College Calculus

Instructor. Mr. Connor

Email:

Course Overview:

Calculus is the gate through which students wanting advanced training in most scientific, mathematical and technical fields must pass. This full year scholar course provides the academically talented high school senior with the equivalent of a semester of college calculus. Since a full year is devoted to the course, more emphasis can be placed on multiple methods of solving problems. Students will explore the traditional, algebraic approach to calculus as well as use graphing calculators to represent functions numerically, graphically, and symbolically. Major topics include the theory and application of limits, continuity, derivatives, and integration.

Resources:

Textbook: Larson, Hostetler, & Edwards, 2006, Calculus of a Single Variable, 8th edition, Houghton Mifflin Co.

Calculator: Texas Instruments TI-84 Plus

Calculators are a very important part of this course. They will be used to interpret results and support conclusions by examining graphs and tables of values. The calculators will also be used to find zeros of a function, compute the derivative of a function numerically, and compute definite integrals numerically. Calculators are provided by the school and may be signed out for at home use

Study Suggestions:

1.  Preview the material that will be covered in the class

2.  Always arrive at the classroom on time and never miss class.

3.  Take detailed classroom notes. Feel free to stop the teacher’s lecture and ask questions when you have trouble to catch up.

4.  Do a thorough review after class and finish all homework on time.

Student Evaluation:

4-9week grades are computed using the following weighted scale.

Tests: 45% - At least on test a marking period will be given covering the information learned throughout the chapter.

Quizzes: 35% - Given usually after each section, some sections are combined.

Mathematica Labs: 15% - There are 13 MATHEMATICA labs that will be completed throughout the course as required by Pitt Bradford. Class time will be provided to work on the labs, if additional time is needed the student must arrange a time with the teacher to complete the labs. All labs will be submitted electronically through course sites.

Practice Problems/Projects – 5% - At various points during each marking period, the class will complete problems sets or projects in class.

Homework: Will be given on a regular basis but will not be graded. Although not graded, students are responsible for the completion of all homework assignments.

Retakes - As this is a college class there will be no retakes of tests and quizzes. If you are absent the day of a test/quiz it is your responsibility to schedule a time to make it up not to exceed 10 days from the day of the absence.

Final: There will be a comprehensive final given at the end of the year, the Pitt Calc Final, that will count as 20% of the students final course grade.

The letter grade for the class is determined using the following Pitt-Bradford scale:

A+ = 98-100 A= 92- 97 A- = 90-91 F = Below 60

B+ = 88-89 B = 82-87 B- = 80-81

C+ = 78-79 C = 72-77 C- = 70-71

D+ = 68-69 D = 62-67 D- = 60-61

Course Topics for College Calculus

Chapter 1: Limits and Their Properties

1.1  A Preview of Calculus

1.2  Finding Limits Graphically and Numerically

·  Definition

·  Properties

1.3  Evaluating Limits Analytically

·  Trig Limits

·  Limits with radicals

·  Limits of composition functions

·  Limits of functions that agree at all but one point

1.4  Continuity and One-Sided Limits

·  One and two sided limits

·  Removable discontinuity

·  Nonremovable discontinuity – Jump, asymptote, or oscillating

·  Intermediate Value Theorem for continuous functions

1.5  Infinite Limits

·  Asymptotic behavior

·  End behavior

·  Visualizing limits

Chapter 2: Differentiation

2.1 The Derivative and the Tangent Line Problem

·  Tangent to a curve

·  Slope of a curve

·  Normal to a curve

2.2 Basic Differentiation Rules and Rates of Change

·  Constant, Power, Sum and Difference, and Constant Multiple Rules

·  Sine and Cosine

·  Rates of Change

2.3 Product and Quotient Rules and Higher-Order Derivatives

·  Product and Quotient Rules

·  Trigonometric functions

·  Second and higher order derivatives

·  Acceleration due to gravity

2.4 The Chain Rule

·  Composition of a function

·  Power Rule

·  Trig functions with the Chain Rule

2.5 Implicit Differentiation

·  Implicit and Explicit functions

·  Differential method

·  Second derivative implicitly

·  Slope, tangent, and normal

2.6 Related Rates

·  Applications to derivatives

·  Guidelines for related rate problems

Chapter 3: Applications of Differentiation

3.1 Extrema on an Interval

·  Relative extrema

·  Critical numbers

·  Finding extrema on a closed interval

·  Absolute extrema

3.2 Rolle’s Theorem and the Mean Value Theorem

·  Illustrating Rolle’s Theorem

·  Tangent line problems and instantaneous rate of change problems with the Mean Value Theorem

3.3 Increasing and Decreasing Functions and the First Derivative Test

·  Testing for increasing and decreasing

·  First Derivative Test for extrema

·  Applications

3.4 Concavity and the Second Derivative Test

·  Testing for concavity

·  Points of inflection

·  Second Derivative Test for extrema

3.5 Limits at Infinity

·  Horizontal Asymptotes

·  Limits at infinity

·  Trig functions

·  Infinite limits at infinity

3.6 A summary of Curve Sketching

·  Rational functions

·  Radical functions

·  Polynomial function

·  Trig function

3.9 Differentials

·  Tangent line approximation

·  Error propagation

Chapter 4: Integration

4.1 Antiderivatives and Indefinite Integration

·  Definition

·  Integration Rules

·  Vertical Motion

4.2 Area

·  Sigma notation

·  Upper and lower sums

4.3 Riemann Sums and Definite Integrals

·  Subintervals with equal and unequal widths

·  Definition

·  Continuity

·  Area of a region

·  Properties of definite integrals

4.4 The Fundamental Theorem of Calculus

·  Guidelines for using FTC

·  Mean Value Theorem for integrals

·  Average Value of a function

·  Second fundamental theorem

4.5 Integration by Substitution

·  Composition function

·  Change of variables

·  Power rule for integration

4.6 Numerical Integration

·  Trapezoidal Rule

Chapter 5: Logarithmic, Exponential, and Other Transcendental Functions, Chapter 6: Differential Equations

5.1 The Natural Logarithmic Function: Differentiation

·  Definition

·  Properties of the Natural Logarithmic Function

·  Definition of e

·  Derivative of ln

5.2 The Natural Logarithmic Function: Integration

·  Log rule for integration

·  Trig functions

5.3 Inverse Function

5.4 Exponential Functions: Differentiation and Integration

·  Definition of

·  Operations and properties with exponential functions

6.1 Slope Fields and Euler’s Method

·  General and particular solutions

·  Slope fields – Visualizing and sketching

·  Approximating solutions with Euler’s method

6.2 Differential Equations: Growth and Decay

·  Growth and decay model

Chapter 7: Applications of Integration

7.1 Area of a Region Between Two Curves

·  Area between two curves

·  Intersecting curves

7.2 Volume: The Disk Method

·  Disk and washer method

7.3 Volume: The Shell Method