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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)− ...
3653 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?
5032 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.
9535 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.
5034 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
5029 05 What is a Derivative?
What is a Derivative? Deriving the Power Rule
5051 Proverbs and Quotes in English
Short sentences and easy words.

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
○ 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｣/ ...
9697 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) ...
9693 09 Derivatives of Logarithmic and Exponential Functions
Derivatives of Logarithmic and Exponential Functions
3387 17 What is Integration?
○ Finding the Area Under a Curve ○ tinker, articulation, solidify
5123 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.
5193 19 Properties of Integrals and Evaluating Definite Integrals
Properties of Integrals and Evaluating Definite Integrals
3416 20 Evaluating Indefinite Integrals
Evaluating Indefinite Integrals
3417 21 Evaluating Integrals With Trigonometric Functions
Evaluating Integrals With Trigonometric Functions
3418 22 Integration Using The Substitution Rule
Integration Using The Substitution Rule
3419 23 Integration By Parts
Integration By Parts
3420 24 Integration by Trigonometric Substitution
Integration by Trigonometric Substitution
3421 25 Advanced Strategy for Integration in Calculus
Advanced Strategy for Integration in Calculus
3422 26 Evaluating Improper Integrals
Evaluating Improper Integrals
3423 27 Finding the Area Between Two Curves by Integration
Finding the Area Between Two Curves by Integration
3424 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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