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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
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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Proverbs and Quotes in English
Short sentences and easy words.
 
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 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
Definition of differentiation f(g(x))▽x = 「(f(g(x+h))−f(g(x)))/h :h⨠0」 = 「(f(g(x+h))−f(g(x)))/(g(x+h)−g(x))·(g(x+h)−g(x))/h :h⨠0」 = 「(f(g(x+h))−f(g(x)))/(g(x+h)−g(x)) :h⨠0」·「(g(x+h)−g(x))/h :h⨠0」 ᐥ Let g(x+h)−g(x) = t, then 「t :h⨠0」 = 0 and g(x+h) = g(x) ...
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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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