# MTH 151 Midterm: MTH 151 - Term Test 1

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Published on 15 Sep 2018
School
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 βΆ'
m!" = y!βy!
x!βx!
'
How'to'find'the'tangent'line'at'a'point?'
Example:))
!&#+)%0-)%*#D-#%).&#-)*%)1'&#%)<3C)3=)'#):"#\$%&'#):<;=)A)π±π4)
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 β1.'
F4F)N&?&%()':)*)!"#\$%&'#)
34 G-:&#&%&'#I
a. lim!β!fx=L is read βthe limit of fx,as x approaches a is Lβ'
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C);)β π.ππ’π§π π₯π’π¦π±βπππ±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
π!π±
π!π±4)
Solution:'
If x β€3,then lim
!β!
!!!
!!!='lim!β!!
!!!
!!!
='lim!β!!
!!!
!!!
=1'
If x>3,then lim
!β!
!!!
!!!='lim!β!!
!!!
!!!
='lim!β!!
!!!
!!!
=β1'
lim
!β!!
3βx
3βx β  lim
!β!!
xβ3
3βx'
lim
!β!
3βx
3βx does not exist.'
J4 >#:&#&%-)N&?&%(
x'that'sufficiently'close'to'a'but'not'equal.'
>?1'2%*#%)@'%-I)
If'lim!β!!fx=+ β,'lim!β!!fx= β β,'then'lim!β!fx does not exist.'
If'lim!β!!fx=+ β,'lim!β!!fx= + β,'then'lim!β!fx= + β.'
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