From Biohack

For all of the notes in one giant swoop, see [1]. These files were taken by me (Bryan) in [http://teacherweb.hayscisd.net/web/spillers/ Spiller's AP Calculus BC class over at Hays High School, mostly in real time. This wiki page is a review of the same content in slightly more formalized syntax, images, etc.

Contents 1 Really, really good resources 2 Lists 2.1 Differentiation rules 2.2 Lists on volume finding in calculus 3 MIT for High School - Calculus BC 3.1 Analysis of Graphs 3.2 Limits of Functions 3.2.1 Intuitive understanding 3.2.2 Algebraic approach to limits 3.3 Asymptotic & Unbounded Behavior 3.4 Continuity: Property of Functions 3.5 Parametric, Polar & Vector Functions / Analysis of Planar Curves 3.6 Concept of the Derivative 3.7 Derivative at a Point 3.8 Derivative as a Function 3.9 Second Derivatives 3.10 Applications of Derivatives 3.10.1 Analysis of Curves 3.10.2 Applications of Derivatives 3.10.3 Optimization: Absolute & Relative Extrema 3.10.4 Modeling Rates of Change 3.10.5 Implicit Differentiation & Derivatives of Inverse Functions 3.10.6 Derivative as a Rate of Change 3.10.7 Geometric Interpretation of Differential Equations 3.10.8 L'Hospital's Rule 3.11 Computation of Derivatives 3.11.1 Derivatives of Basic Functions 3.11.2 Rules: Derivative of Sums, Products & Quotients of Functions 3.11.3 Chain Rule & Implicit Differentiation 3.11.4 Derivatives: Parametric, Polar & Vector Functions 3.12 Interpretations & Properties of Definite Integrals 3.12.1 Definite Integral - Limit of Riemann Sums 3.12.2 Rate of Change of a Quantity/Change of Quantity over Interval 3.12.3 Properties of Definite Integrals 3.13 Applications of Integrals 3.14 Fundamental Theorem of Calculus 3.15 Antidifferentiation Techniques 3.15.1 Antiderivatives From Derivatives of Basic Functions 3.15.2 Integration by Substitution, Parts & Partial Fractions 3.15.3 Improper Integrals 3.16 Applications of Antidifferentiation (separable equations & modeling) 3.17 Numerical Approximations to Definite Integrals (Riemann & Trapezoidal Sums) 3.18 Concept of Series 3.19 Series of Constants 3.19.1 Introduction 3.19.2 Convergence 3.19.2.1 Absolute convergence 3.19.2.2 Ratio test 3.19.2.3 Integral test 3.19.2.4 Limit Comparison 3.19.2.5 Alternating series 3.19.3 Geometric series 3.19.4 Telescoping series 3.19.5 Motivating Examples 3.19.6 Geometric Series with Applications 3.19.7 The Harmonic Series 3.19.8 Alternating Series - Error Bound 3.19.9 Series as Riemann Sums & the Integral Test 3.19.10 Ratio Test - Convergence/Divergence 3.19.11 Comparison Test 3.20 Taylor Series 3.20.1 Taylor Polynomial Approximation 3.20.2 Maclaurin & Taylor Series 3.20.3 Maclaurin Series for Basic Functions 3.20.4 Manipulating Taylor Series 3.20.5 Power Series Defining Functions 3.20.6 Radius/Interval of Convergence 3.20.7 Lagrange Error Bound 4 Calculus BC 4.1 Functions, Graphs, and Limits 4.1.1 Analysis of graphs 4.1.2 Limits of functions (including one-sided limits) 4.1.3 Asymptotic and unbounded behavior 4.1.4 Continuity as a property of functions 4.1.5 Parametric, polar and vector functions 4.2 Derivatives 4.2.1 Concept of the derivative 4.2.2 Table of derivatives 4.2.2.1 General differentiation rules 4.2.2.2 Derivatives of simple functions 4.2.2.3 Derivatives of exponential and logarithmic functions 4.2.2.4 Derivatives of trigonometric functions 4.2.2.5 Derivatives of hyperbolic functions 4.2.3 Derivative at a point 4.2.4 Derivative as a function 4.2.5 Second derivatives 4.2.6 Application of derivatives 4.2.6.1 Analysis of curves 4.2.6.2 Notions of monotonicity and concavity 4.2.6.3 Analaysis of planar curves given in parametric form, polar form, and vector form 4.2.6.4 Optimization for absolute and relative extrema 4.2.6.5 Related rates 4.2.6.6 Modeling rates of change 4.2.6.7 Implicit differentiation (to find the derivative of an inverse function) 4.2.6.7.1 For use with two variables 4.2.6.8 Slope fields and solution curves 4.2.6.9 L'Hôpital's Rule 4.2.7 Computation of derivatives 4.2.7.1 Derivatives of basic functions 4.3 Integrals 4.3.1 Interpretations and properties of definite integrals 4.3.2 Applications of integrals 4.3.3 Fundamental theorem of calculus 4.3.4 Techniques of antidifferentiation 4.3.5 Applications of antidifferentiation 4.3.6 Numerical approximations to deifnite integrals 4.4 Polynomial Approximations and Series 4.4.1 Concept of Series 4.4.2 Series of constants 4.4.3 Taylor Series 4.4.3.1 Definition 4.4.3.2 Examples 4.4.3.3 Convergence 5 Today's Nonsense: extended edition

Really, really good resources

• Karl's Calculus (more for first-year calculus)
• Paul's online math notes
  • [pdf] complete calculus cheat-sheet
  • cal1
  • cal2
  • cal3 - this is more about PDEs or partial differential equations.
• (less so) [2] - calculus java applets for visualization
• youtube vids aren't that bad for (basic) calculus, you can even learn the basics of partial differential equations on youtube. There's also the beginnings of a mathstube out there, so that's going to be interesting, it's a mathcast kind of like a podcast.

Here is a listing and brief description of the material in this set of notes.

Integration Techniques

Integration by Parts[[Image:&%7BDSMP.gEmptySrc%7D;]] Of all the integration techniques covered in this chapter this is probably the one that students are most likely to run into down the road in other classes.

Integrals Involving Trig Functions[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we look at integrating certain products and quotients of trig functions.

Trig Substitutions[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will look using substitutions involving trig functions and how they can be used to simplify certain integrals.

Partial Fractions[[Image:&%7BDSMP.gEmptySrc%7D;]] We will use partial fractions to allow us to do integrals involving some rational functions.

Integrals Involving Roots[[Image:&%7BDSMP.gEmptySrc%7D;]] We will take a look at a substitution that can, on occasion, be used with integrals involving roots.

Integrals Involving Quadratics[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we are going to look at some integrals that involve quadratics.

Using Integral Tables[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we look at using Integral Tables as well as relating new integrals back to integrals that we already know how to do.

Integration Strategy[[Image:&%7BDSMP.gEmptySrc%7D;]] We give a general set of guidelines for determining how to evaluate an integral.

Improper Integrals[[Image:&%7BDSMP.gEmptySrc%7D;]] We will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section.

Comparison Test for Improper Integrals[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will use the Comparison Test to determine if improper integrals converge or diverge.

Approximating Definite Integrals[[Image:&%7BDSMP.gEmptySrc%7D;]] There are many ways to approximate the value of a definite integral. We will look at three of them in this section.

Applications of Integrals

Arc Length[[Image:&%7BDSMP.gEmptySrc%7D;]] We’ll determine the length of a curve in this section.

Surface Area[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we’ll determine the surface area of a solid of revolution.

Center of Mass[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will determine the center of mass or centroid of a thin plate.

Hydrostatic Pressure and Force[[Image:&%7BDSMP.gEmptySrc%7D;]] We’ll determine the hydrostatic pressure and force on a vertical plate submerged in water.

Probability[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will look at probability density functions and computing the mean of a probability density function.

Parametric Equations and Polar Coordinates

Parametric Equations and Curves[[Image:&%7BDSMP.gEmptySrc%7D;]] An introduction to parametric equations and parametric curves (i.e. graphs of parametric equations)

Tangents with Parametric Equations[[Image:&%7BDSMP.gEmptySrc%7D;]] Finding tangent lines to parametric curves.

Area with Parametric Equations[[Image:&%7BDSMP.gEmptySrc%7D;]] Finding the area under a parametric curve.

Arc Length with Parametric Equations[[Image:&%7BDSMP.gEmptySrc%7D;]] Determining the length of a parametric curve.

Surface Area with Parametric Equations[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will determine the surface area of a solid obtained by rotating a parametric curve about an axis.

Polar Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] We’ll introduce polar coordinates in this section. We’ll look at converting between polar coordinates and Cartesian coordinates as well as some basic graphs in polar coordinates.

Tangents with Polar Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] Finding tangent lines of polar curves.

Area with Polar Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] Finding the area enclosed by a polar curve.

Arc Length with Polar Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] Determining the length of a polar curve.

Surface Area with Polar Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will determine the surface area of a solid obtained by rotating a polar curve about an axis.

Arc Length and Surface Area Revisited[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will summarize all the arc length and surface area formulas from the last two chapters.

Sequences and Series

Sequences[[Image:&%7BDSMP.gEmptySrc%7D;]] We will start the chapter off with a brief discussion of sequences. This section will focus on the basic terminology and convergence of sequences

More on Sequences[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will take a quick look about monotonic and bounded sequences.

Series '[[Image:&%7BDSMP.gEmptySrc%7D;]' The Basics][[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will discuss some of the basics of infinite series.

Series '[[Image:&%7BDSMP.gEmptySrc%7D;]' Convergence/Divergence][[Image:&%7BDSMP.gEmptySrc%7D;]] Most of this chapter will be about the convergence/divergence of a series so we will give the basic ideas and definitions in this section.

Series '[[Image:&%7BDSMP.gEmptySrc%7D;]' Special Series][[Image:&%7BDSMP.gEmptySrc%7D;]] We will look at the Geometric Series, Telescoping Series, and Harmonic Series in this section.

Integral Test[[Image:&%7BDSMP.gEmptySrc%7D;]] Using the Integral Test to determine if a series converges or diverges.

Comparison Test/Limit Comparison Test[[Image:&%7BDSMP.gEmptySrc%7D;]] Using the Comparison Test and Limit Comparison Tests to determine if a series converges or diverges.

Alternating Series Test[[Image:&%7BDSMP.gEmptySrc%7D;]] Using the Alternating Series Test to determine if a series converges or diverges.

Absolute Convergence[[Image:&%7BDSMP.gEmptySrc%7D;]] A brief discussion on absolute convergence and how it differs from convergence.

Ratio Test[[Image:&%7BDSMP.gEmptySrc%7D;]] Using the Ratio Test to determine if a series converges or diverges.

Root Test[[Image:&%7BDSMP.gEmptySrc%7D;]] Using the Root Test to determine if a series converges or diverges.

Strategy for Series[[Image:&%7BDSMP.gEmptySrc%7D;]] A set of general guidelines to use when deciding which test to use.

Estimating the Value of a Series[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will look at estimating the value of an infinite series.

Power Series[[Image:&%7BDSMP.gEmptySrc%7D;]] An introduction to power series and some of the basic concepts.

Power Series and Functions[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will start looking at how to find a power series representation of a function.

Taylor Series[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will discuss how to find the Taylor/Maclaurin Series for a function.

Applications of Series[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will take a quick look at a couple of applications of series.

Binomial Series[[Image:&%7BDSMP.gEmptySrc%7D;]] A brief look at binomial series.

Vectors

Vectors '[[Image:&%7BDSMP.gEmptySrc%7D;]' The Basics][[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will introduce some of the basic concepts about vectors.

Vector Arithmetic[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will give the basic arithmetic operations for vectors.

Dot Product[[Image:&%7BDSMP.gEmptySrc%7D;]] We will discuss the dot product in this section as well as an application or two.

Cross Product[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we’ll discuss the cross product and see a quick application.

Three Dimensional Space

This is the only chapter that exists in two places in my notes. When I originally wrote these notes all of these topics were covered in Calculus II however, we have since moved several of them into Calculus III. So, rather than split the chapter up I have kept it in the Calculus II notes and also put a copy in the Calculus III notes.

The 3-D Coordinate System[[Image:&%7BDSMP.gEmptySrc%7D;]] We will introduce the concepts and notation for the three dimensional coordinate system in this section.

Equations of Lines[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will develop the various forms for the equation of lines in three dimensional space.

Equations of Planes[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will develop the equation of a plane.

Quadric Surfaces[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will be looking at some examples of quadric surfaces.

Functions of Several Variables[[Image:&%7BDSMP.gEmptySrc%7D;]] A quick review of some important topics about functions of several variables.

Vector Functions[[Image:&%7BDSMP.gEmptySrc%7D;]] We introduce the concept of vector functions in this section. We concentrate primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well.

Calculus with Vector Functions[[Image:&%7BDSMP.gEmptySrc%7D;]] Here we will take a quick look at limits, derivatives, and integrals with vector functions.

Tangent, Normal and Binormal Vectors[[Image:&%7BDSMP.gEmptySrc%7D;]] We will define the tangent, normal and binormal vectors in this section.

Arc Length with Vector Functions[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will find the arc length of a vector function.

Curvature[[Image:&%7BDSMP.gEmptySrc%7D;]] We will determine the curvature of a function in this section.

Velocity and Acceleration[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will revisit a standard application of derivatives. We will look at the velocity and acceleration of an object whose position function is given by a vector function.

Cylindrical Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] We will define the cylindrical coordinate system in this section. The cylindrical coordinate system is an alternate coordinate system for the three dimensional coordinate system.

Spherical Coordinates[[Image:&%7BDSMP.gEmptySrc%7D;]] In this section we will define the spherical coordinate system. The spherical coordinate system is yet another alternate coordinate system for the three dimensional coordinate system.

Lists

Differentiation rules

See the list of differentiation identities. This is included on this page some ways down, but basically includes linearity, the product rule, reciprocal rule, quotient rule, chain rule, derivation of inverse functions, exponential funcs, logarthmic funcs, trig funcs, and some hyperbolic funcs.

Lists on volume finding in calculus

• Volume, washer, shell method on physicsforums.com (which, btw, just had a merger with SciAm).
• Volume by washer method
• When to use washer, shell or disk
• Cavalieri solids

MIT for High School - Calculus BC

• [3]
• textbook (online)

Analysis of Graphs

• [4] - methods for changing a function to shift it left, right, up, or down. Includes three examples.
• Changing scale - ways to stretch or shrink a function by changing the expression used to define it, with an example.
• Even and odd functions - how to reflect a function across either of the coordinate axes, including definitions for even and odd functions. Rules for the behavior of even and odd functions are given, along with examples.
• Trigonometric functions - graphs of the sine, cosine, and tangent functions, including definitions of periodicity and the general sinusoidal wave, with examples.
• Inverses - reflecting a graph across the line y=x to create an inverse function. Includes examples and discussion of the need to restrict the domain of the inverse function in some cases.
• Complete graph analysis - graphing a function and finding its asymptotes, maxima, minima, inflection points, and regions where the graph is concave up or concave down.
• Practice questions: [5] and [6].

Limits of Functions

Intuitive understanding

• [7] w/ a java applet (meh)
• One-sided limits - definition of right- and left-handed limits, with diagrams and examples.

Algebraic approach to limits

• Some trig limits
• Practice questions: [8] [9] [10] [11]

Asymptotic & Unbounded Behavior

• asymptotes

Continuity: Property of Functions

• continuity & types of discontunities
• [12] - definition of points of discontinuity, including removable, jump, infinite, and essential discontinuities. Includes diagrams and four examples.
• Practice questions: [13] [14] [15]

Parametric, Polar & Vector Functions / Analysis of Planar Curves

• parametric equations - representing Cartesian coordinate curves using explicit and implicit forms. Representing curves using parametric equations which define x and y in terms of a third variable. Includes examples of parametric equations for a circle, ellipse, and projectile fired at an angle.
• implicitization - finding implicit forms for parameterized curves. Uses examples from the previous section of the notes.
• polar coordinate curves - definition, with examples of circles and a horizontal line defined in polar coordinates.
• textbook chapter
• Practice questions: [16] [17] [18] area inside a polar curve [19] [20] [21] [22] [23] [24]
• Java applets: polar plotter, curves in 2D

Concept of the Derivative

• [25], the tangent-line linear approximation, java applet
• instantaneous interpretation
• limit of the difference quotient
• Practice problems: [26] [27] [28] [29] [30] [31] [32] [33]
• differentiability - [34] and examples
• Practice questiosn: [35] piecewise [36] [37] [38]

Derivative at a Point

• local-linear approximation (slope-point form) - [39] [40] [[41] [42] review probs [43] algebraic view of linearization apps
• textbook - [44] function estimation determining inverse funcs
• Practice problems: [45] [46] implicit differentiation [47] w/ hyperbola [48] [49] [50] hawk chasing a mouse [51] [52]
• Java applets - [53] and constant, linear, quadratic and cubic approximations
• instantaneous velocity [54]
• approximate derivatives [55] [56] symmetric approximation extrapolation

Derivative as a Function

• Practice problems - [57] [58]
• first derivative test - increasing, decreasing, non-increasing, and non-decreasing functions are defined. First Derivative Test is explained and an example is given.
• mean-value theorem and geometric consequences - [59] [60] [61] including Rolle's Theorem [62]
• Practice problems - [63] [64] [65] [66]

Second Derivatives

• complete graph analysis - [67] - sketching a graph and finding the maxima, minima, points of inflection, and regions where the graph is concave up and concave down. See also some questions.
• second derivative test for concavity - [68]
• inflection points - [69]

Applications of Derivatives

Analysis of Curves

• first derivative test - [70]
• extremal points
• concavity and the second derivative test
• Practice problems - [71] [72] [73] [74] curve sketching [75]

Applications of Derivatives

• [76] examples from economics, thermodynamics, biology, and geometry.

Optimization: Absolute & Relative Extrema

• applied maxima/minima probs - [77] [78] review probs - problems and answers without full explanation. Finding tangent lines to an ellipse, minimizing surface area of a grain silo, finding the volume of a solid of revolution, computing an antiderivative using trig substitution, and computing an antiderivative using integration by parts. Prior Knowledge: Tangent Lines (section 1 of lecture 2), Max/Min Problems (section 2 of lecture 10), Volume of Solids of Revolution (section 3 of lecture 19), Inverse Substitution (section 3 of lecture 25), Integration by Parts (section 1 of lecture 27)
• textbook - quadratic behavior of critical points, general conditions for maxima and minima, divide and conquer method for finding one-dimensional extremas
• Practice problems - [79] [80] [81] [82] [83] [84] [85] [86] [87] [88]

Modeling Rates of Change

• Related rates - [89] step-by-step guide [90]
• Practice problems - [91] related rates w/ polar coordinates [92] [93] radioactive decay [94] [95] viewing angle for a launching rocket [96] [97]

Implicit Differentiation & Derivatives of Inverse Functions

• inverse functions - [98] [99] [100]
• Practice problems - [101]

Derivative as a Rate of Change

• [102]
• Practice problems - [103] [104] [105]

Geometric Interpretation of Differential Equations

• solving an equation in one variable
• Newton's method
• poor man's Newton
• another linear method
• divide & conquer method
• Java applet re: Newton's method - [106]

L'Hospital's Rule

• indeterminate forms - definition, with explanation that limits leading to indeterminate forms sometimes have values that can be found using calculus. Example of finding the value of a limit that leads to the indeterminate form infinity - infinity.
• L'Hopital's rule
• and for other indeterminate forms
• Practice problems - [107] [108] [109] [110]

Computation of Derivatives

Derivatives of Basic Functions

• various rules - [111] [112] [113] [114] exponentials and logarithms [115] [116] logarithmic differentiation trig funcs trig identities [117] derivatives of other trig funcs inverse trig funcs [118]
• textbook -
  • exponential func
  • properties of trig funcs
  • philosophic implications ?
  • derivatives of basic funcs
• Practice problems - [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136]

Rules: Derivative of Sums, Products & Quotients of Functions

• linearity - [137]
• Liebniz rule / product rule - [138] [139]
• quotient rule - [140]
• logarithmic differentiation - [141]
• differentials - [142]
• textbook - derivatives of combinations of equations
• Practice problems - [143] [144] [145] [146] [147] differentials and indefinite integration

Chain Rule & Implicit Differentiation

• chain rule - [148] [149]
• implicit differentiation - [150] [151]
• textbook - [152]
• continuously compounded interest - [153]
• Practice problems - [154] [155] [156] [157] [158] [159]

Derivatives: Parametric, Polar & Vector Functions

• tangent lines in parametric curves - [160]
• tangent lines for polar curves - [161]

Interpretations & Properties of Definite Integrals

Definite Integral - Limit of Riemann Sums

• problem of areas - Riemann integrals are introduced as a concept using the example of finding the area of a circle from the areas of N-sided polygons inscribed in the circle. Signed area (positive above the x-axis, negative below) is introduced.
• interval partitions - interval partitions are defined, including the concepts of mesh size and fine vs. coarse partitions.
• Riemann sums and one for Riemann sums and exponential functions, [162]
• Riemann integral
• Textbook - [163] [164]
• Practice problems - evaluating a Riemann integral, Riemann sums and integrals, limit definition of integral, definite integrals

Rate of Change of a Quantity/Change of Quantity over Interval

• geometric area & algebraic area - [165], [166]