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What is Calculus? Calculus is concerned with change and motion; it
deals with the quantities that approach other quantities.
Sir Isaac Newton invented his version of calculus in order to
explain the motion of planets around the sun. Today, calculus is
used in calculating the orbits of satellites and spacecrafts, in
predicting population sizes, in estimating how fast prices rise,
in forecasting weather, in calculating life insurance premiums,
and in many other areas.
The topics covered are: Differential Calculus, Integral Calculus,
Sequences and Series, Parametric Curves and Polar Coordinates,
Multivariable Calculus, and Differential Equations.
Highlights of Calculus (Part 1)
Big Picture of Calculus Big Picture: Derivatives Max and Min and Second Derivative The Exponential Function Big Picture: Integrals Derivative of sin x and cos x Product Rule and Quotient Rule Chains f(g(x)) and the Chain Rule |
Highlights
of Calculus (Part 2)
Limits and Continuous Functions Inverse Functions f^{-1}(y) & x = ln y Derivatives of ln y & sin^{-1}(y) Growth Rates & Log Graphs Linear Approximation/Newton's Method Power Series/Euler's Great Formula Differential Equations of Motion Differential Equations of Growth |
The study of differential calculus is concerned with how one quantity changes in relation to another quantity. The central concept of differential calculus is the derivative.
Introduction to Calculus | The Limits of a Function Definition and Techniques to find Limits |
Derivatives Definition and Slope of Tangent | Power Rule Constant Multiple Rule, Sum Rule, Difference Rule |
Product Rule Find the derivative of the product of two functions | Quotient Rule Find the derivative of the quotient of two functions |
Chain Rule Find the derivative of composite functions | More Chain Rule Examples |
Derivative Rules Product Rule, Quotient Rule, Chain Rule, Power Rule, Exponential and Logarithmic Functions, Trigonometric Functions, Inverse Trigonometric Functions, Hyperbolic Functions and Inverse Hyperbolic Functions | Examples using the Derivative Rules |
More Examples using the Derivative Rules | Trigonometric Derivatives Derivatives of sin, cos, tan, scs, sec, cot |
More Derivatives Involving Trigonometric Functions | Inverse Trigonometric Derivatives Derivatives of the inverse of sin, cos, tan, scs, sec, cot |
Trigonometric Substitution Videos | Deriving the Derivative Formulas for Tan, Cot, Sec, Cosecant, Arctan |
Derivatives of Exponential Functions Derivatives of e^{x}, a^{x}, a^{g(x)} | Derivative of the Natural Log Derivatives of ln(x), ln[g(x)] |
Examples of Logarithmic Differentiation | Implicit Differentiation Find the derivatives of non-functions |
Examples using Implicit Differentiation | Second Derivative and Higher Derivatives Find the derivatives of derivatives and their applications |
Maxima and Minima Maximum amd Minimum Values, Fermat's Theorem, Critical Number, Extreme Value Theorem, Closed Interval Method | Maximum and Minimum Videos |
Finding Critical Numbers | Derivative Test First Derivative Test, Second Derivative Test, Minima, Maxima, Increasing/Decreasing Test |
Concavity Problems I Analyze functions through increasing, decreasing, concavity, inflection points, critical points, and extremum | Concavity Problems II Analyze functions through increasing, decreasing, concavity, inflection points, critical points, and extremum |
Curve Sketching using Calculus | Mean Value Theorem What is Mean Value Theorem? |
Asymptotes Vertical Asymptote, Horizontal Asymptote, Oblique Asymptote | Hyperbolic Functions Hyperbolic Identities, Derivatives of Hyperbolic Functions and Derivatives of Inverse Hyperbolic Functions |
L'Hopitals Rule Indeterminate Quotients, Product, Differences, and Power | Optimization Problems using Derivatives |
More Optimization Problems using Derivatives | Related Rates using Derivatives |
More Related Rates using Derivatives | Newton's Method Used to approximate a root |
The following are solutions to sample questions of the CollegeBoard AP/AB and AP/BC Calculus examination
Parametric Curves | Calculus with Parametric Curves |
Polar Coordinates | Calculus and Area in Polar Coordinates |
Graphing Polar Curves |
Vector Functions | Derivatives of Vector Functions |
Arc Length of Vector Functions |
Big Picture of Calculus
Calculus is about change. One function tells how quickly another
function is changing. Professor Strang shows how calculus applies
to ordinary life situations, such as: * driving a car * climbing a
mountain * growing to full adult height
Big Picture: Derivatives
Calculus finds the relationship between the distance traveled and
the speed - easy for constant speed, not so easy for changing
speed. Professor Strang is finding the "rate of change" and the
"slope of a curve" and the "derivative of a function."
Big Picture: Integrals
The second half of calculus looks for the distance traveled even
when the speed is changing. Finding this "integral" is the
opposite of finding the derivative. Professor Strang explains how
the integral adds up little pieces to recover the total distance.
I know the speed at each moment of my trip, so how far did I go?