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Properties
of continuous functions |
|
Continuous function definition |
|
Intermediate value theorem - Bolzano's theorem |
|
Existence of
roots |
|
Extreme value
theorem |
|
Monotone function |
|
Differential
Calculus
- Derivatives and differentials |
|
The
derivative of a function |
|
Definition of the derivative of a function |
|
Tangent to
a curve |
|
The
equation of the line tangent to the given curve at the given point |
|
Determining the derivative of a function as the limit of the
difference quotient |
|
The
equation of the line tangent to the given curve at the given point,
example |
|
Determining
the lines tangent to the graph of a function from a point outside the
function |
| Determining
the lines tangent to the graph of a function from a point outside the
function, example |
|
The
derivative as a function |
| The
derivative function |
| Differentiation,
determining (or deriving) derivative of a function |
|
Derivatives
of basic or elementary functions |
| Determining
the derivative of a function as the limit of the difference
quotient |
|
The
derivative of the power function
|
|
The
derivative of the linear function
|
|
The
derivative of a constant
|
|
The
derivative of the sine function |
|
The
derivative of the cosine function |
|
The
derivative of the exponential function |
|
The
derivative of the logarithmic function |
|
Differentiation
rules |
|
The
derivative of the sum or difference of two functions |
|
The
product rule |
|
A
constant times a function rule |
|
Derivatives of functions, examples |
|
The
quotient rule |
|
The
derivative of the tangent function, use of the quotient rule |
|
The
derivative of the cotangent function, use of the quotient rule |
|
The chain rule |
|
Differentiation using the chain rule, examples |
|
Table
of derivatives of elementary functions |
|
Differentiation
rules |
|
Table of derivatives |
|
The chain rule
applications |
|
Implicit differentiation |
|
Implicit differentiation examples |
|
Generalized power rule |
|
Generalized power rule examples |
|
Logarithmic differentiation |
|
Logarithmic differentiation examples |
|
Derivative of a composite exponential function |
|
Use of the logarithmic differentiation |
|
Derivatives of composite functions examples |
|
Derivatives of the hyperbolic functions |
|
Derivatives of inverse hyperbolic functions |
|
Derivative of
the inverse function |
|
Derivatives of the inverse trigonometric functions |
|
Derivative of
parametric functions, parametric derivatives |
|
Derivative of
parametric functions example
|
|
Derivative
of vector-valued functions |
|
Applications of the derivative |
|
Tangent,
normal, subtangent and subnormal |
|
Property of the parabola |
|
Property
of power functions |
|
Property
of the exponential function |
|
Angle
between two curves |
|
Angle
between two curves, examples |
|
Differential
of a function |
|
Use
of differential to approximate the value of a function |
|
Rules
for differentials |
|
Differentials
of some basic functions |
|
Higher order derivatives and higher order differentials |
|
Higher order derivatives |
|
Higher order derivatives examples |
|
Higher derivative formula for the product - Leibniz formula |
|
Higher derivatives of composite functions |
|
Higher derivatives of composite functions examples |
|
Higher derivatives of implicit functions |
|
Higher derivatives of implicit functions examples |
|
Higher derivatives of parametric functions |
|
Higher derivatives of parametric functions examples |
|
Higher order differentials |
|
Higher order differentials examples |
|
Applications
of differentiation - the graph of a function and its derivative |
|
Definition
of increasing and decreasing |
|
Increasing/decreasing
test |
|
Rolle's theorem
|
|
The mean value theorem |
|
Generalization
of the mean value theorem
|
|
Cauchy's mean value theorem or generalized mean value theorem |
|
L'Hospital's
rule - limits of indeterminate forms |
|
Applications
of L'Hospital's
rule - evaluation of limits of indeterminate forms, examples |
|
a)
Applications
of differentiation - the graph of a function and its derivatives |
|
Generalization
of the mean value theorem, concavity of the graph of a function
|
|
Concavity of the graph of a
function |
|
Concave
up and concave down definition |
|
Points
of inflection |
|
Points
of inflection and concavity of the sine function |
|
Points
of inflection and concavity of the cubic polynomial |
|
Approximate
solution to an equation, Newton's method (or the Newton-Raphson method) |
|
Use of
Newton's method, example |
|
b)
Applications
of differentiation - the graph of a function and its derivatives |
|
Taylor's
theorem (Taylor's
formula) - The extended mean value theorem |
|
The proof of Thaylor's theorem |
|
Maclaurin's
formula or Maclaurin's theorem |
|
Representing
polynomial using Maclaurin's and Taylor's formula |
|
Representing
polynomial using Maclaurin's and Taylor's formula examples |
|
The approximation of the
exponential function by polynomial using Taylor's or Maclaurin's formula |
|
Properties
of the power series expansion of the exponential function |
|
The
approximation of the sine function
by polynomial using Taylor's or Maclaurin's formula |
|
Properties
of the power series expansion of the sine function |
|
Extreme
points, local (or relative) maximum and local minimum |
|
The
first derivative test |
|
The
second derivative test and concavity |
|
Finding and classifying
critical (or stationary) points |
|
Finding extreme points |
|
Points
of inflection |
|
Finding points
of inflection |
|
Finding and classifying
critical (or stationary) points, examples |
|
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