Chapter 1  Limits
and Rates of Change


The Tangent and
Velocity Problem
·
Intuitive understanding of the limiting
process.
·
The tangent viewed as a limit of secant lines.
·
Average vs. instantaneous velocity and local
linearity.

We will analyze and discuss what information the tangent
to a curve gives us about a function.

The Limit of a
Function
·
Estimating limits from graphs and tables.
·
Understanding asymptotes and describing
asymptotic behavior in terms of limits involving infinity.
·
Using computing devices and limits.

We will use calculators and Excel to help us understand
the concept of a limit and to estimate limiting values for functions.

The Limit Laws
·
Calculating limits using limit laws and
algebra.
·
Using algebraic and graphical techniques to
explore where limits don't exist.

We will learn how to use the limit laws to determine exact
values of limits analytically using algebra.

Continuity
·
Understanding continuity intuitively and in
terms of limits.
·
The Intermediate Value Theorem and geometric
interpretation of graphs of continuous functions.

You will understand the concept of continuity, and its formal mathematical definition.

Tangents,
Velocities and Other Rates of Change
·
The derivative defined as the limit of a
difference quotient.
·
Instantaneous rate of change defined as the
limit of the average rate of change.

Using our understanding of limits, we will revisit the
concept of a tangent line, and rigorously define it in mathematical language.

Chapter 2  Derivatives


Derivatives:
·
Derivatives presented geometrically,
numerically, and analytically.
·
Derivatives interpreted as an instantaneous
rate of change, and defined as the limit of a difference quotient.
·
Slope of a curve at a point, vertical
tangents, no tangents.
·
Tangent lines to curves and local linear
approximations.

We will extend the concept of tangent lines into the
calculus concept of a derivative.

Derivatives as
Functions
·
Relationship between differentiability and
continuity.
·
Corresponding characteristics of f and f
'.

Armed with the definition of the derivative, we can now
discuss what the derivative of a function can tell us about the function,
using graphs, values, and equations.

Differentiation
Formulas:
·
Derivatives of power functions, basic rules
for the derivative of sums, products, and quotients of functions.

We will discover mathematical patterns that help us find
derivatives quickly.

Rates of Change in
the Natural and Social Sciences:
·
Instantaneous rate of change as the limit of
average rate of change.
·
Equations involving derivatives, and verbal
descriptions translated into equations involving derivatives (and vice
versa).
·
Interpretation of the derivative as a rate of
change in varied applied contexts, including velocity, speed, and
acceleration.

We can now apply the concept of derivatives to solving
realworld problems, given written descriptions, data, and graphs. We will
explain and interpret our answers in the context of our realworld problems.

Derivatives of
Trigonometric Functions
·
Derivatives of basic trig functions, such as sin(x),
cos(x), and tan(x).

We will learn how to find the derivatives of cyclical or
periodic functions. We will analyze the functions and their derivatives
graphically and analytically.

The Chain Rule
·
Derivatives of compound functions.

Functions in nature are often complex. We will complete an
indepth exploration using Excel’s numerical and graphical features to derive
techniques for finding derivatives of compound functions.

Implicit
Differentiation
·
The concept of implicit functions and implicit
curves.
·
The technique of implicit differentiation.

Many functions and models cannot be express in y=f(x) form. Some models may not even
be functions (e.g. x^{2}+y^{2}=8).
We will learn how to find derivatives of implicit functions.

Higher Derivatives
·
The derivative of a derivative.
·
Corresponding characteristics of the graphs of
f, f', and, f".
·
Interpretation of the derivative as a rate of
change in varied applied contexts.

What is the derivative of a derivative? What can it tell
us about the function?

Related Rates
·
Modeling rates of change, including related
rates problems, numerically, graphically, and analytically.
·
Verbal descriptions are translated into
equations involving derivatives and vice versa.
·
The value of careful diagrams and good
notation.

Nature is dynamic. Inputs to functions can change with
time. How do changing inputs affect the output of a function and how does the
derivative help us to analyze rates of change in the natural world?

Linear Approximations
and Differentials
·
Tangent line to a curve and local linear
approximation.

We have very powerful tools to analyze linear functions.
How can they be used to approximate and analyze nonlinear models? We will
see how linear approximations are especially powerful when used with
technology.

Chapter 3  Applications
of Differentiation


Maximum and Minimum
Values
·
Geometric understanding of graphs of
continuous functions.
·
Optimization, both global, and relative
extrema.
·
Critical values and closed interval method.
Extreme Value Theorem, and Fermat's Theorem.

We learn how to analyze graphs of functions in great detail,
and interpret the results. We’ll learn how calculus lets us determine exactly
where graphs reach minimum and maximum values, and why these values are
important.

The Mean Value
Theorem
·
The MVT and its geometric consequences.
·
Rolle's Theorem, what it says, and when it's
applicable.
·
How to obtain information about a function
from its derivative.

We’ll learn how to determine the average rate of change of
a function using calculus, and what it means numerically and graphically.

How Derivatives
Affect the Shape of a Graph
·
Relationship between increasing and decreasing
behavior of f and the sign of f'.
·
Relationship between the concavity of f and
sign of f". Points of Inflection as places where concavity changes.
·
Analysis of curves, including monotonicity and
concavity.
·
Optimization.

We’ll explore how the shape of a graph and its rate of
growth or decay are related. We’ll learn how to do this by just looking at
the graph, and find out how to exactly determine where the rate of change
itself changes.

Limits at Infinity
 Horizontal Asymptotes
·
Understanding asymptotes in terms of graphical
behavior.
·
Describing asymptotic behavior in terms of
limits involving infinity.

We’ll study how functions behave as both the dependent and
independent variables approach infinity.

Graphing with
Calculus and Calculators
·
With the aid of technology, graphs of
functions are easy to produce.
·
The emphasis on the interplay between
geometric and analytic information and the use of calculus both predict and
explain the local and global behavior of a function.
·
The use of graphing calculators for estimation
of local extrema and inflection points, contrasted with calculus for precise
computation of such points.
·
The need for special care when using graphing
technology.

We’ll learn how to use technology with calculus to find
solutions to interesting problems. We’ll learn how to use technology appropriately,
and understand common problems we can run into if we simply rely on the
machine for everything.

Optimization
Problems
·
Setting up and solving optimization problems.

Having studied maxima and minima of functions and graphs,
we’ll now apply these skills to finding the optimum solutions to real world
problems. We will reinterpret minmax problems in the context of the real
world.

Antiderivatives
·
Corresponding characteristics of the graphs of
f, f', and, f".
·
Geometric interpretation of differential
equations via slope fields and the relationship between slope fields and
solution curves for differential equations.
·
Antiderivatives following directly from
derivatives of basic functions.
·
Finding specific antiderivatives using
initial conditions, including applications to motion along a line.

If we know the how a function changes (i.e. its
derivative) how can we find the function itself? Why would we want to? How
can we do it graphically, numerically, and analytically? What do the phrases
“family of curves” and “family of solutions” mean in the context of calculus?

Chapter 4  Integrals


Areas and Distances
·
Computation of Riemann sums using left, right,
and midpoint evaluation.
·
Using the integral of a rate of change to find
accumulated change.

If we know how something is changing, how can we find out
how much the function has change in total?

The Definite
Integral
·
Definite integral as limit of Riemann sums
over equal subdivisions.
·
Basic properties of definite integral, such as
additivity, and linearity.
·
Use of Riemann sums to approximate definite
integrals of functions represented algebraically, geometrically, and by
tables of values.

Just as we used the limit to precisely study and define
the derivative, we’ll now use it to precisely define the integral – a
function that tells us how much something has changed.
Get ready to add an infinite number of numbers!

The Fundamental
Theorem of Calculus
·
The FTC, using the integral of a rate of
change to give accumulated change. Use of FTC to evaluate definite integrals.
·
Use of FTC to represent a particular
antiderivative and the analytical and graphical analysis of functions so
defined.

How are the derivative and the integral related?

Indefinite
Integrals and the Net Change Theorem
·
Finding specific antiderivatives using
initial conditions, including applications to motion along a line.
·
Interpretation of FTC2 as the total change of
a rate function.
·
Representation of an indefinite integral as a
family of functions.

Using integrals, we can now find out exactly how much a
physical system has changed over time. We’ll see how the initial values of
variables describing a system can impact how the system changes over time.

The Substitution
Rule
·
Antiderivatives by substitution of variables,
including change of limits for definite integrals.
·
The two methods of using substitution to
compute definite integrals.

Real world functions are often complicated. We will learn
how to simplify functions using analytical substitution, and see how this
impacts the system graphically.

Chapter 5  Applications
of Integration


More About Areas
·
Appropriate integrals are used in a variety of
applications to model physical, social, or economic situations.
·
Specific application should include finding
the area of a region.

The area under a curve and between two curves can be
interpreted in many powerful ways. We’ll learn how to apply this simple area
concept to very different real world situations.

Volume
·
Finding the volume of a solid with known
crosssections.
·
Volumes of solids of revolution.

We’ll learn how to find the volumes of complex shapes using
calculus. These shapes can often times be hard to pictures, so we’ll build
realworld models, and 3D computer models to better understand them.

Volumes by
Cylindrical Shells
·
The method of cylindrical shells to compute
volumes of revolution.
·
Comparisons between the shell and washer
methods.

We’ll use the natural symmetries of rotation to find the
volumes of even more solids.

The Average Value
of a Function
·
Finding the average value of a function and
the Mean Value Theorem.
·
A geometric interpretation of the MVT for
integrals.

We know how to find the average rate of change of a
function. What does it mean to find the average value of a function?

Chapter 6  Logarithmic
and Exponential Functions


Inverse Functions
·
The use of multiple representations (verbal,
numeric, visual, algebraic) to understand inverse functions, always returning
to the central idea of reversing inputs and outputs.
·
Use of implicit differentiation to find the
derivative of an inverse function.

Functions relate inputs to outputs. What if we’re given
the output and we need to find the input? Most functions have inverse functions, and we’ll learn how
inverse functions are treated in calculus.

Exponential
Functions and Their Derivatives
·
Comparing relative magnitudes of functions,
their rates of change (e.g. exponential vs. polynomial vs. logarithmic).
·
Algebraic and geometric properties of
exponential functions.
·
Translation and reflection of exponential
functions, from both symbolic and geometric perspectives.
·
Derivatives of exponential functions. The
definition of e.

Exponential functions are everywhere in nature, from
biology, to nuclear physics. We need to understand their behavior, and how to
study the rate of change of exponential functions.

Logarithmic
Functions
·
Comparing magnitudes of functions, their rates
of change.
·
Logarithmic functions and their properties,
including their geometric properties as inverses of exponentials.
·
Graphs of logarithmic functions, including
asymptotic behavior.

The logarithm is the inverse of the exponential. What are
logarithmic functions, and why are they useful?

Derivatives of
Logarithmic Functions
·
Derivatives of logarithmic functions.

We will discover the huge importance of the derivative of
logarithm functions.

Chapter 9  Differential
Equations


Modeling with
Differential Equations
·
Equations involving derivatives.
·
Verbal descriptions are translated into
equations involving derivatives and vice versa.
·
Contrasting the solution of an Initial Value
Problem with the general solution of a differential equation.
·
Solving logistic differential equations and
using them in modeling.

It turns out that natural world can be described using
equations using derivatives. How do we interpret these differential
equations?

Direction Fields
and Euler's Method
·
Geometric interpretation of DiffEQs via slope
fields and the relationship between slope fields and solution curves of
DiffEQs.
·
Deriving Euler's method, using linear
approximations, and the care that must be taken in its use (appropriate step
size, interpreting results, etc.)

How can we graphically analyze and interpret differential
equations? How can we study them numerically using technology?

Separable Equations
·
Solving separable DiffEQs and using them in
modeling.
·
In particular, studying the equation and
exponential growth.
·
The importance of the constant of integration.

How do we find the exact solution to a differential
equation analytically?

Exponential Growth
and Decay
·
The equation y'=ky and exponential growth.
·
The importance of the sign of the growth
constant k.

What does exponential growth/decay mean? What are examples
of exponential growth and decay from nature?
