 7x = 21
x!3
Example 9:
6
lim
x!36 5(36
p
x
x)
1
x6 7x5
lim
x!1 12x4 + 73
p
4
x
lim
x!16 10(16
x)
p
x 2 4
lim
x!18
x 18
3
4
1
lim
x!36
5(36
lim
x)
x!1
b)
w to writeb)(aa+thesis
x
10
a2
b2 = (a
4
lim
)(x
x!1 x2)+ 6x
10:
+ bx Example
+ c = (x
vs.
x2 + 6x
x3 + 15
lim
x!1 x6 + 76x
Daria
Kurzanova
lim (3x
7)
x3 + 15
lim
x!5
x!1
x6 + 76x
How to write a thesis
x 2
March,2015
lim
x!2
)0
4x
8
Daria Kurzanova
E) doesn’t exist
F) 73
March,2015
8
2
L| <>
whenever
+ B |x
if x c|
<<
2
<"Ax
= 3.5x
if 2  x 8
6Ax2 + B if x < 2
>
<
>
lim f (x) = L
: x!c
if 2  x
Bx + 3A if x f>(x)6 = >3.5x
:
Bx + 3A if x > 6
works for " = 0.7
f (x) = 3x62 8x5 10 f (x) = 3x2 8x 10
x
7x
2
2
lim
a2 b2 = (a b)(a + b)
a x!1
bQuestion
=12x
(a#4 +b)(a
+
b)
73
x2 + bx + c = (x
)(x
p
+ bx + c =4 (x x )(x
) lim (3x 7)
lim
x!5
x!16 10(16
x)
x 2
lim (3x 7)
lim
2
x!5
A)
1
8
8
4x 8
3
D) 10
E) 0 F) doesn’t exist "
L| < " whenever |x c| <
E) 2
F)
3
50
B)
x 161 2 C) 801
lim
8" > 09 s.t. |f (x)
4x
1
D) 0
Find the largest
L| < " whenever |x
c| <
lim f (x) = L
Question
works
for "# = 0.7
A) doesn’t
x6 exist7x5B) 0
lim
3 lim f (x) = L
4 x!c
that works for " = 0.7
x!c
x!1
)2
x!2
x!2
C)
6
x6 7x5
lim
x!1 12x4 + 73
p
4
x
lim
x!16 10(16
x)
p
x 2 4
lim
x!16
x 18
1
1
C) 18
D) 14 E) 16
F)
1
15
12x4 + 73
A) doesn’t exist
1
B) 0
C)
1
18
1
D) 1
E)
1
16
F)
1
15
Daria Kurzanova
Quiz 4 types problems
March,2015
8
>
>
3x
if x < 10
>
Example
11: Find A and B such that f(x) is continuous at both points 2 and 6
<
f (x) = 50 8
if x = 10
>
>
2
>
>
+ B if x < 2
:6x
<Ax
Daria
Kurzanova
30 if x > 10
f (x) = 3.5x
if 2  x 6
>
:x 4
March,2015
lim Bx + 3A if x > 6
x!4 |x
4|
ow to write a thesis
that works for ✏ = 0.7
8
2
>
B= 21
if x < 2
lim+7x
<Axx!3
x) = 3.5x
if 2  x
>
p
:
6
x
Bx
lim+ 3A if x > 6
x!36
5(36
f (x) = 3x
a2
2
6
x)
8x
10
x4 10
lim
b2 x!1
= (ax2 +b)(a
6x + b)
x2 + bx + c = x(x
)(x
3
+ 15
)
lim 6 12: Define function f(x) at 10, such that f(x) becomes continuous at 10
Example
x!1
x + 76x
lim
(3x
7)
x!5 p
x 1 3
f (x) = x 2
lim x 10
4x
E) 0
8
3
D) 10
F) doesn’t exist "
1
1
1
D) 4 E) 16 F) 15
< " whenever |x c| <
x!2
A) doesn’t
lim f (x) = L
1
x!c
at works for " = 0.7
x6 7x5
lim
x!1 12x4 + 73
p
4
x
lim
Question
#
x!16
10(16
x)
p 2
x 2 4
lim
x!18
x 18
1
A) doesn’t exist
B) 0
C)
1
18
D)
1
4
E)
1
8
F)
1
15