Topical help resources – Frisbie

Perimeter and area of plane figures. Properties of triangles and quadrilaterals, including parallelograms, rhombuses, rectangles, squares, kites and trapezoids; compound shapes

Quadrilaterals (Math Is Fun)
Geometry Worksheets (EdHelper)

Familiarity with three-dimensional shapes (prisms, pyramids, spheres, cylinders and cones)

3D Objects (Shmoop)

Volumes and surface areas of cuboids, prisms, cylinders, and compound three-dimensional shapes

Volume of a Cuboid (Math Is Fun)
Cone vs Sphere vs Cylinder (Math Is Fun)
Prisms (Math Is Fun)
Composite solids (Bitesize)

Statistics and probability (prior)

The collection of data and its representation in bar charts, pie charts, pictograms, and line graphs

Stem and Leaf, Box Plots (Math Planet)
Pie Charts (Math Is Fun)
Line Graphs (Math Is Fun)

Obtaining simple statistics from discrete data, including mean, median, mode, range

Central Measures (Math Is Fun)
The Range (Math Is Fun)
Quartiles (Math Is Fun)

Calculating probabilities of simple events

Simple Probability (Math Is Fun)

Venn diagrams for sorting data

Venn Diagrams videos (JMT) *

Tree diagrams

Tree diagrams (Bitesize)
Probability Tree Diagrams (Math Is Fun)

Calculus (prior)

$\text{Speed} = \frac{\text{distance}}{\text{time}}$

Speed and Velocity (Math Is Fun)

IB Math Analysis and Approaches SL Syllabus

Topic 1: Number and algebra

1.1 Operations with numbers in the form a × 10^k where 1 ≤ a < 10 and k is an integer.

Operations with Numbers in Scientific Notation (CK12)

1.2 Arithmetic sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for sums of arithmetic sequences. Applications. Analysis, interpretation and prediction where a model is not perfectly arithmetic in real life.

Summation Notation (Paul)
Arithmetic Sequences videos (JMT) *
Compound Interest videos (JMT) *

1.3 Geometric sequences and series. Use of the formulae for the nth term and the sum of the first n terms of the sequence. Use of sigma notation for the sums of geometric sequences. Applications.

Summation Notation (Paul)

1.4 Financial applications of geometric sequences and series: compound interest, annual depreciation.

Compound Interest videos (JMT) *
Compound interest (Math Is Fun)
Working with appreciation and depreciation (Bitesize)

1.5 Laws of exponents with integer exponents. Introduction to logarithms with base 10 and e. Numerical evaluation of logarithms using technology.

Laws of Exponents (SOS)
Simplifying Logarithms Review (Paul)
Inverses of Logarithmic Functions (SOS)
Exponents and Logs videos (JMT) *

1.6 Simple deductive proof, numerical and algebraic; how to lay out a left-hand side to right-hand side (LHS to RHS) proof. The symbols and notation for equality and identity.

Proof by Deduction (StudyWell)
Mathematical Proof: Definition and Examples (Study)

1.7 Laws of exponents with rational exponents. Laws of logarithms. $\log_a{xy}=\log_a{x}+\log_a{y}; \log_a{\displaystyle{\frac{x}{y}}} = \log_a{x}- \log_a{y}; \log_a{x^m}=m\log_a{x}$ for $a, x, y > 0$ . Change of base of a logarithm. $\log_a{x}=\displaystyle{\frac{\log_b{x}}{\log_b{a}}}$ , for $a, b, x > 0$ . Solving exponential equations, including using logarithms.

Properties of Logarithms (SOS)
Logarithm Properties Review (Paul)
Change of Base (SOS)
Laws of Logarithms (SOS)
Exponents and Logs videos (JMT) *

1.8 Sum of infinite convergent geometric sequences.

Convergent geometric series (Khan)
Divergent geometric series (Khan)
Infinite Series (Math Is Fun)

1.9 The binomial theorem; expansion of $(a+b)^n$ , $n \in$ ℕ. Use of Pascal’s triangle and $^nC_r$ .

Binomial Theorem videos (JMT) *

Topic 2: Functions

2.1 Different forms of the equation of a straight line. Gradient; intercepts. Lines with gradients $m_1$ and $m_2$ . Parallel lines $m_1 = m_2$ . Perpendicular lines $m_1 \times m_2 = -1$ .

Graphing Lines (Paul)
Graphing Lines videos (JMT) *
Lines (Paul)

2.2 Concept of a function, domain, range and graph. Function notation, for example $f(x), \, v(t),\, C(n)$ . The concept of a function as a mathematical model. Informal concept that an inverse function reverses or undoes the effect of a function. Inverse function as a reflection in the line $y = x$ , and the notation $f^{-1}(x)$ .

Definition of a Function (Paul)
Combining Functions (Paul)
Inverse Functions (Paul)
Function Evaluation Review (Paul)
Inverse Functions Introduction (SOS)
Inverse Functions, Same Domain (SOS)
Inverses of Composite Functions (SOS)
Inverse Functions, Restricted Domains (SOS)
Finding Domains videos (JMT) *

2.3 The graph of a function; its equation $y = f(x)$ . Creating a sketch from information given or a context, including transferring a graph from screen to paper. Using technology to graph functions including their sums and differences.

Graphing Functions (Paul)
Graphing videos (JMT) *

2.4 Determine key features of graphs. Finding the point of intersection of two curves or lines using technology.

Graphing Functions (Paul)
How to Find Points of Intersection (TI Education)

2.5 Composite functions. Identity function. Finding the inverse function $f^{-1}(x)$ .

Composition of Functions (Purple)
Inverse Functions (Purple)

2.6 The quadratic function $f(x) = ax^2 + bx + c$ : its graph, y-intercept $(0, c)$ . Axis of symmetry. The form $f(x) = a(x - p)(x - q)$ , x– intercepts $(p, 0)$ and $(q, 0)$ . The form $f(x) = a(x - h)^2 + k$ , vertex $(h , k)$ .

Graphing Parabolas (Paul)
Parabolas videos (JMT) *

2.7 Solution of quadratic equations and inequalities. The quadratic formula. The discriminant $\Delta = b^2 - 4ac$ and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots.

Completing the Square (SOS)
Quadratic Formula (SOS)
Solving Quadratics by Factoring (SOS)
Solving Quadratic Equations (SOS)
Solving Quadratics by Factoring (Paul)
Solving Quadratics by Formula (Paul)
Equations in Quadratic Form (Paul)
Use of the Discriminant (Paul)

2.8 The reciprocal function $f(x) = \displaystyle{\frac{1}{x}} ,\, x \neq 0$ : its graph and self-inverse nature. Rational functions of the form $f (x) = \displaystyle{\frac{ax + b}{cx+d}}$ and their graphs. Equations of vertical and horizontal asymptotes.

Solving Rational Equations (SOS)
Rational Expressions Review (Paul)
Rational Expressions (Paul)
Rational Functions videos (JMT) *

2.9 Exponential functions and their graphs: $f(x)=a^x,\, a>0,\, f(x)=e^x$ Logarithmic functions and their graphs: $f(x)=\log_a x,\, x>0, f(x)=\ln x,\, x>0$ .

Definition of Exponential Function (SOS)
Exponential Functions (Paul)
Exponential Functions Review (Paul)
Definition of Logarithmic Function (SOS)
Logarithm Functions (Paul)
Logarithmic Functions Review (Paul)
Graphs of Exponential and Logarithmic Functions (SOS)
Solving Exponential Equations (SOS)
Solving Logarithmic Equations (SOS)
Exponents and Logs videos (JMT) *

2.10 Solving equations, both graphically and analytically. Use of technology to solve a variety of equations, including those where there is no appropriate analytic approach. Applications of graphing skills and solving equations that relate to real-life situations.

Exponential Equations Review (Paul)
Solving Exponential Equations (Paul)
Logarithmic Equations Review (Paul)
Solving Logarithm Equations (Paul)
Solving Exponential, Log Equations (Paul)
Equations with Radicals (Paul)
Solving Equations videos (JMT) *
Exponential and Log Word Problems (SOS)
Applications of Linear Equations (Paul)
Applications of Quadratic Equations (Paul)
Applications of Exponentials (Paul)
Word Problems videos (JMT) *

2.11 Transformations of graphs. Translations: $y = f(x) + b; \, y = f(x - a)$ . Reflections (in both axes): $y = -f(x); y = f(-x)$ . Vertical stretch with scale factor $p:\, y = pf (x)$ . Horizontal stretch with scale factor $\displaystyle{\frac{1}{q}} :\, y = f (qx)$ . Composite transformations.

Summary of Transformations (Leckie)
Transformations (Paul)
Graphical Transformations videos (JMT) *
Graphical Transformations (Ellermeyer)

Topic 3: Geometry and trigonometry

3.1 The distance between two points in three- dimensional space, and their midpoint. Volume and surface area of three-dimensional solids including right-pyramid, right cone, sphere, hemisphere and combinations of these solids. The size of an angle between two intersecting lines or between a line and a plane.

The Distance Formula in 3-Dimensions (JMT)
The Midpoint and Distance Formulas in 3D (Prokup)
Geometry: Volume and surface area (Khan)
Angle in Cube Using Trigonometry (OnMaths)
Angle in Pyramid Using Trigonometry (OnMaths)
Angle in Cuboid Using Trigonometry (OnMaths)

3.2 Use of sine, cosine and tangent ratios to find the sides and angles of right-angled triangles. The sine rule: $\displaystyle{\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}}$ . The cosine rule: $c^2 = a^2 + b^2 - 2ab \cos C; \cos C = \displaystyle{\frac{a^2+b^2-c^2}{2ab}}$ . Area of a triangle as $\frac{1}{2} ab \sin C$ .

Law of Sines and Cosines videos (JMT) *

3.3 Applications of right and non-right angled trigonometry, including Pythagoras’s theorem. Angles of elevation and depression. Construction of labelled diagrams from written statements.

Applications of Right Triangle Trigonometry (Wikibooks)
Applications of Basic Triangle Trigonometery (CK12)
Laws of Sines and Cosines (LibreTexts)

3.4 The circle: radian measure of angles; length of an arc; area of a sector.

Trig Errors to Avoid (Paul)
Radians videos (JMT) *

3.5 Definition of cos θ, sin θ in terms of the unit circle. Definition of tan θ as $\displaystyle{\frac{\sin \theta}{\cos \theta}}$ . Exact values of trigonometric ratios of $\displaystyle{0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}}$ and their multiples. Extension of the sine rule to the ambiguous case.

Unit Circle Practice (Weinberg)
Trig Values Review (Paul)
Evaluating Trig Functions videos (JMT) *
Amazing Unit Circle (Mistakes)

3.6 The Pythagorean identity $\cos^2 \theta + \sin^2 \theta = 1$ . Double angle identities for sine and cosine. The relationship between trigonometric ratios.

Pythagorean Identities (SOS)
Trig Identities Review (Paul)
Trig Identities videos (JMT) *

3.7 The circular functions sin x, cos x, and tan x; amplitude, their periodic nature, and their graphs. Composite functions of the form $f(x) = a \sin (b (x+c))+d$ . Transformations. Real-life contexts.

Graphing Trig Functions Review (Paul)
Graphing Trig Functions videos (JMT) *

3.8 Solving trigonometric equations in a finite interval, both graphically and analytically. Equations leading to quadratic equations in sin x, cos x or tan x.

Solving Trigonometric Equations (SOS)
More Solving Trig Equations (SOS)
Still More Solving Trig Equations (SOS)
Solving Trig Equations 1 (Paul)
Solving Trig Equations 2 (Paul)
Solving Trig Equations 3 (Paul)
Solving Trig Equations videos (JMT) *

Topic 4: Statistics and probability

4.1 Concepts of population, sample, random sample, discrete and continuous data. Reliability of data sources and bias in sampling. Interpretation of outliers. Sampling techniques and their effectiveness.

4.2 Presentation of data (discrete and continuous): frequency distributions (tables). Histograms. Cumulative frequency; cumulative frequency graphs; use to find median, quartiles, percentiles, range and interquartile range (IQR). Production and understanding of box and whisker diagrams.

Box-and-Whisker Plots video (JMT) *
Frequency Distributions (Australia)
Frequency Distribution Tables (StatCan)
Cumulative Frequency (Bitesize)

4.3 Measures of central tendency (mean, median and mode). Estimation of mean from grouped data. Modal class. Measures of dispersion (interquartile range, standard deviation and variance). Effect of constant changes on the original data. Quartiles of discrete data.

Variance video (JMT) *
Interquartile Range, Outliers (Purple)
Standard Deviation (StatCan)

4.4 Linear correlation of bivariate data. Pearson’s product-moment correlation coefficient, r. Scatter diagrams; lines of best fit, by eye, passing through the mean point. Equation of the regression line of y on x. Use of the equation of the regression line for prediction purposes. Interpret the meaning of the parameters, a and b, in a linear regression $y = ax + b$ .

Guessing Correlations (OnlineStat)
Linear Regression (OnlineStat)

4.5 Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event. The probability of an event A is $\displaystyle{P(A) = \frac{n(A)}{n(U)}}$ . The complementary events A and A′ (not A). Expected number of occurrences.

Basic Probability (OnlineStat)
Basic Probability (CK12)

4.6 Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities. Combined events: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$ . Mutually exclusive events: $P(A \cap B) = 0$ . Conditional probability: $\displaystyle{P(A | B) = \frac{P(A \cap B)}{P(B)}}$ . Independent events: $P(A \cap B) = P(A)P(B)$ .

Probability videos (JMT) *
Independent Events (Math Is Fun)
Conditional Probability (Math Is Fun)
Conditional Probability (Math Goodies)

4.7 Concept of discrete random variables and their probability distributions. Expected value (mean), for discrete data. Applications.

Constructing a probability distribution (Khan)
Valid discrete probability distribution (Khan)
Mean (expected value) of a discrete random variable (Khan)
Expected Value video (JMT) *

4.8 Binomial distribution. Mean and variance of the binomial distribution.

Binomial Distribution video (JMT) *
Binomial Distribution (StatTrek)

4.9 The normal distribution and curve. Properties of the normal distribution. Diagrammatic representation. Normal probability calculations. Inverse normal calculations.

Normal Distribution video (JMT) *

4.10 Equation of the regression line of x on y. Use of the equation for prediction purposes.

Regression y on x versus x on y (Tam)

4.11 Formal definition and use of the formulae: $\displaystyle{P(A|B) = \frac{P(A \cap B)}{P(B)}}$ for conditional probabilities, and $P(A | B) = P(A) = P(A | B')$ for independent events.

Conditional Probability (Math Is Fun)
Conditional Probability (Math Goodies)

4.12 Standardization of normal variables (z-values). Inverse normal calculations where mean and standard deviation are unknown.

Z-scores review (Khan)
Finding the mean and standard deviation (Wills)

Topic 5: Calculus

Many, many more resources are listed in the AP Calculus section below.

5.1 Introduction to the concept of a limit. Derivative interpreted as gradient function and as rate of change.

Definition of Derivative video (JMT) *

5.2 Increasing and decreasing functions. Graphical interpretation of $f'(x) > 0, f'(x) = 0, f'(x) < 0$ .

Increasing and Decreasing videos (JMT)

5.3 Derivative of $f(x) = ax^n$ is $f'(x) = anx^{n-1}, n \in$ ℤ. The derivative of functions of the form $f(x) = ax^n + bx^{n-1} + \dots$ where all exponents are integers.

Derivatives videos (JMT) *

5.4 Tangents and normals at a given point, and their equations.

Rates of Change and Tangent Lines (Paul)
How to find equations of tangent lines and normal lines (Warehouse)

5.5 Introduction to integration as anti-differentiation of functions of the form $f(x) = ax^n + bx^{n-1}+ \dots$ , where $n \in$ ℤ, $n \neq -1$ . Anti-differentiation with a boundary condition to determine the constant term. Definite integrals using technology. Area of a region enclosed by a curve $y = f(x)$ and the x-axis, where $f(x) > 0$ .

Integration videos (JMT) *

5.6 Derivative of $x^n\,(n \in$ ℚ $), \sin x, \cos x, e^x \text{ and }\ln x$ . Differentiation of a sum and a multiple of these functions. The chain rule for composite functions. The product and quotient rules.

Derivatives videos (JMT) *

5.7 The second derivative. Graphical behaviour of functions, including the relationship between the graphs of $f, \, f' \text{ and } f''$ .

Relationships of f, f ′, and f ″ p. 11 (Explained)

5.8 Local maximum and minimum points. Testing for maximum and minimum. Optimization. Points of inflexion with zero and non-zero gradients.

Maximum/Minimum videos (JMT) *
Concavity and Points of Inflection (SOS)

5.9 Kinematic problems involving displacement s, velocity v, acceleration a and total distance travelled.

Position, Velocity, Acceleration Relationship (Leckie)

5.10 Indefinite integral of $x^n \, (n \in$ ℚ $), \sin x, \cos x, \displaystyle{ \frac{1}{x}}$ and $e^x$ . The composites of any of these with the linear function $ax + b$ . Integration by inspection (reverse chain rule) or by substitution for expressions of the form: $\int kg'(x)f(g(x))dx$ .

Integration videos (JMT) *

5.11 Definite integrals, including analytical approach. Areas of a region enclosed by a curve $y = f (x)$ and the x-axis, where $f (x)$ can be positive or negative, without the use of technology. Areas between curves.

Area Between Curves (Paul)

AP Calculus AB Syllabus

Prerequisites

Linear, polynomial, rational, exponential, logarithmic, trigonometric, inverse trigonometric, and piecewise-defined functions.
Exponential Functions Review (Paul)
Logarithmic Functions Review (Paul)
Graphing Review (Paul)
Inverse Trig Functions (Paul)

Properties of functions, the algebra of functions, and the graphs of functions
Functions Review (Paul)
Inverse Functions Review (Paul)
Common Graphs (Paul)

The language of functions (domain and range, odd and even, periodic, symmetry, zeros, intercepts, and so on)
Odd and Even Functions (Leckie)
Symmetry (Paul)

Values of the trigonometric functions at the numbers 0, $\frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2}$ and their multiples
Trig Cheat Sheet (Paul)
Trig Functions Review (Paul)
Solving Trig Equations Review (Paul)
Unit Circle videos (JMT) *

Limits

Analysis of graphs

Interplay between geometric and analytic information
Limits (Paul)

Use of calculus to predict and explain observed local and global behavior of a function
Common Calculus Errors to Avoid (Paul)

Limits of functions (including one-sided limits)

An intuitive understanding of the limiting process
Informal Limits applet (Open)
Intuitive Notion of Limit applet (Renault)
Intuitive One-Sided Limits applet (Renault)
One-Sided Limits, Limits that Do Not Exist applet (Open)
Ways a Limit Can Fail to Exist (Leckie)

Calculating limits using algebra
Limits Cheat Sheet (Paul)
Limit Properties (Paul)
Computing Limits (Paul)
Limit Laws applet (Renault)
Limits videos (JMT) *
Calculating Limits video (McMullin)

Estimating limits from graphs or tables of data
Limits and Tables applet (Open)
One-Sided Limits (Paul)

Asymptotic and unbounded behavior

Understanding asymptotes in terms of graphical behavior
Graphs of Rational Functions (Paul)

Describing asymptotic behavior in terms of limits involving infinity
Limits Involving Infinity (SOS)
Limits at Infinity applet (Open)
Definition of Horizontal Asymptote (Leckie)
Definition of Vertical Asymptote (Leckie)
Infinite Limits (Paul)
Limits at Infinity 1 (Paul)

Comparing relative magnitudes of functions and their rates of change (e. g., contrasting exponential growth, polynomial growth, and logarithmic growth)
Limits at Infinity 2 (Paul)
Limits at Infinity, Dominance (McMullin)

Continuity as a property of functions

An intuitive understanding of continuity (close values of the domain lead to close values of the range)
Continuity: Informal Approach applet (Open)

Understanding continuity in terms of limits
Continuity (SOS)
Continuity applet (Renault)
Continuity: Formal Approach applet (Open)
Continuity (Paul)
Continuity videos (JMT) *
Continuity video (McMullin)

Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem)
Intermediate Value Theorem applet (Open)
Extreme Value Theorem applet (Open)

Derivatives

Equations involving derivatives; translating words to symbols
Business Applications (Paul)

Second derivatives

Corresponding characteristics of the graphs of f, f ′, and f ″
Relationships of f, f ′, and f ″ p. 11 (Explained)
Second Derivative applet (Open)
Twice-Differentiable applet (Open)
Matching f, f ′, and f ″ applet (Renault)
Matching Antiderivatives applet (Renault)
Multiple Derivatives applet (Renault)
Reconstruct f from f ″ applet (Renault)
Derivatives from Shape of Graph applet (Renault)
Derivatives videos (JMT) *

Relationship between the concavity of f and the sign of f ″
Curve Analysis Basic applet (Open)
Curve Analysis Special Cases applet (Open)
Derivatives videos (JMT) *
Graphing 2 video (McMullin)

Points of inflection as places where concavity changes
Concavity and Points of Inflection (SOS)
Justifying Points of Inflection video (McMullin)

Applications of derivatives

Analysis of curves, including monotonicity and concavity
Monotonicity and the First Derivative (SOS)
Critical Points (Paul)
The Shape of a Graph 1 (Paul)
The Shape of a Graph 2 (Paul)
Derivatives videos (JMT) *

Optimization, both absolute (global) and relative (local) extrema
Justifying Extreme Values video (McMullin)
Where to Find Extrema (Leckie)
First Derivative Test (Leckie)
Second Derivative Test (Leckie)
Critical Points and Local Extrema (SOS)
Global Extrema (SOS)
Global Extrema applet (Open)
Maximizing Volume applet (Open)
Maximum and Minimum Values (Paul)
Finding Absolute Extrema (Paul)
Optimization (Paul)
More Optimization Problems (Paul)
Derivatives videos (JMT) *
Extreme Values video (McMullin)
Critical Points video (McMullin)
Optimization applets (Renault): Bending a Wire, Rectangle in Parabola,Three Pens, Power to an Island, Another Wire Problem, Function and Rectangle 1, Function and Rectangle 2, Triangle Circumscribing Circle

Modeling rates of change, including related rates problems
Related Rates p. 12-13 (Explained)
Related Rates applet (Open)
Related Rates (Paul)
Rates of Change (Paul)
Related Rates videos (JMT) *
Applications applets (Renault): Oil Slick, Falling Ladder, Conical Tank, Car Driving Past House, Lamppost, Two Trains, Searchlight, Rocket Launch

Use of implicit differentiation to find the derivative of an inverse
Differentiating Inverse Functions (SOS)
Implicit Differentiation (SOS)
Derivatives of Inverse Functions applet (Renault)

Interpretation of derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration
Kinematics p. 15 (Explained)
Position, Velocity, Acceleration Relationship (Leckie)
Motion on a Line applet (Open)
Motion Problems video (McMullin)

Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations
Slope Fields applet (Open)
Definition of Slope Field (Leckie)
Slope Fields Revisited (Leckie)
Slope Fields video (McMullin)

L’Hôpital’s rule for evaluating indeterminate limits.
L’Hôpital’s Rule applet (Open)
Indeterminate Forms and L’Hospital’s Rule (Paul)
L’Hospital’s Rule videos (JMT) *

Computation of derivatives

Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trig
Derivative Formulas, p. 6-7 (Explained)
Power Rule applet (Open)
Exponential Function Rule applet (Open)
Trigonometric Derivatives applet (Open)
Derivatives of Exponentials applet (Renault)
Derivatives Cheat Sheet (Paul)
Derivatives videos (JMT) *
Computing Derivatives 1 video (McMullin)
Exponential Functions video (McMullin)
Fractional Exponents, Logs video (McMullin)

Basic rules for derivatives of sums, products, and quotients
Derivatives and Transformations applet (Renault)
Constant Multiple Rule applet (Open)
Sum and Difference Rule applet (Open)
Product and Quotient Rules applet (Open)
Differentiation Formulas (Paul)
Product and Quotient Rules (Paul)
Derivatives of Trig Functions (Paul)
Exponent and Log Derivatives (Paul)
Inverse Trig Derivatives (Paul)
Higher Order Derivatives (Paul)
Derivatives videos (JMT) *
Computing Derivatives 2 video (McMullin)

Chain rule and implicit differentiation
Intuitive Chain Rule applet (Renault)
Implicit Differentiation applet (Renault)
Chain Rule (Paul)
Implicit Differentiation (Paul)
Chain rule and implicit differentiation videos (JMT) *
Chain Rule video (McMullin)
Implicit Differentiation video (McMullin)
Higher Derivatives video (McMullin)

Integrals

Interpretations and properties of definite integrals

Definite integral as a limit of Riemann sums
Area Problem (Paul)
Definition of Definite Integral (Paul)
Introduction to Integration 1 video (McMullin)
Introduction to Integration 3 video (McMullin)

Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: $\int_{a}^{b} f'(x)dx = f(b)-f(a)$ .
Computing Definite Integrals (Paul)
Meaning of a Definite Integral video (McMullin)
Rate and Accumulation video (McMullin)

Basic properties of definite integrals (incl. additivity, linearity)
Properties of Definite Integrals applet (Open)

Applications of integrals

Knowledge, techniques adapted to many applications; emphasis is on setting up approximating Riemann sum and representing its limit as a definite integral
Exercise Bike 1 applet (Renault)
Exercise Bike 2 applet (Renault)

Specific applications include finding the area of a region, volume of a solid with known cross sections, average value of a function, distance traveled by a particle along a line, and accumulated change from a rate of change
Kinematics p. 15 (Explained)
Volumes of Solids p. 20-21 (Explained)
Average Value of a Function applet (Open)
Average Function Value (Paul)
Area Between Curves (Paul)
Areas by Slicing applet (Open)
Volumes of Revolution applet (Open)
Volumes of Solids of Revolution (Paul)
Volumes of Known Cross Sections applet (Open)
Cross-Section Volumes (Paul)
Area Between Two Graphs video (McMullin)
Volumes from Cross-Sections video (McMullin)
Volumes of Revolution video (McMullin)
Average Value of a Function video (McMullin)

Fundamental Theorem of Calculus

Use of the Fundamental Theorem to evaluate definite integrals
FTC Evaluating Integrals applet (Renault)
Integrals videos (JMT) *
Fundamental Theorem of Calculus video (McMullin)

Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis
Area Function applet (Renault)
FTC Derivative of Integral applet (Renault)
Derivative of an Integral applet (Open)
Functions Defined Using Integrals applet (Open)

Techniques of antidifferentiation

Antiderivatives following from derivatives of basic functions
Basic Antiderivatives applet (Open)
Integrals of Trig Functions (Leckie)
Integrals Cheat Sheet (Paul)
Indefinite Integrals (Paul)
Computing Indefinite Integrals (Paul)
Integrals videos (JMT) *
Antiderivatives 1 video (McMullin)

Antiderivatives by substitution of variables, incl. change of limits for definite integrals
u-Substitution p. 19 (Explained)
Integration by Substitution applet (Open)
Substitution with Indefinite Integrals (Paul)
More Substitution Rule (Paul)
Substitution with Definite Integrals (Paul)
u-substitution videos (JMT) *
Antiderivatives u-substitution video (McMullin)
Antiderivatives with Logarithms video (McMullin)

Applications of antidifferentiation

Finding specific antiderivatives using initial conditions, incl. applications to motion along a line
Differential Equations video (McMullin)
Motion Problems video (McMullin)

Solving separable differential equations and using them in modeling. In particular, studying the equation y’ = ky and exponential growth
Solving Differential Equations p. 16 (Explained)
Separation of Variables applet (Open)
Separation of Variables (Leckie)
Growth and Decay video (McMullin)

Numerical approximations to definite integrals

Use of Riemann sums (using left, right, midpoint values) and trapezoidal sums to approximate definite integrals of functions represented algebraically, geometrically, and by tables of values
Gaining Geometric Intuition (Renault)
Riemann Sum applet (Renault)
Numerical Integration Rules applet (Furman)
Approximating Distance from Velocity Table applet (Open)
Approximating Distance from Velocity Graph applet (Open)
Equations of Motion applet (Open)
Riemann Sum applet (Open)
Midpoint and Trapezoidal Rules applet (Open)
LRAM, RRAM, MRAM rectangular approximations (Leckie)
Riemann sum videos (JMT) *
Introduction to Integrals 2 video (McMullin)

*Note: For all videos marked JMT, when you get to the page, use CTRL-F (or Command-F) and type in the topic to locate the particular videos on that topic. There are often several relevant ones.

A Key to the Sources

Australia: Australian Bureau of Statistics, http://www.abs.gov.au/
Bitesize: BBC GCSE Bitesize, http://www.bbc.co.uk/schools/gcsebitesize/
Blyth: Russell Blyth, http://mathmistakes.info/index.html
ChiliMath: Mike Estella, https://www.chilimath.com
CK-12: https://www.ck12.org/student/
EdHelper: http://www.edhelper.com/
Ellermeyer: Sean Ellermeyer, Kennesaw State University, http://math.kennesaw.edu/~sellerme/
ExamSolutions: Exam Solutions: Maths Made Easy, https://www.examsolutions.net/
Explained: Peggy Frisbie, Some Calculus Concepts, Explained
FAQ: Peggy Frisbie, Some Frequently Asked Questions, https://pfrisbie.com/topical-help-resources/calculus-faq/
Furman: Dan Sloughter, Furman University, http://dananne.org/dw/doku.php
IBV: IBVodcasting (YouTube account) http://www.youtube.com/user/ibpodcasting/videos
Investopedia: https://www.investopedia.com
Ishan: Ishan Quicknotes, https://www.youtube.com/channel/UCYfMpRHIqgp1QB1p8KytvcQ
JMT: Patrick Jones, Patrick JMT videos, http://patrickjmt.com/ *
Khan: Khan Academy, https://www.khanacademy.org/
Kuta: Kuta Software, Test and Worksheet Generators for Math Teachers, https://www.kutasoftware.com/
Leckie: Chad Leckie and colleague, Air Academy High School, http://www.chaoticgolf.com/tutorials_calc.html
LibreTexts: https://math.libretexts.org/
LOLA: Learning Objects for Linear Algebra, http://thejuniverse.org/PUBLIC/LinearAlgebra/LOLA/index.html
Math Is Fun: Math Is Fun, http://www.mathsisfun.com/
Math Goodies: Math Goodies, http://www.mathgoodies.com/
Math Planet: Math Planet, http://www.mathplanet.com/
MathBits: Math Bits, http://mathbits.com/
MathsRevision: Matthew Pinckney, http://www.mathsrevision.net/
McMullin: Lin McMullin, National Math + Science Initiative videos , https://vimeo.com/user4579133
Morgan: Stephen L. Morgan, University of South Carolina, http://sc.edu/study/colleges_schools/chemistry_and_biochemistry/our_people/morgan_stephen.php
OnlineStat: Online Statistics Education, http://onlinestatbook.com/
OnMaths: https://www.onmaths.com/
Open: John D. Page, Math Open Reference, http://www.mathopenref.com/
Paul: Paul Dawkins, Paul’s Online Notes, http://tutorial.math.lamar.edu/
Prokup: Norm Prokup, Brightstorm, https://www.brightstorm.com/
Purple: Elizabeth Stapel, PurpleMath, http://www.purplemath.com/modules/index.htm
Rosengarten: Mark Rosengarten, http://www.youtube.com/user/MarkRosengarten
Renault: Marc Renault, Shippensburg University, http://webspace.ship.edu/msrenault/GeoGebraCalculus/GeoGebraCalculusApplets.html
Shmoop: Shmoop, http://www.shmoop.com/
SOS: SOS Mathematics, http://www.sosmath.com/index.html
Sparrow: Adrian Sparrow, http://www.ibmaths.com/
StatCan: Statistics Canada, http://www.statcan.gc.ca/
Stat Trek: Stat Trek, http://stattrek.com/
Study: Study.com, https://study.com/
StudyWell: https://studywell.com/
Tam: https://www.youtube.com/@ssltam
TI Education: Texas Instruments Education YouTube channel, https://www.youtube.com/channel/UCAykbCi_wrR10EskHm71fJw
Warehouse: Math Warehouse https://www.mathwarehouse.com
Weinberg: Evan Weinberg, Gealgerobophysiculus, http://evanweinberg.com/
Wikibooks: https://en.wikibooks.org/wiki/Main_Page
Wills: MathsNZStudents, https://students.mathsnz.com/