EngliSea > M > math > 60 Calculus

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
9444 SHARED
 
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 SHARED
 
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 SHARED
 
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 SHARED
 
02 The Slope of a Tangent Line
5033 SHARED
 
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 SHARED
 
Pillow English Listening
 
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 SHARED
 
05 What is a Derivative?
What is a Derivative? Deriving the Power Rule
5051 SHARED
 
06 Derivatives of Polynomial Functions
5052 SHARED
 
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˄ ...
9687 SHARED
 
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 ┅ ...
9689 SHARED
 
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( ...
9692 SHARED
 
06 Taylor and Maclaurin Series
5104 SHARED
 
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 SHARED
 
07 Derivatives of Trigonometric Functions
5053 SHARED
 
08 Chain Rule
5054 SHARED
 
08 Chain Rule
Derivatives of composite functions
9695 SHARED
 
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 SHARED
 
09 Derivatives of Logarithmic and Exponential Functions
Derivatives of Logarithmic and Exponential Functions
3387 SHARED
 
17 What is Integration?
○ Finding the Area Under a Curve ○ tinker, articulation, solidify
5123 SHARED
 
18 The Fundamental Theorem of Calculus
Redefining Integration
3415 SHARED
 
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 SHARED
 
19 Properties of Integrals and Evaluating Definite Integrals
Properties of Integrals and Evaluating Definite Integrals
3416 SHARED
 
20 Evaluating Indefinite Integrals
Evaluating Indefinite Integrals
3417 SHARED
 
21 Evaluating Integrals With Trigonometric Functions
Evaluating Integrals With Trigonometric Functions
3418 SHARED
 
22 Integration Using The Substitution Rule
Integration Using The Substitution Rule
3419 SHARED
 
23 Integration By Parts
Integration By Parts
3420 SHARED
 
24 Integration by Trigonometric Substitution
Integration by Trigonometric Substitution
3421 SHARED
 
25 Advanced Strategy for Integration in Calculus
Advanced Strategy for Integration in Calculus
3422 SHARED
 
26 Evaluating Improper Integrals
Evaluating Improper Integrals
3423 SHARED
 
27 Finding the Area Between Two Curves by Integration
Finding the Area Between Two Curves by Integration
3424 SHARED
 
28 Calculating the Volume of a Solid of Revolution by Integration
3425 SHARED
 
29 Calculating Volume by Cylindrical Shells
3426 SHARED
 
30 The Mean Value Theorem For Integrals: Average Value of a Function
The Mean Value Theorem For Integrals: Average Value of a Function
3427 SHARED
 
Double and Triple Integrals - YouTube
6899 SHARED
 
How to read
How to read calculus termsIn-line Notationread out loudsay out loudverbally

-