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 real-world problems, given written descriptions, data, and graphs. We will explain and interpret our answers in the context of our real-world 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 in-depth 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. x²+y²=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 non-linear 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 re-interpret min-max problems in the context of the real world.

Anti-derivatives

·    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.

·    Anti-derivatives following directly from derivatives of basic functions.

·    Finding specific anti-derivatives 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 anti-derivative 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 anti-derivatives 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

·    Anti-derivatives 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 cross-sections.

·    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 real-world 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?