  Calculus

Open course material Calculus offered by KdVI, SMASH and TLC-FNWI

Auteurs: André Heck, Marthe Schut

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Available languages:  Course content
Functions and graphs
Relations and functions
THEORY
T
1.
Example 1 of a linear relation: Height prediction
THEORY
T
2.
Example 2: The Widmark model
THEORY
T
3.
Example 3: Colour models
THEORY
T
4.
An implicit relationship
THEORY
T
5.
Isolation of a variable
PRACTICE
P
6.
Converting a relation into a functional relationship
3
Function machines and composition of functions
THEORY
T
1.
Function definition and the argument of a function
PRACTICE
P
2.
Computing function values
2
THEORY
T
3.
Combining functions
PRACTICE
P
4.
Composing functions
4
THEORY
T
5.
Domain and range of a function and interval notation
PRACTICE
P
6.
Interval notation
3
PRACTICE
P
7.
Domain and range of a function
4
Data and graphs
THEORY
T
1.
Coordinate system
PRACTICE
P
2.
Coordinate system
1
THEORY
T
3.
Data graphs
PRACTICE
P
4.
A scatter plot
1
THEORY
T
5.
Graph of a function
PRACTICE
P
6.
Graph of a function
3
Properties of functions
THEORY
T
1.
Perforations, limits, and continuous or discontinuous functions
PRACTICE
P
2.
Perforations and contininuous or discontinuous functions
4
THEORY
T
3.
Even and odd functions
PRACTICE
P
4.
Even and odd functions
1
THEORY
T
5.
Periodic functions
PRACTICE
P
6.
Periodic functons
2
Transformations of graphs and functions
THEORY
T
1.
Vertical translation
THEORY
T
2.
Horizontal translation
THEORY
T
3.
Vertical multiplication
THEORY
T
4.
Horizontal multiplication
THEORY
T
5.
Combination of transformations
PRACTICE
P
6.
Transformations of graphs and functions
4
THEORY
T
7.
Reflection in the line y = x
PRACTICE
P
8.
Graph of an inverse function
3
Families of functions
THEORY
T
1.
A function with a parameter in it
PRACTICE
P
2.
Solving parametric problems
3
Basic functions
Power functions
THEORY
T
1.
Definition
THEORY
T
2.
Power functions with negative integer exponents
THEORY
T
3.
Power functions with positive fractional exponents
PRACTICE
P
4.
Computing with power functions
6
PRACTICE
P
5.
Graphs of power functions
4
THEORY
T
6.
Context 1: Kleiber's law and allometry between brain weight and body weight
THEORY
T
7.
Context 2: Psychophysics
PRACTICE
P
8.
Computing with power laws
3
THEORY
T
9.
Transformations of power functions
PRACTICE
P
10.
Transformations of power functions
4
THEORY
T
11.
Equations with power functions
PRACTICE
P
12.
Solving equations with power functions
5
THEORY
T
13.
Context 3: Equivalent doses of antipsychotics
PRACTICE
P
14.
Computing the equivalent dose of antipsychotics
1
THEORY
T
15.
Context 3: Solubility of chemical substances
PRACTICE
P
16.
Computing solubility
1
THEORY
T
17.
Sums of power functions
PRACTICE
P
18.
Sums of power functions
2
Linear functions
THEORY
T
1.
Linear functions
PRACTICE
P
2.
Defining a linear function
1
THEORY
T
3.
A linear relationship on the basis of two data points
PRACTICE
P
4.
Finding the straight line through two data points
2
Polynomial functions
THEORY
T
1.
PRACTICE
P
2.
2
THEORY
T
3.
THEORY
T
4.
The notion of a quadratic equation in one unknown
PRACTICE
P
5.
The notion of a quadratic equation in one unknown
2
THEORY
T
6.
PRACTICE
P
7.
9
THEORY
T
8.
Completing the square
PRACTICE
P
9.
Completing the square
4
THEORY
T
10.
Factorisation of a quadratic equation by inspection
PRACTICE
P
11.
Factorisation by inspection
2
THEORY
T
12.
PRACTICE
P
13.
7
THEORY
T
14.
A quadratic inequality in basic form
THEORY
T
15.
Solving quadratic inequalities via reduction and/or factorisation
PRACTICE
P
16.
Solving quadratic inequalities via reduction and/or factorisation
3
THEORY
T
17.
PRACTICE
P
18.
3
THEORY
T
19.
PRACTICE
P
20.
6
THEORY
T
21.
Root equations
PRACTICE
P
22.
Solving root equations
4
THEORY
T
23.
Polynomial functions of degree three and higher
THEORY
T
24.
Three properties of polynomial functions
PRACTICE
P
25.
Zeros of third degree polynomial functions
1
THEORY
T
26.
Applications of polynomial functions of degree three
PRACTICE
P
27.
Computing the solubility or solubility product
3
THEORY
T
28.
A quadratic relation on the basis of three data points
PRACTICE
P
29.
Finding a parabola through 3 points
1
THEORY
T
30.
Lagrange interpolation
PRACTICE
P
31.
Lagrange interpolation
2
Rational functions
THEORY
T
1.
Definition and basic properties
PRACTICE
P
2.
Determining the behaviour of a fractional linear function
8
THEORY
T
3.
Applications
PRACTICE
P
4.
Computing the degree of protolysis of a weak acid
1
THEORY
T
5.
Equations with rational functions
PRACTICE
P
6.
Equations with rational functions
4
THEORY
T
7.
Division with remainder for polynomials
PRACTICE
P
8.
Polynomial division with remainder
4
THEORY
T
9.
An oblique asymptote of a rational function
PRACTICE
P
10.
An oblique asymptote of a rational function
2
THEORY
T
11.
Greatest common divisor and least common multiple of polynomials
PRACTICE
P
12.
Greatest common divisor of polynomials
4
THEORY
T
13.
Normal form of a rational function
PRACTICE
P
14.
Normal form of a rational function
2
THEORY
T
15.
Partial fraction decomposition: distinct linear factors for the denominator
PRACTICE
P
16.
Partial fraction decomposition: distinct linear factors for the denominator
2
THEORY
T
17.
Partial fraction decomposition: repeated linear factors for the denominator
PRACTICE
P
18.
Partial fraction decomposition: repeated linear factors for the denominator
3
THEORY
T
19.
Partial fraction decomposition: irreducible quadratic factors for the denominator
PRACTICE
P
20.
Partial fraction decomposition: irreducible quadratic factors for the denominator
2
Exponential functions and logarithms
Exponential functions
THEORY
T
1.
Definition and basic properties
PRACTICE
P
2.
Graphs of exponential functions
2
THEORY
T
3.
Computational rules for exponential functions
THEORY
T
4.
The exponential function exp(x)
PRACTICE
P
5.
Symplifying exponential expressions
5
THEORY
T
6.
Transformations of exponential functions
PRACTICE
P
7.
Transformations of exponential functions
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T
8.
Equations and inequalities with exponential functions that have the same base
PRACTICE
P
9.
Equations and inequalities with exponential functions that have the same base
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T
10.
Sums of exponential functions
PRACTICE
P
11.
Asymptotics of sums of exponential functions
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T
12.
Applications
THEORY
T
13.
Hyperbolic functions
Logarithms
THEORY
T
1.
The natural logarithm
PRACTICE
P
2.
Working with expressions that contain the natural logarithm
13
THEORY
T
3.
Applications of ln
THEORY
T
4.
Logarithmic functions
PRACTICE
P
5.
Computing logarithms on the basis of their definition
3
THEORY
T
6.
Calculation rules for logarithmic functions
THEORY
T
8.
Transformations of logarithmic functions
PRACTICE
P
9.
Transformations of logarithmic functions
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T
10.
Equations and inequalities with logarithms that have the same base
PRACTICE
P
11.
Solving equations and inequalities with logarithms that have the same base
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T
12.
Equations and inequalities containing logarithms with different bases
PRACTICE
P
13.
Solving equations and inequalities that contain logarithms with different bases
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T
14.
Applications of the logarithm with base 10
Trigonometry
Trigonometric functions
THEORY
T
1.
PRACTICE
P
2.
Practising with angles of rotation
4
THEORY
T
3.
Trigonometry in right-angled triangles
PRACTICE
P
4.
Calculating trigonometric values
3
THEORY
T
5.
Trigonometry in arbitrary triangles
PRACTICE
P
6.
Triangulation
2
THEORY
T
7.
Definitions of trigonometric functions
THEORY
T
8.
Basic properties of trigonometric functions
PRACTICE
P
9.
Computing function values
4
THEORY
T
10.
Graphs of trigonometric functions
THEORY
T
11.
Transformations of trigonometric functions
THEORY
T
12.
Movie (in Dutch): Determining a sinusoid based on a graph
THEORY
T
14.
Addition formulas, double angle formulas, and other trigonometric identities
THEORY
T
16.
Signals in the time domain
THEORY
T
18.
Arbitrary periodic signals
THEORY
T
20.
Signal analysis
Inverse trigonometric functions
THEORY
T
1.
The arcsine, arccosine, and arctangent
THEORY
T
3.
Trigonometric equations
Parametric curves
Plane curves
THEORY
T
1.
Plane curves
THEORY
T
2.
Video: parameter curves
Polar coordinates and polar curves
THEORY
T
1.
Polar coordinates
Space curves
THEORY
T
1.
Parametric curves in space
Numerical methods for solving a nonlinear equation
non-linear equations
THEORY
T
1.
Non-linear equations
The bisection method
THEORY
T
1.
The bisection method
THEORY
T
2.
Implementation of the bisection method (programming assignment)
THEORY
T
3.
Convergence of the bisection method (programming assignment)
THEORY
T
4.
Stopping criterion for the bisection method (programming assignment)
Regula falsi method
THEORY
T
1.
The regula falsi method
THEORY
T
2.
Implementation of the regula falsi method (programming assignment)
The secant method
THEORY
T
1.
The secant method
THEORY
T
2.
Implementation of the secant method (programming assignment)
Differentiation, derivatives and Taylor approximations
Tangent line
THEORY
T
1.
Difference quotient and tangent line
PRACTICE
P
2.
Calculating a difference quotient
5
THEORY
T
3.
Difference quotient at a point
PRACTICE
P
4.
Difference quotient in a point
1
THEORY
T
5.
Tangent line and slope function
PRACTICE
P
6.
Tangent line and slope function
2
THEORY
T
7.
Derivative of a function
THEORY
T
8.
A simple derivative
PRACTICE
P
9.
Drawing a tangent line
2
Differentiating power functions
THEORY
T
1.
The derivative of a power function
Rules for differentiating functions
THEORY
T
1.
The constant factor rule and the sum and difference rule
PRACTICE
P
2.
Applying the constant factor, sum and difference rule
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T
3.
The product rule
THEORY
T
5.
The quotient rule
THEORY
T
7.
The chain rule
Differentiating implicit functions
THEORY
T
1.
implicit differentiation
Differentiating exponential and logarithmic functions
THEORY
T
1.
Derivatives of exponential functions
THEORY
T
3.
Derivatives of logarithmic functions
THEORY
T
5.
Logarithmic differentiation
THEORY
T
7.
Derivatives of hyperbolic functions and their inverses
Differentiating trigonometric functions and their inverse functions
THEORY
T
1.
Derivatives of trigonometric functions and their inverse functions
Higher-Order derivatives
THEORY
T
1.
Higher derivatives
Applications of derivatives
THEORY
T
1.
Increasing, decreasing, and extreme values of mathematical functions
THEORY
T
2.
Application 1: Change of behaviour of a function
THEORY
T
3.
Application 2: Quantitative pharmacokinetics
Taylor approximations
THEORY
T
1.
Taylor polynomials
THEORY
T
2.
Taylor series
THEORY
T
4.
Approximation of π and a Taylor series of arctan
Numerical differentiation
Time series
THEORY
T
1.
Terminology and examples
THEORY
T
2.
Filters
THEORY
T
3.
Cross-correlation and convolution of time series
THEORY
T
4.
Calculation of cross-correlation (programming assignment)
THEORY
T
5.
Processing of an EMG signal via convolution (programming assignment)
THEORY
T
6.
Exponential data smoothing (programming assignment)
Difference formulas for the first derivative
THEORY
T
1.
Simple difference formulas
THEORY
T
2.
The 3-point central difference approximation
THEORY
T
3.
The central difference method via cross-correlation and applied to measurement data
THEORY
T
5.
General difference formulas
THEORY
T
6.
Computing a numerical first derivative (programming assignment)
THEORY
T
7.
Derivative via a 5-point difference method (programming assignment)
Difference formulas for the second derivative
THEORY
T
1.
The 3-point central difference formula
THEORY
T
2.
General difference formulas for the second derivative
PRACTICE
P
3.
Computing a numerical 1st and 2nd derivative
Unlock full access THEORY
T
4.
Numerical derivatives of noisy data (programming assignment)
THEORY
T
5.
Pezzack's benchmark data (programming assignment)
Central difference formulas for higher derivatives
THEORY
T
1.
Difference formulas for higher derivatives
Iteration of functions
Iteration of functions and fixed points
THEORY
T
1.
Iteration of a function and fixed points
THEORY
T
2.
Héron's method (programming assignment)
Finding zeros via iterations
THEORY
T
1.
Stability of a fixed point
THEORY
T
2.
Finding zeros via iteration of a function (programming assignment)
Finding zeros via the Newton-Raphson method
THEORY
T
1.
Deriving the Newton-Raphson method
PRACTICE
P
2.
Computing the iteration function in the Newton-Raphson method
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T
3.
Implementation of the Newton-Raphson method
Limits part 1: Infinite sequences
Introduction
THEORY
T
1.
Introduction
PRACTICE
P
2.
Introduction exercise
1
THEORY
T
3.
The limit of 1/n
Basic calculation rules for limits
PRACTICE
P
1.
Introduction exercises about rules for limits
2
THEORY
T
2.
Calculation rules for limits
PRACTICE
P
3.
Exercises about calculation rules for limits
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T
4.
The limit n^r
THEORY
T
5.
Convergence and divergence
THEORY
T
6.
Calculation rules for limits (continuation)
PRACTICE
P
7.
Exercises about calculation rules for limits (continuation)
Unlock full access Techniques
THEORY
T
1.
Conjugate multiplication
THEORY
T
2.
Estimation
THEORY
T
3.
The squeeze lemma
Standard limits
THEORY
T
1.
Geometric sequences
THEORY
T
2.
Polynomials versus exponential functions
THEORY
T
3.
An extraordinary limit
THEORY
T
4.
Overview of standard limits
Final paragraph
THEORY
T
2.
* Extra: all video summaries together (only in Dutch)
Limits part 2: Functions
Introduction
THEORY
T
1.
Introduction
All kinds of limits
THEORY
T
1.
Limits at infinity
THEORY
T
2.
Limits at a point
THEORY
T
3.
One-sided limits
THEORY
T
4.
Continuity of functions
THEORY
T
5.
Limits equal to infinity
PRACTICE
P
6.
Exercises about different types of limits
Unlock full access Techniques
THEORY
T
1.
Calculation rules for limits
THEORY
T
2.
Squeeze lemma
THEORY
T
3.
Factorisation
THEORY
T
4.
Polynomial division
THEORY
T
5.
L'Hôpital's rule
THEORY
T
6.
Substitution rule
THEORY
T
7.
Substitution and one-sided limits
Infinite series
Introduction
THEORY
T
1.
Introduction
THEORY
T
2.
Convergence of series
THEORY
T
3.
Minima, maxima, infima and suprema
THEORY
T
4.
Lim inf and lim sup
Common series
THEORY
T
1.
Geometric series
THEORY
T
2.
The harmonic series
Convergence tests
THEORY
T
1.
The comparison test
THEORY
T
2.
The root test
THEORY
T
3.
The ratio test
THEORY
T
4.
The integral test
THEORY
T
5.
Alternernating series
Final chapter
THEORY
T
1.
Overview common series and convergence tests
Differentials and integrals
Differentials
THEORY
T
1.
What is a differential?
THEORY
T
2.
Differential of a function
THEORY
T
3.
Calculation rules for differentials
THEORY
T
5.
Application: error analysis
Area and primitive function
THEORY
T
1.
Calculating the area under the curve of a function
THEORY
T
2.
Area and primitive function
THEORY
T
3.
The relation between area and integral
THEORY
T
4.
General definition and basic rules of integrals
Indefinite integrals
THEORY
T
1.
Antiderivatives of standard functions
THEORY
T
2.
Standard antiderivatives
PRACTICE
P
3.
Calculating antiderivatives of functions
Unlock full access Integration techniques
THEORY
T
1.
Introduction
THEORY
T
2.
The method of substitution
THEORY
T
4.
Integration by parts
THEORY
T
6.
Partial fraction decomposition
PRACTICE
P
7.
Practicing partial fraction decomposition
Unlock full access THEORY
T
8.
Reduction formulas
PRACTICE
P
9.
Reduction formulas
2
THEORY
T
10.
More black magic in determining an antiderivative
PRACTICE
P
11.
More black magic in determining an antiderivative
5
THEORY
T
12.
Integrating trigonometric expressions
PRACTICE
P
13.
Integrating trigonometric expressions
2
Applications of integration
THEORY
T
1.
Calculating an area under the curve
PRACTICE
P
3.
Length of a curve
6
THEORY
T
4.
Line integral of a function
PRACTICE
P
5.
Line integral of a function
6
THEORY
T
6.
Volume of a soldid of revolution
THEORY
T
7.
Surface area of a solid of revolution
THEORY
T
9.
Average value of a function on an interval
PRACTICE
P
10.
Average value of a function on an interval
1
THEORY
T
11.
Simple differential equations
PRACTICE
P
12.
Solving simple differentiation equations
Unlock full access Improper integrals
THEORY
T
1.
Introduction
THEORY
T
2.
Improper integrals of type 1
THEORY
T
4.
Improper integrals of type 2
Numerical Integration
Introduction
THEORY
T
1.
Why numerical integration?
THEORY
T
2.
Some Riemann sums
THEORY
T
1.
Left, right, and midpoint Riemann sums
THEORY
T
2.
Truncation error in Riemann sums
The trapezoidal rule
THEORY
T
1.
The trapezoidal rule
THEORY
T
2.
Truncation error in the trapezoidal rule
Simpson's rule
THEORY
T
1.
Simpson's rule
THEORY
T
2.
Truncation error in Simpson's rule
Computer inquiry into efficiency of numerical integration methods
THEORY
T
1.
Efficiency of numeric integration (programming assignment)
Monte Carlo integration
THEORY
T
1.
A Monte Carlo method
THEORY
T
2.
Efficiency of Monte Carlo integration (programming assignment)
Unlimited growth
THEORY
T
1.
Introduction
THEORY
T
2.
Linear growth
THEORY
T
3.
Examples of linear growth
THEORY
T
5.
THEORY
T
6.
Example: average height growth of girls with Turner syndrome
THEORY
T
8.
Cubic growth and beyond
Exponential growth
THEORY
T
1.
Introduction
THEORY
T
2.
Binary division
THEORY
T
3.
Exponential growth model
PRACTICE
P
4.
Basic calculations with a model of exponential growth
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T
5.
Solving exponential equations
THEORY
T
8.
Growth factors at different time scales
PRACTICE
P
11.
Computing the intensity of a light bulb
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T
12.
Doubling time and half-life for exponential growth
THEORY
T
14.
Formulas for exponential growth
THEORY
T
16.
The differential equation of exponential growth
THEORY
T
17.
COVID-19, exponential growth and R0
PRACTICE
P
18.
Solving initial value problem of unlimited growth
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T
19.
Geometrical "solution" via a slope field
Applications of exponential growth models
THEORY
T
1.
A video clip - quantitative pharmacokinectics (in Dutch)
THEORY
T
2.
Blood drug concentration after a single intravenous bolus injection
THEORY
T
4.
Reaction kinetics of the chemical reaction
A → B
THEORY
T
5.
Simulation of first-order reaction kinetics
THEORY
T
6.
First-order chemical reaction kinetics
THEORY
T
7.
Discharge of a capacitor
THEORY
T
9.
Poisson process of neural firing
Limited exponential growth
Introduction
THEORY
T
1.
Why limited growth?
THEORY
T
2.
The von Bertalanffy model of length growth of fish
THEORY
T
3.
Administration of a drug via a constant rate intravenous infusion
PRACTICE
P
6.
Computing in the context of a constant rate intravenous infusion
Unlock full access Limited exponential growth functions
THEORY
T
1.
The limited exponential growth function
THEORY
T
2.
The differential equation of limited exponential growth
PRACTICE
P
4.
Computing in the context of plant physiology
Unlock full access Applications of limited exponential growth models
THEORY
T
1.
Reaction kinetics of a unimolecular equilibrium reaction
THEORY
T
2.
Simulation of the kinetics of a unimolecular equilibrium reaction
THEORY
T
3.
The Keller model of sprinting
THEORY
T
4.
A simplified model for the membrane potential
PRACTICE
P
5.
Modelling a vertically falling shutlecock
Unlock full access Logistic growth
Introduction
THEORY
T
1.
Introduction to logistic growth
The logistic function
THEORY
T
1.
Definition and basic properties
PRACTICE
P
2.
Computing the logistic growth model fro several initial values
Unlock full access THEORY
T
5.
Behaviour of the logistic function
More examples of logistic growth
THEORY
T
1.
Modellling microbial growth
THEORY
T
2.
Epidemiology
THEORY
T
3.
Autocatalysis in chemistry
Unlock full access Open course material Calculus offered by KdVI, SMASH and TLC-FNWI

Auteurs: André Heck, Marthe Schut

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