A. P. Calculus - BC Course Outline

I Preparation for the course 1. We start preparing students for AP BC Calculus while they are in Pre-Calculus H. The students learn Mathematics the best by solving interesting, thought-provoking problems. The problem-solving approach is implemented in my classroom. 2. Students always have cumulative assessments rather than chapter by chapter tests. 3. Comprehensive Summer Take-Home Assignment is given to the future Calculus students to assure their readiness for BC Calculus. This assignment covers the prerequisites for learning calculus. This allows us to begin the course without the necessity for review. The summer assignment also requires a certain level of commitment from the students. During the first week of school in September, the students will have to take a test based on the summer assignment. II Course Philosophy and Classroom Dynamic Students learn and retain what they learn most effectively when they understand the "why" and "how" behind what they are studying. The problem-solving approach is implemented in my classroom. My students are at the white boards before I come to class. The students put out all questions, concerns, and solutions on a board. We discuss problems in several different ways and choose the most effective ways of solving them. When students do the problems on their own they are in charge of their knowledge. Each new concept presented in class is discussed using analytical, graphical, and numerical approaches. The students learn to use a tabular representation. They also have to verbalize what does the solution mean. III Technology component Technology is woven into the calculus curriculum. We use a TI-83/84 calculator. Students know how to graph functions, their derivatives, find the intersection of two curves, evaluate derivatives and integrals as a result of our every day calculator activities. Graphing Calculator software developed by Pacific Tech (2002) is used to graph polar and parametric equations, as well functions in 3 variables. The latter we do after the AP Exam. Calculus in Motion by Audrey Weeks is used to demonstrate volumes of revolution, area problems and other concepts. Technology removes the necessity for tedious calculation and table manipulation, allowing more time to develop concepts and interpret results. IV Assessments (a) Quizzes/ pop-Quizzes; (b) Tests (calculator and non-calculator parts); (c) Projects after AP Exam (usually in pairs); (d) Partner quizzes or tests with a Take-Home Part; (e) Working at a white board (I assess it too). (f) Take-Home Tests

Quizzes and tests are all cumulative. Students expect that all topics covered before a test could be given on the test. On each give test, there are two relatively equal parts – a calculator part and a non-calculator part. Starting at the end of October, I give Take-Home Tests every 10 days. There are 5-6 old AP problems in each Take-Home Test. On all oral or written assignments the students are asked to use precise language in describing their answer, and always justify their answer. V Topics after AP Exam 1. Hyperbolic Functions; 2. Parametrics in Physics World: Three-Dimensional Approach; 3. Vectors in 3-D; 4. Second-Order Linear Homogeneous Differential Equations; 5. As a culmination of my students’ learning of Calculus, they do the final project using their knowledge of Calculus. They choose a topic that should include Calculus. Among the topics were History of Calculus, Models that demonstrate the Volumes of revolutions, Volumes by Discs and Washers, TI-84 Calculator programs designed to improve Numerical evaluations of Integrals (in particular, using left, right, midpoint, and trapezoidal and Simpson’s approximations). VI Course texts: Main Textbook: Calculus by Anton, Bivens, Davis; Seventh edition. Other sources: 1. Instructor’s Resourse CD-ROM, Single and Multivariable Calculus by

Hughes-Hallett, Gleason, McCallum et al.,Third Edition; 2. Mathematics 4C, Math Department, Phillips Exeter Academy; 3. Multiple-Choice & Free-Response Questions in Preparation for the AP Calculus BC Examination, David Lederman; 4. Solutions, AP Calculus Problems Part II AB and BC 1987 – 2001 J. Broadwin, G. Lenchner, and M. Rudolph; 5. All Free-Response Questions from the recent years (2002 – 2006); 6. Calculus in Motion by Audrey Weeks (software); 7. Graphing Calculator Version 2 by Pacific Tech. VII AP Calculus Course Outline Unit 1 Review of Functions (Week 1) The students receive the comprehensive Summer Assignment before the course starts and turn in the Take –Home Test on the first day of school. They also take the in-school test to show the mastery of the functions described below. 1. Linear, Quadratic, Logarithmic, Exponential, Polynomial, Rational, Six Trigonometric,

Absolute Value, Radical Functions and their graphs are reviewed; 2. Inverse Functions and Their Graphs; 3. Inverse Trigonometric Functions and Their Graphs; 4. Transformations, Composition of Functions; 5. Parametric Equations and Graphs. 6. Vectors in plane. Students study vectors in Pre-Calculus class and review the properties of vectors at the beginning of the year. We make connection between parametric equations and vectors, such as: if the position of a particle in the plane is x(t) = 2+3t and y(t) = -1 + t, then the displacement vector can be represented as [2 + 3t, -1 + t], or r = (2 + 3t)i + (-1 + t)j with the velocity vector to be [3,1]. Later in the course (Unit 5 - parametric functions revisited and Unit 10 - tangent lines and arc length for parametric curves), we discuss more deeply how we represent a position, velocity and acceleration vectors (in plane) in general. Unit 2 Limits and Continuity (Week 2 - Week 4) 1. Intuitive Approach to Limits; 2. One-sided Limits; 3. Infinite Limits and Vertical Asymptotes; 4. Limits at Infinity and Horizontal Asymptote; End Behavior of a Function; 5. Techniques of Computing Limits: (a) Properties of Limits; Some Basic Limits;

(b) Limits of Polynomial and rational Functions as x! a ;

(c) Indeterminate forms: 0

0, !

!, ! " ! , 0! ;

(d) Limits Involving Radicals; (e) Limits of Piecewise-Defined Functions; (f) Limits of Polynomials as x!" ; (g) Limits of Rational Functions as x!" . 6. Limits discussed rigorously with the formal definition and "epsilon and delta" notation; 7. Continuity:

(a) Definition of Continuity; (b) Continuity on an interval; (c) Continuity of Polynomials; (d) Properties of Continuous Functions; (e) Continuity of Rational Functions; (f) Continuity of Compositions; (g) Continuity on a closed interval [a,b]; (h) The Intermediate Value Theorem. (i) Continuity of Trigonometric Functions;

8. The Squeezing Theorem.

Unit 3 The derivative (Week 5 – Week 9) 1. Secant Lines, Tangent Lines, Velocity, Slopes, Rate of Change; 2. Average Rate of Change and Instantaneous Rate of Change; 3. The Derivative Function:

(a) Difference Quotient; (b) Slope of a Curve and Tangent Lines; (c) Definition of the derivative of f at x= x

0;

(d) Tangent Line to the graph of f at the point ( x0

,P( x0

)); (e) Interpretation of the Derivative; (f) Differentiability (corner, vertical tangent line, and

discontinuity); (g) Differentiability and Continuity; (h) Different Notations for the Derivative.

4. Techniques of Differentiation: we prove most of the rules of differentiation (a) Derivative of a constant; (b) Derivative of a constant times a function; (c) The Power rule; (d) Derivative of Sums and Difference;

(e) The Derivatives of a Product; (f) The Derivatives of a Quotient; (f) Higher Derivatives.

6. Derivatives of Trigonometric Functions; 7. The Chain Rule; 8. Related Rates; 9. Local Linear Approximation, Differentials; 10. Implicit Differentiation; 11. Derivatives of Logarithmic and Exponential Functions; 12. Derivatives of Inverse Trigonometric Functions. The following is a an example of class notes with additional exercises for better understanding a concept of the derivatives Class Notes

1. Average Velocity (geometric interpretation) 3.1.1 2. Instantaneous Velocity see 3.1.2

3. Average rate of change (over an interval) vs. instantaneous rate of change at a

point

rave =f (x) ! f (x

0)

x ! x0

or rave =f (x + h) ! f (x)

h

this is a slope of secant line between two points on a graph (x, f(x)) and (

�

x0, f (x

0) )

rinst =x!x0

limf (x) " f (x0 )

x " x0or rinst =

x!hlim

f (x + h) " f (x)

h

this is the slope of the tangent line at the point (x0, f (x

0)) .

We will do ##2,3,6,8,10 12 on page 176 in class.

4. Difference quotient, symmetric difference quotient

Difference Quotient f (x + h) ! f (x)

h

Symmetric Difference Quotient f (x + h) ! f (x ! h)

2h

5. Derivative of f with respect to x, notation: y / , dy

dx, df (x)

dx, f

/(x)

6. In class ##6,8,26,28,30,44 of section 3.1.

7. Tangent line to the graph of f at a point x=

�

x0

Using the definition of the derivative f'(x

0) =

f (x) ! f (x0)

x ! x0

, we write

the equation of the tangent line to the graph of f at the point (x0, f (x

0)) as

y = f (xo ) + f/(x

0)(x ! x

0) .

Use a calculator to ‘linearize’ the curve of

�

y = x2 at the point x=2

Zoom in several times until you have a locally linear graph, then Evaluate the slope. For instance, we have two points (2.0345745, 4.1394933) and (1.9780585, 3.9127155), and the slope is m=4.01263 8. Differentiability a)

�

f (x0) has to exist;

b)

�

x!x0

limf (x) " f (x0)

x " x0has to exist.

If f is differentiable at every point

�

x ! (a,b) , then we say that f is differentiable on (a, b).

9. A function has to be locally linear in the neighborhood of

�

x = x0 to be

differentiable at

�

x = x0.

10. A function is not differentiable at a) a point of vertical tangency; See

Ex.:

�

y = x

1

3 b) corner; See Ex: y=|x| c) at a point of discontinuity (obvious?!)

11. Derivative<0 corresponds to a decreasing function;

Derivative>0 corresponds to an increasing function; Derivative=0 max or min of a function, or an inflection point.

12. Ex.:

a) f (x) =| x | +1

b) g(x) = x2+ .0001 + .99

(c) Graph both functions f and g in a standard square viewing window, then zoom in. d) Which function is differentiable?

13. Differentiability implies continuity (the reverse is not true), proof.

14. Ex.: Let

�

f (x) =3! x, x <1

ax2

+ bx, x "1

#

$

%

a. If the function is continuous for all x, what is the relationship between a and b?

b. Find the unique values for a and b that will make f both continuous and differentiable.

Unit 4 Derivatives in Graphing and Applications (Week 10 – Week 11) 1. Analysis of Functions: Increase, Decrease, Concavity, Relative Extrema; Graphing Polynomials; 2. More on Curve Sketching: Rational Functions; Curves with Asymptotes, Cusps, Corners. Vertical Tangent Lines; 3. Absolute Maxima and Minima 4. Applied Maximum and Minimum Problems; Optimization Problems; 5. Newton's Method 6. Rolle's Theorem; Mean-Value Theorem 7. Rectilinear Motion: Position, Velocity, Acceleration Functions and their Graphs

Unit 5 Integration (Week 12 – Week 15) 1. Overview of the Area problem 2. The Indefinite Integral 3. Integration by Substitution 4 Sigma Notation. The Definition of the Area as a Limit; 5. Numerical approximations of areas (a) Left endpoint approximation; (b) Right endpoint approximation; (c) Midpoint approximation; (d) Trapezoidal approximation; 6. The Definite Integral

7. The Fundamental Theorem of Calculus and its Applications: (a) Distance as the definite integral of speed; (b) Rectilinear Motion using Integration (see examples 1,3 below); (c) Problems using the integral of a rate of change to determine total or accumulated change (see example 1&2 below) (d) Average Value of a Function (see example 3 below). 8. Parametric Equations and Vector functions revisited (see the examples below). At this point in time, we discuss the motion of a particle in plane (extending it to space, too). A position function is introduced parametrically as x(t) = f(t), and y(t) = g(t) and by using a vector notation r(t) = [f(t), g(t)]. The instantaneous velocity and acceleration of the particle at time t are given as r / (t) = v(t) = [f / (t), g / (t)] and r / / (t) = a(t) = [f / / (t), g / / (t)], respectively. 9. Length of the Plane Curve: (a) Arc length L of the curve y = f(x) over the interval [a,b];

(b) Arc length L of the curve x = f(y) over the interval [c,d];

(c) Arc length of parametric curves. (d) Finding Arc length by numerical methods. As a good example of Vector Functions, I discuss Question 1 of AP Calculus BC (2004, Form B). Another example that I always use is as follows: The position of a particle at any time t ≥ 0 is given by a vector function

r = ( t 2 ! 3 )i + (2

3t3 )j

(a) Find the magnitude of the velocity vector at t = 4. (b) Find the acceleration vector for t = 4. (c) Find the total distance traveled by the particle from t = 0 to t = 4. (d) Write an equation for the line tangent to the curve at t = 4.

Below are the examples of the class work that the students have to do in groups for 10-15 minutes. Ex 1 BC Calculus Class Work (working an groups) Names:

Ex 2 BC Calculus Class work (in pairs) Names: Definite Integral of a Rate of Change.

2.

Ex 3 BC Calculus Short Quiz Name: Average Value of a Function 1.

2.

3. Suppose that the position function of a particle moving along a coordinate line is s(t) = 6t 2 + t . Find the average velocity over the time interval 1 ≤ t ≤ 4. Unit 6 More Applications of the Definite Integral (Week 16 – Week 17) 1. Area between two curves: (a) Area between y = f(x) and y = g(x) (b) Area between x = v(y) and x = w(y) 2. Volumes by slicing: (a) Solids of Revolution; (b) Volumes by disks perpendicular to the x-axis; (c) Volumes by washers perpendicular to the x-axis; (d) Volumes by disks and washers perpendicular to the y-axis; 3. Volumes by cylindrical Shells (a) Volume by Cylindrical Shells about the y-axis; (b) Volume by Cylindrical Shells about the x-axis; 5. Area of a Surface of Revolution; 6. Work. The Role of Work in Physics and Engineering.

Unit 7 Advanced Integration (Week 18 – Week 19) 1. Integration by Parts; 2. Trigonometric Integrals; 3. Trigonometric Substitutions; 4. Integrating Rational Function by Using Partial Fractions Decomposition Technique; 5. Numerical Integration; Simpson's Rule; Error Estimates; 6. Improper Integrals

(a) Improper Integrals with infinite intervals of integration; (b) Improper integrals with infinite discontinuities in the interval

of integration; (c) Improper integrals with infinite discontinuity and infinite

intervals of integration. Unit 8 Differential Equations (Week 20 – Week 22) 1. Classification of Differential Equations; 2. Solving First -Order Differential Equation Using Separation of Variables Technique; 3. Solving First-Order Differential Equations Using the Method of Integrating Factors; 4. Solving Simple Second-Order Differential Equations; 5. Direction Fields; Euler's Method; 6. Modeling with First-Order Differential Equations:

(a) Exponential Growth and Decay Models; (b) Doubling Time and a Half-Life; (c) Logistic Model.

Unit 9 Infinite Series (Week 23 – Week 28) 1. Sequences

(a) Sigma Notation revisited; (b) Limit of a Sequence; (c) Properties of Limits; (d) The Squeezing Theorem for Sequences; (e) Sequence defined recursively; (f) Monotone Sequences;

2. Infinite Series (a) Sigma notation; (b) Divergence Test; (c) Convergence Tests: Integral, Ratio, Root, Comparison, Limit Comparison Tests; We abbreviate it for a better remembering as I (Integral) Really (Ratio) Really (Root) Love (Limit Comparison) Calculus (Comparison)

(d) Alternate Series; Conditional Convergence, Absolute Convergence (e) McLaurin and Taylor Series; (f) Functions defined by Power Series; (g) Approximating Trigonometric, Logarithmic, Exponential, and other functions; (f) Convergence of Taylor Series; (g) Differentiating and Integrating of series to determine new series; (h) Interval and Radius of Convergence; (i) Error bounds - convergent geometric series; - convergent alternating series; - Lagrange remainder.

Example of one of the assessments (quiz) for this chapter is below BC Calculus Quiz on section 10.1 1.

2.

3.

4.

Unit 10 Analytic Geometry in Calculus (Week 29 – Week 30) 1. Polar Coordinates Revisited 2. Parametric Equations and Vectors Revisited 3. Tangent Lines and Arc Length for Parametric and Polar Curves 4. Area in Polar Coordinates Review for the Advanced Placement (week 31 to week 33) Comprehensive review of all topics necessary for the Advanced Placement exam; Practice using questions from old AP exams; Taking an old AP exam for three hours on a Saturday morning and analyzing the results.