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Calculus Summary
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Calculus Summary
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01 Essence of calculus
Essence of calculus, chapter 1
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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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01 What Is a Derivative?
What Is a Derivative?
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01 What is Calculus?
In calculus you start with two big questions about functions. First how steep is a function at a point? Second what is the area underneath the graph over some region?
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02 The paradox of the derivative
The paradox of the derivative | Essence of calculus, chapter 2
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02 The Slope of a Tangent Line
Understanding Differentiation Part 1: The Slope of a Tangent Line
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02 The Tangent Line and the Derivative
A tangent line to a point A is the limit of the secant lines as P approaches A.
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03 Derivative formulas through geometry
Derivative formulas through geometry | Essence of calculus, chapter 3
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03 Product Rule for Derivatives
Product Rule for Derivatives (Calculus)
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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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04 The Quotient Rule
The Quotient Rule (Calculus)
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04 Visualizing the chain rule and product rule
Visualizing the chain rule and product rule | Essence of calculus, chapter 4
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05 Derivatives of exponentials
Derivatives of exponentials | Essence of calculus, chapter 5
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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
Power Rule, Product Rule, and Quotient Rule
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06 Implicit differentiation, what's going on here?
Implicit differentiation, what's going on here? | Essence of calculus, chapter 6
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07 Derivatives of Trigonometric Functions
Derivatives of Trigonometric Functions
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07 Limits
Limits | Essence of calculus, chapter 7
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08 Derivatives of Composite Functions: The Chain Rule
Derivatives of Composite Functions: The Chain Rule
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08 Integration and the fundamental theorem of calculus
The integral equals the antiderivative evaluated at the top bound, minus its value at the bottom bound. This fact is called 'the fundamental theorem of calculus'.
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09 Derivatives of Logarithmic and Exponential Functions
Derivatives of Logarithmic and Exponential Functions
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09 What does area have to do with slope?
What does area have to do with slope? | Essence of calculus, chapter 9
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10 Higher order derivatives
Higher order derivatives | Essence of calculus, chapter 10
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11 Taylor series
Taylor series | Essence of calculus, chapter 11
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17 What is Integration?
Finding the Area Under a Curve
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18 The Fundamental Theorem of Calculus
Redefining Integration
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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
Calculating the Volume of a Solid of Revolution by Integration
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29 Calculating Volume by Cylindrical Shells
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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Taylor and Maclaurin Series
Taylor and Maclaurin Series
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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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What Is a Derivative?
What Is a Derivative?
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