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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.

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)− ...

｢(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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○ 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

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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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○ 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

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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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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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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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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˄ ...

(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˄ ...

(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

┅ ...

= ｢(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

┅ ...

(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( ...

= ｢(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( ...

○ 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｣/ ...

○ 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｣/ ...

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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09 Derivatives of Logarithmic and Exponential Functions

Derivatives of Logarithmic and Exponential Functions

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Derivatives of Logarithmic and Exponential Functions

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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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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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Properties of Integrals and Evaluating Definite Integrals

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21 Evaluating Integrals With Trigonometric Functions

Evaluating Integrals With Trigonometric Functions

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Evaluating Integrals With Trigonometric Functions

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25 Advanced Strategy for Integration in Calculus

Advanced Strategy for Integration in Calculus

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Advanced Strategy for Integration in Calculus

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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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Finding the Area Between Two Curves by Integration

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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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The Mean Value Theorem For Integrals: Average Value of a Function

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