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Whether differential or integral both concepts involve the idea that we can do something infinately many times and get a finite answer that is useful.
 
Calculus summary
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Calculus summary
When x changes by h, the ratio of a change in F(x) to the change in x that caused it, is equal to (F(x+h)−F(x))/h. As h approaches 0, the limit of the ratio becomes a value. A function that maps x to this limit is called the derivative of F(x).

「(F(x+h)− ...
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01 The Greeks, Newton and Leibniz
○ Many students quit math entirely when they get up to this point, mainly out of fear. ○ We will get to some of those in due time as well. ○ What challenge could possibly have necessitated the development of calculus?
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01 The Greeks, Newton and Leibniz
○ Whether differential or integral both concepts involve the idea that we can do something infinately many times and get a finite answer that is useful.
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02 The Slope of a Tangent Line
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03 Rates of Change
Galileo had already discovered some years prior that the distance traveled by a falling object is represented by a function of time. Newton wondered how one could calculate the velocity of the object at any particular instance during the fall.
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04 Limits and Limit Laws in Calculus
Asymptote: a straight line approached by a given curve as one of the variables in the equation of the curve approaches infinity
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05 What is a Derivative?
What is a Derivative? Deriving the Power Rule
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06 Derivatives of Polynomial Functions
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06 Power Rule proof
Definition of differentiation
(x˄n)▽x = 「((x+h)˄−x˄n)/h :h⨠0」

Binomial Expansion
(x+h)˄n
= 「n!/((n−k)!·k!)·x˄(n−k)·h˄k Σk=0,n」

(x+h)˄n − x˄n
= 「n!/((n−k)!·k!)·x˄(n−k)·h˄k Σk=1,n」

((x+h)˄n−x˄n) / h
= 「n!/((n−k)!·k!)·x˄(n−k)·h˄(k−1) Σk=1,n」

「((x+h)˄n−x˄ ...
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06 Product Rule proof
(f(x)·g(x))▽x
= 「(f(x+h)·g(x+h)−f(x)·g(x))/h :h⨠0」
= 「(f(x+h)·g(x+h)−f(x+h)·g(x)+f(x+h)·g(x)−f(x)·g(x))/h :h⨠0」


−f(x+h)·g(x)+f(x+h)·g(x) inserted


= 「(f(x+h)·g(x+h)−f(x+h)·g(x))/h :h⨠0」 + 「(f(x+h)·g(x)−f(x)·g(x))/h :h⨠0」


split up into two limits
┅ ...
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06 Product Rule visualized
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06 Quotient Rule proof
(f(x)/g(x))▽x
= 「(f(x+h)/g(x+h)−f(x)/g(x))/h :h⨠0」
= 「(f(x+h)·g(x)−f(x)·g(x+h))/(g(x+h)·g(x))/h :h⨠0」 ➊
= 「(f(x+h)·g(x)−f(x)·g(x+h))/h/(g(x+h)·g(x)) :h⨠0」
= 「(f(x+h)·g(x)−f(x)·g(x+h))/h :h⨠0」
· 「1/(g(x+h)·g(x)) :h⨠0」
= 「((f(x+h)−f(x))·g(x)−f(x)·(g(x+h)−g( ...
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06 Taylor and Maclaurin Series
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06 Taylor and Maclaurin Series
○ Power Series 「cₙ·x˄n Σn=0,∞」
○ Taylor Series
「f(x)▽ⁿx「a」/n!·(x−a)˄n Σn=0,∞」
○ Maclaurin Series
「f(x)▽ⁿx「0」/n!·x˄n Σn=0,∞」


f(x)▽⁰x
= c₀·(x−a)˄0
+ c₁·(x−a)˄1
+ c₂·(x−a)˄2
+ c₃·(x−a)˄3 + ...
= 「cₙ·(x−a)˄n Σn=0,∞」
f(x)▽⁰x「a」 = c₀ = c₀·0!
c₀ = f(x)▽⁰x「a」/ ...
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07 Derivatives of Trigonometric Functions
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08 Chain Rule
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08 Chain Rule
Derivatives of composite functions
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08 Chain Rule proof

Let g(x+h)−g(x) = t, then 「t :h⨠0」 = 0 and g(x+h) = g(x) + t.
「(f(g(x+h))−f(g(x)))/(g(x+h)−g(x)) :h⨠0」
= 「(f(g(x)+t)−f(g(x)))/t :t⨠0」
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08 Chain Rule visualized
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09 Derivatives of Logarithmic and Exponential Functions
Derivatives of Logarithmic and Exponential Functions
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17 What is Integration?
○ Finding the Area Under a Curve ○ tinker, articulation, solidify
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18 The Fundamental Theorem of Calculus
Redefining Integration
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18 The fundamental theorem of calculus
If f is Riemann integrable on [a,b] and F(x) is the integral of f(t) from a to b then F is continuous on [a,b]. Furthermore, if f is continuous on [a,b] then F is differentiable on [a,b] and F' = f.
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19 Properties of Integrals and Evaluating Definite Integrals
Properties of Integrals and Evaluating Definite Integrals
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20 Evaluating Indefinite Integrals
Evaluating Indefinite Integrals
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21 Evaluating Integrals With Trigonometric Functions
Evaluating Integrals With Trigonometric Functions
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22 Integration Using The Substitution Rule
Integration Using The Substitution Rule
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23 Integration By Parts
Integration By Parts
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24 Integration by Trigonometric Substitution
Integration by Trigonometric Substitution
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25 Advanced Strategy for Integration in Calculus
Advanced Strategy for Integration in Calculus
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26 Evaluating Improper Integrals
Evaluating Improper Integrals
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27 Finding the Area Between Two Curves by Integration
Finding the Area Between Two Curves by Integration
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28 Calculating the Volume of a Solid of Revolution by Integration
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29 Calculating Volume by Cylindrical Shells
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30 The Mean Value Theorem For Integrals: Average Value of a Function
The Mean Value Theorem For Integrals: Average Value of a Function
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Double and Triple Integrals - YouTube
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multivariable
01 domain  
 
Single Variable Calculus
Lec 1 | MIT 18.01 Single Variable Calculus, Fall 2007
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