Topic 1: Limits and Their Properties
1. Rates of Change and Limits: Average and Instantaneous Speed, Definition of Limit, Properties
of Limits, One-Sided and Two-Sided Limits, Sandwich Theorem
2. Limits Involving Infinity: Finite Limits as x
,Infinite Limits as x a,End Behavior
Models, Seeing Limits as x
3. Continuity: Continuity at a Point, Continuous Functions, Algebraic Combinations, Composites,
Intermediate Value Theorem for Continuous Functions
4. Rates of Change and Tangent Lines: Average Rate of Change, Tangent to a Curve, Slope of a
Curve, Normal to a Curve
Topic 2: Derivatives
1. Derivative of a Function: Definition and Notation of a Derivative, Relationship between the
graphs of f and f, Graphing the Derivative from Data, One-Sided Derivatives
2. Differentiability: How f(a)Might Fail to Exist, Differentiability Implies Local Linearity,
Derivatives on a Calculator, Differentiability Implies Continuity, Intermediate Value Theorem
for Derivatives
3. Rules for Differentiation: Positive, Multiples, Sums, and Differences, Products and Quotients,
Negative Integer Power of x, Second and High Order Derivatives
4. Velocity and Other Rates of Change: Instantaneous Rates of Changes, Motion along a Line,
Sensitivity to Change, Derivatives in Economics
5. Derivatives of Trigonometric Functions: Derivative of the Sine, Cosine, and Other Basic Trig.
Functions, Simple Harmonic Motion, Jerk
6. Chain Rule: Derivative of a Composite Function, Outside-Inside Rule, Slopes of Parameterized
Curves, Power Chain Rule
7. Implicit Differentiation: Implicitly Defined Functions, Lenses, Tangents, and Normal Lines,
Derivatives of Higher Order, Rational Powers of Differentiable Functions
8. Derivatives of Inverse Trigonometric Functions: Derivatives of Inverse Functions, Derivative
of the Inverse Trigonometric Functions
9. Derivatives of Exponential and Logarithmic Functions: Derivatives of ex and ax, Derivative
of ln x and loga x, Power Rule for Arbitrary Real Powers
Topic 3: Applications of Derivatives
1. Extreme Values of Functions: Absolute (Global) Extreme Values, Local (Relative) Extreme
Values. Finding Extreme Values
2. Mean Value Theorem: Physical Interpretation, Increasing and Decreasing Functions, Other
Consequences
3. Connecting f and f with the Graph of f: First and Second Test for Local Extrema,
Concavity, Points of Inflection, Learning about Functions from Derivatives
4. Modeling and Optimization: Examples from Mathematics, Business, Industry, and Economics,
Modeling Discrete Phenomena with Differentiable Functions
5. Linearization and Newtons Method: Linear Approximation, Newtons Method, Differentials,
Estimating Change with Differentials, Absolute, Relative, and Percentage of Change, Sensitivity
to Change
6. Related Rates: Related Rate Equations, Solution Strategy, Simulating Related Motion
Topic 4: The Definite Integral
1. Estimating with Finite Sums: Distance Traveled, Rectangular Approximation Method, Volume
of a Sphere, Cardiac Output
2. Definite Integrals: Riemann Sums, Terminology and Notation of Integration, Definite Integral
and Area, Constant Functions, Integrals on a Calculator, Discontinuous Integrable Functions
3. Definite Integrals and Antiderivatives: Properties of Definite Integrals, Average Value of a
Function, Mean Value Theorem for Definite Integrals, Connecting Differential and Integral
Calculus
4. Fundamental Theorem of Calculus: Fundamental Theorem Part I and Part II, Graphing the
x
Function f (t)dt, Area Connection. Analyzing Antidervatives Graphically
a
5. Trapezoidal Rule: Trapezoidal Approximation, Other Algorithms, Error Analysis
Topic 5: Differential Equations and Mathematical Modeling
1. Slope Fields and Eulers Method
2. Antidifferentation by Substitution: Indefinite Integrals, Leibniz Notation and Antiderivatives,
Substitution in Indefinite Integrals
3. Antidifferentation by Parts: Product Rule in Integral Form, Solving for the Unknown Integral,
Tabular Integration, Inverse Trigonometric and Logarithmic Functions
4. Exponential Growth and Decay: Separable Differential Equations, Law of Exponential
Change, Continuously Compound Interest, Radioactivity and Newtons Law of Cooling,
Modeling Growth with Other Bases
5. Logistic Growth: How Population Grow, Partial Fractions, The Logistic Differential Equations,
The Logistic Growth Models
Topic 6: Application of Definite Integrals
1. Integral as Net Change: Consumption Over Time, Net Change from Data, Work
2. Areas in the Plane: Area Between Curves, Area Enclosed by Interesting Curves, Boundaries
with Changing Functions, Integrating with Respect to y, Saving Time with Geometry Formulas
3. Volumes: Volume as an Integral, Square Cross Sections, Circular Cross Sections, Cylindrical
Shells, Other Cross Sections
4. Lengths of Curves: A Sine Wave, Length of a Smooth Curve, Vertical Tangents, Corners, and
Cusps
5. Applications from Science and Statistics: Work, Fluid Force and Fluid Pressure, Normal
Probabilities
Topic 7: Sequences, LHospitals Rule, and Improper Integrals
1. Sequences: Defining a Sequence, Arithmetic and Geometric Sequences, Graphing a Sequence,
Limit of a Sequence
2. LHospitals Rule: Indeterminate Form
0
Indeterminate Forms 1 ,00,
0, and
, Indeterminate Forms
3. Relative Rates of Growth: Comparing Rates of Growth, Using LHospitals Rule to Compare
Growth Rates, Sequential versus Binary Search
4. Improper Integrals: Infinite Limits of Integration, Integrands with Infinite Discontinuities, Test
for Convergence and Divergence, Applications
Topic 8: Infinite Series
1. Power Series: Geometric Series, Representing Functions by Series, Differentiation and
Integration, Identifying Series
2. Taylor Series: Constructing a Series, Series for sinx and cosx,Maclaurin and Taylor Series,
Combining Taylor Series, Table of Maclaurin Series
3. Taylors Theorem: Taylor Polynomials, The Remainder, Remainder Estimation Theorem,
Eulers Formula
4. Radius of Convergence: Convergence, nth Test, Comparing Nonnegative Series, Ratio Test,
Endpoint Convergence
5. Testing Convergence at Endpoints: Integral Test, Harmonic Series and p-series, Comparison
Tests, Alternating Series, Absolute and Conditional Convergence, Intervals of Convergence
Topic 9: Parametric, Vector, and Polar Functions
1. Parametric Functions: Parametric Curves in the Plane, Slope and Concavity, Arc Length,
Cycloids
2. Vectors in the Plane: Two-Dimensional Vectors, Vector Operations, Modeling Planar Motion,
Velocity, Acceleration, and Speed, Displacement and Distance Traveled
3. Polar Functions: Polar Coordinates, Polar Curves, Slopes of Polar Curves, Areas Enclosed by
Polar Curves, A Small Polar Gallery
