Course Outline: Advanced Calculus Honors 2020 - 2021 Course Objectives: 2021 Course ...

Course Outline: Advanced Calculus Honors
2020 - 2021

Course Objectives:
     This course consists of a full high school academic year of work
that is comparable to calculus courses in colleges and universities. It is
expected that students who take this course will seek college credit.
This course explores the “mathematics of change”. The goal of this
class is to have students:
            Work with and understand the connections among functions
            represented graphically, numerically, analytically, and/or
            verbally.
            Understand the meaning of the derivative in terms of a rate
            of change and local linear approximation.
            Use graphs of 1st and 2nd derivatives to approximate the
            graph of f(x) and likewise use the graph of f)x) to
            approximate the graphs of the corresponding derivatives.
            Understand the meaning of the definite integral both as a
            limit of Riemann sums and as the net accumulation of a rate
            of change.
            Understand the relationship between the derivative and the
            definite integral as expressed in both parts of the
            Fundamental Theorem of Calculus.
            Communicate mathematics both orally and in well-written
            sentences and should be able to explain solutions to
            problems, specifically optimization and related rate
            problems.
            Model a written description of a physical situation with a
            function, a differential equation, or an integral and if
            applicable graphs and tables.

Determine the reasonableness of solutions, including sign,
          size, relative accuracy, units of measure and be able to
          explain solutions orally with adequate mathematical
          terminology.

Dual-Credit:
      This course is designed to be a dual-credit course. This allows
students to earn high school credit and college credit at the same
time. Students enrolled in a Dual Credit course take the course at the
high school, and if the student earns a passing grade, he/she then
earns college credit. Students must pay the tuition and fees if they do
not qualify for an ACE scholarship. Dual Credit courses are more
challenging and stimulating. The dual credit classes require energetic,
involved and self-motivated students.

How to Study Math:
          Take good notes.
          Review notes after class.
          Read the text when assigned.
          Get a study buddy.
          Have a scheduled time to do your math homework (as soon
          after class as possible).

The Logic of Problem Solving:
          Read the entire problem to get a general idea.
          Read the problem again, this time answer the following
          three questions
            1) What is given?
            2) What am I asked to find?
            3) How am I going to do it?

Course Topics:
      There is flexibility in the order and time allotted to each of the
topics below, but all topics must be covered by the instructor and
understood by the student.

     Unit 1: “Review of Basics”
           Lines and Functions
           Functions … inverse, greatest integer, piecewise, odd, even
           Trigonometric functions
           Function transformations

     Unit 2: “Limits and Continuity”
          Rates of change and limits, including one-sided limits,
          removable discontinuities, jump discontinuities
          Limits involving infinity, including sandwich theorem,
          horizontal asymptotes, end behavior
          Continuity and differentiable
          Rates of change and tangent lines, average velocity
          Graphical and analytical computation of limits

     Unit 3: “Derivatives”
          Derivative of a function … definition and derivation of
          polynomial, rational, and radical functions
          Differentiability and local linearity, introduction of corners
          and cusps, vertical tangents
          Instantaneous velocity, acceleration and other rates of
          change with an introduction to analyzing graphs of the
          derivative
          Derivatives of trigonometric functions
          Chain Rule for composite functions
          Implicit differentiation

Analyzing derivative graphs and sketching the first
     derivative from the graph of the function

Unit 4: “Applications of Derivatives”
     Finding global and local extrema, analyzing critical points
     Rolle’s Theorem and the Mean Value Theorem
     Connecting f’ and f” with the graph of f(x), the first
     derivative test, concavity, increasing and decreasing
     functions
     Modeling and optimization
     Linearizations and Newton’s Method
     Related rates
     Graph analysis, applications problems

Unit 5: “The Definite Integral”
     Estimating with finite sums
     Definite integrals
     Definite integral rules, antiderivatives and average value of
     a function
     Fundamental Theorem of Calculus and area under a curve
     Trapezoidal Rule
     Volume: disk, washer, shell, cross-sections

Unit 6: “Exponential and Logarithmic Functions”
     Derivatives of exponential and logarithmic functions,
     including logarithmic differentiation
     Applications of exponential growth and decay
     Population growth: logistic growth, carrying capacity, slope
     fields
     Differential equations

Unit 7: “Additional Topics
          Derivatives of inverse trigonometric functions including
          antidifferentiation
          L’Hopital’s Rule and why it works, to use with taking limits
          Relative rates of growth, to help in finding limits as ‘x’
          approaches infinity

Final Thoughts
      Many students get to Precalculus Honors through perseverance
and diligence, while others have really not had to struggle much to get
here. I have discovered that the individuals who fall into the first
group tend to do better in this course (at least, perhaps, initially). Be
advised that regardless of what road or what habits of math study
have brought you to this course, you are about to embark on a
challenging and rewarding journey unlike any you have expected. Much
of the material covered this year, as well as my expectations for you,
will be entirely new to you. Growing pains are to be expected and are
minimized by redoubled efforts, patience, and perseverance. Don’t be
that proud student who disregards this advice. As your teacher, I will
challenge each of you, but I will also provide you with the instruction
and extra assistance you need to rise to the challenge. In the end,
though, I cannot do the work for you, and there is unfortunately no
royal road to mastering the material.

      Success also requires excellent class attendance and an alert,
active, focused, supportive, and courteous engagement in class every
day. Try to come to class rested and ready. When possible read the
next day’s topic in the textbook prior to class. Please try not to miss
class due to other activities --- I’ve seen too many students fall behind
early, never to fully recover.