MTH 151 Midterm: MTH 151 - Term Test 1

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Published on 15 Sep 2018
Department
Course
Professor
Miami University
MTH 151
Calculus I
Winter 2018
Term Test 1
Exam Guide
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N&?&%()*#+)G-2&Z*%&Z-()
Chapter(2(
F4 3)V0-)V*#D-#%)P2'T.-?(
The'slope'of'the'line'passing'through'the'point'A'('x!, y! ) and B x!, y! is ∢'
m!" = y!βˆ’y!
x!βˆ’x!
'
How'to'find'the'tangent'line'at'a'point?'
Example:))
!&#+)%0-)%*#D-#%).&#-)*%)1'&#%)<3C)3=)'#):"#$%&'#):<;=)A)𝐱𝟐4)
Solution:'
y'='f(x)'='x!,all the points on the function can be express as x,x!.'
The'slope'of'the'line'passing'through'(1,'1)'and' x,x! is = !! !!
! !!'='x+1'
The'tangent'line'slope'='lim!β†’!x+1=2'
Then'the'tangent'line'function'is'y'='2x'+'b.''
Since'(1,'1)'is'on'the'tangent'line,'then'we'have:'
1'='2x1'+b'
b'='-1'
So'the'tangent'line'at'point'(1,'1)'on'function'f(x)'='x! is y=2x βˆ’1.'
F4F)N&?&%()':)*)!"#$%&'#)
34 G-:&#&%&'#I
a. lim!β†’!fx=L is read β€œthe limit of fx,as x approaches a is L”'
b. >#%"&%&Z-).&?&%'means'as'x'gets'very'close'to'the'number'a'('from'both'sides'),'f(x)'will'get'very
close'to'the'number'L.
@'%-I)
lim!β†’!fx'may'exist,'even'f(a)'is'undefined.'If'
lim!β†’!fx and faboth exist,and lim!β†’!fx=
fa,then the function fxis continous at a.'
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Example:))
>:)6)A):<;=)A)π±πŸ‘C);)β‰ πŸ.𝐅𝐒𝐧𝐝 π₯π’π¦π±β†’πŸπŸπ±4)
Solution:''
lim!β†’!fx'='lim!β†’!x!'='8'
However,'since'x'β‰ 2,f2is undefined.'
F4 _#-`Y&+-)N&?&%(
Left-hand'side'limit:''lim!β†’!!fx=L'
Right-hand'side'limit:''lim!β†’!!fx=L'
lim
!β†’!
fx=L if and only if lim
!β†’!!
fx= lim
!β†’!!
fx=L'
Past'Exam'Question:'
!&#+)π₯π’π¦π±β†’πŸ‘
πŸ‘!𝐱
πŸ‘!𝐱4)
Solution:'
If x ≀3,then lim
!β†’!
!!!
!!!='lim!β†’!!
!!!
!!!
='lim!β†’!!
!!!
!!!
=1'
If x>3,then lim
!β†’!
!!!
!!!='lim!β†’!!
!!!
!!!
='lim!β†’!!
!!!
!!!
=βˆ’1'
lim
!β†’!!
3βˆ’x
3βˆ’x β‰  lim
!β†’!!
xβˆ’3
3βˆ’x'
lim
!β†’!
3βˆ’x
3βˆ’x does not exist.'
J4 >#:&#&%-)N&?&%(
We'write'lim!β†’!fx= ±∞'if'the'value'of'f(x)'can'be'made'arbitrarily'large,'by'taking'value'of'
x'that'sufficiently'close'to'a'but'not'equal.'
>?1'2%*#%)@'%-I)
If'lim!β†’!!fx=+ ∞,'lim!β†’!!fx= βˆ’ ∞,'then'lim!β†’!fx does not exist.'
If'lim!β†’!!fx=+ ∞,'lim!β†’!!fx= + ∞,'then'lim!β†’!fx= + ∞.'
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