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Class notes Calculus I

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38 Pages of Notes taken in my spare time following Professor Leonard’s calculus 1 course on YouTube. Formally defining the Fundamental Theorem of Calculus. Beginning with an introduction to Limits, followed by definition of a derivative. Continues with Chain Rule, Product Rule, Quotient Rule, Pow...

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  • December 28, 2023
  • 38
  • 2022/2023
  • Class notes
  • Professor leonard
  • All classes
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isaacjc08
CalculusI -
Intro to Limits


Goals of Calculus Ex : f(x) = x2


(x xy) slope

thearea
· (x y) m =




# findi S
T +
point
,
,
of ,
>
-
function
a
& As Q + P

P(1 1)
Mse
· >
Mean
-
,




Eg for tan : & X & I -> undefined

mcn(Xc )
> Site Y y = - X
, Q + P
points two
between
(
Man =
X -
I



X
I -
1

Tangent Problem
↳ becaus X already cannot and will not equl 1.




#decant
- X+ )
ms =




live What ?
happens to
msen as X-1
↳ More
Q mely close to P (but not 4) ↳
Msec-> 2
the secant live becomes (essentially) a




tangent line Because More-> 2
(slope of sea live approaches 2)
...
BUT ↳
Mean = 2 (slope of tan lime actually is .
2)
5For a line , you
need two points (P + Q) 1 = 2(x D -




y
-




↳ As Q approaches P ,
the secant
gets closer to the
y -
1 = 2x -
2

F2x- 11
>

tampant of P
Ein - >




& If Q -P but Q & P
,
the secant will be identical
t the tangent A limit must approach the same value from
↳ THIS A
IS "LIMIT" either side.


I General
limf(x) = L



*I
Spill in 4973 4 .




Sided Limit
Right
↳im f(x) ,
->

/im f(x) ,




↳im f(x) =
(imf(x)

No
For a limit to exist at a

↳ limf(x) =
limf(x)
ab
X> -
2 x -

, 2




Computing Limits
Lim (x-2x
Basics Lim
: c = c (constant

[imx] (in2) (in ** ·


Sim
Lim x
-a



23 -
2 .

207

*
Els ↓im (x2xx) =
= o
-




-


D


Xin = D

fla
↳im f(x)
=




Properties :
Lim f(x) =L
lim X- 4

#(x
2x2
(X -
2
2
2)
+




Xim g(x) L This isskay
x + 2 -


X
=




because crib




i
we

X bo = 2

1
m [f(x) g(x)] = imf(x) my
+
allowing
anyway




2 .
xina(f(x) g(x)] Ximf(x) /lim g(x)
· = ·
-

Lim

3
.
lim( x , Lim g(x lim
x+
3
x2 - 3x-10
x 25 -
- Limitdis

4 .

/ima [f(x)]" =

[limf(x)]" Note
↳ If 8 factor simplify
↳ Lim
and

= f(x) ↳ If you
,



can't simplify the Sinlysis
& I It



:
problem away
. .. It isn't a -
2

hole .




Evaluate
min
↳ sign analysis
>
-




N
Xim 1 1 2
= + =
+




lim M I ·
/
=
16X -




-

X- y X




lim
+ o ) = i

,
,

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