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# Calculus

Assessment of the students is based primarily on test grades and
quizzes; homework and class

participation are also factored into all grades. The assessment for
participation is based mostly

on the students responses to questions in class, not just in responding to
one-word

response questions, but, more importantly, to questions that require a detailed
explanation of

mathematical concepts that were taught in class. Also, most, but not all,
quizzes and tests

require the students to explain in writing concepts and theorems that were
taught in class. For

example, on one exam, the students had to explain whether a Riemann sum
approximation

underestimated or overestimated the exact value of a definite integral. On
another, they had to

express the mean value theorem in writing. Finally, many homework, class work,
and review

problems required the student to offer written explanation and/or justification
of their answers

in writing, some of which are indicated below.

## SYLLABUS

Unit I) Precalculus Review--1 week

1) The equation of a straight line in point/slope and slope/intercept form

2) Review of functions including domain and range of functions

3) Composition of functions

4) Odd and even functions/ Symmetries

5) Asymptotes

Unit II) Limits and Continuity--3 weeks

1) Introduction to limits—What is a limit, the lim x->0 (1+x)^1/x.

2) Analytical approach to solving limits

3) Limits involving infinity

4) One-sided limits

5) Simple trigonometric limits

6) More Trigonometric Limits— Lim x->0 Sin(x)/x (Using the graphing calculator)

7) Horizontal Asymptotes

8) Continuity

9) Intermediate Value Theorem

Unit III) The Derivative3

weeks

1) Intro to the slope of a line tangent to curve vs. the slope secant line that
passes through a curve

2) Definition of the derivative lim h->0 [f(x+h)-f(x)]/h

3) Equation of a tangent line

4) Alternative definitions of the derivative including the symmetric difference
quotient

5) Power rule

6) Product rule

7) Quotient rule

8) Chain rule

9) Higher Order derivatives

10)Numerical derivatives using the graphing calculator

11) Implicit differentiation

12) Local linearity

Unit IV) Related Rates--1 week

Unit V) Applications of the derivative--3 weeks

1) Intervals on which a function is decreasing or increasing

2) 1^{st} derivative test

3) Concavity

4) 2^{nd} derivative test

5) Sketching the graph of a function given the graph of its derivative

6) Rolle’s Theorem

7) The Mean Value Theorem

8) Position, velocity and acceleration

9) Optimization

Unit VI) Antidifferentiation--4—5 weeks

1) Indefinite integration

2) Power Rule

3) Integrals in the form f(x)^ n dx

4) Trigonometric Integrals

5) Reimann sums as numerical approximations using Left endpoint Right endpoint
and Midpoint approximations

6) Trapezoidal rule

7) Computing the definite integral using simple area formulas

8) The Fundamental Theorem of Calculus Part I

9) The Fundamental Theorem of Calculus Part II

10)Using the graphing calculator to compute the definite integral

11) Computing trigonometric definite integrals

12) Computing the definite integral by u-substitution—changing the variable and
the limits of the integral

13)Average value of a function

14) Mean value theorem for integrals

Unit VII) Areas and Volumes--2 weeks

1) Area bounded by a function and the x-axis

2) Area bounded by 2 curves

3) Volumes by the disk method

4) Volumes by the method of washers

5) Volumes of regions rotated about lines other than the x or y-axes

6) Volume by slicing

Unit VIII) Natural Log and e^(x) functions and Differential Equations--3 weeks

1) Review of inverse functions

2) Differentiation of log functions

3) Logarithmic differentiation

4) Integrals in the form u’/u in which u= f(x)

5) Integrals of tan(x) and cot(x) and other trig functions involving the “ln”
function

6) Exponential functions and their properties

7) Derivatives and integrals of expo. Functions

8) Exponential growth and decay models

9) Solving differential equations using separation of variables

10) Slope fields

Unit IX) Inverse Trig functions and their derivatives--1 week

Unit X) Review--3 weeks. Here are some of the review
questions that were done either in

class, or for homework, that require the students to offer Written explanation
or justification of

their answers: 2006 AB2, 2006 AB4, 2005 AB2, 2005 AB3, 2005 AB2 (Form B), 2004
AB1,

2003 AB3, and 2002 AB6.