AMTH122 Survey of Calculus with Applications

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AMTH122Survey of Calculus with Applications

• Dr. Yilian Zhang
• 6413796 yilianz_at_usca.edu

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About the course and instructor

• AMTH A 122 Survey of Calculus
• Fall 2009 (3 credit hours)
• Instructor Yilian Zhang
• Telephone [phone] (3796 on campus)
• Email YilianZ_at_usca.edu
• Meeting Time MW 100pm-215pm, ADMN 233

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Pieces of information

• Websites of the class
• Mathlab
• USCA MATH LABPenland ADMN 221(803) 641-3470
• Monday-Thursday
• 900 a.m. 600 p.m.
• Friday
• 900 a.m. 200 p.m.
• Office Hour

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Course Description

• AMTH 122  is an introduction to the concepts and
  applications of differential calculus through the
  study of rates of change and their
  interpretations, using data-driven,
  technology-based modeling. (Not intended for
  those who plan to take more than one calculus
  course).

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Text Supplies

• Concepts of Calculus with Applications ,  by
  Martha Goshaw
• TI-83 plus or TI-84 Graphing Calculator.

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COURSE Goals

• Analytical thinking and problem solving the
  ability to analyze problems, and to use the
  concepts and techniques of calculus to model and
  solve these problems. The emphasis will be on
  applied problems, primarily from business and
  economics.
• Written and oral communication the ability to
  communicate effectively using the language of
  mathematics, both symbolic and verbal.
• Stimulate students to become critical consumers
  of quantitative arguments .

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COURSE PHILOSOPHY and ORGANIZATION

• Functions and Regression (Unit 0) Linear,
  Nonlinear (polynomial, rational, piecewise,
  exponential and logarithmic).
• Theory of Derivatives (Unit 1) Limits and
  Continuity, Rates of Change, Rules for
  Derivatives, Higher-order Derivatives.
• Applications of Derivatives (Unit 2) Find
  critical values, Assess concavity, Optimization,
  Implicit differentiation and related rates.

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Policy

• GRADING POLICY
• Three major tests 50
• Quizzes/Homework  30
• Cumulative Final Exam 25
• GRADING SCALE
• 90-100 A, 87-89 B
• 80-86 B, 77-79 C
• 70-76 C, 67-69 D
• 60-66 D, 0-59 F

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Homework Quizzes

• Homework
• Homework will be collected on the class period
  following the assignment ! Random homework
  problems will be graded.
• Quizzes are based on homework and activities.
• NO MAKE-UPS ON QUIZZES.
• Lowest grades will be dropped.

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Exam

• Test ISept 23,2009
• Test IIOct  26, 2009
• Test IIINov 23, 2009
• Final Exam
• Dec 7, 2009
• NO MAKE-UP on tests will be given. If your
  absence is excused, final exam grade will replace
  that test grade.
• Final exam is a cumulative exam and will cover
  all of the material covered in class

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Suggestion

• Read it (before the lecture)!
• Prior to reading a section, do the Warm Up
  Exercises
• After each lecture, Use Check Your
  Understanding exercises to see if you really
  understand
• Do the homework as soon as possible
• Ask questions in class about concepts or HW
  problems

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Goal of the course

• Derivative
• Changes of Function
• Any question?

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Basic Algebra Review

• Given equation, solve for y when x2.
• Solve for x

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Functions

• Calculus is the study of change
• Change can be expressed in many ways
• With numerical data, such as a table
• With a graph
• With words
• With an equation
• Change is often expressed as a function.

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Joes weight change (ordered pair)
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Change with a graph
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With words and with formulas

• W(t) is Joes weight (in pounds) t weeks after he
  begins to diet if he weighs 165 pounds at the
  start and loses 2 pounds each week.
• W(t) 165 - 2t pounds,
• after t weeks of dieting

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Functions

• A function is a correspondence between two sets
  that assigns to each member of the first
  set(Domain) one and only one element from the
  second set(Range). Input/output diagram
• Input t
• weeks
• W(t)
  Output
• 
  Pounds

Rule W
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Function, Domain, Range

• A function is a rule that assigns exactly one
  output to each input.
• Domain - set of all possible inputs (independent
  variable, controlled variable)
• eg.0, 1, 2, 3, 4, 5 weeks 0lt weekslt5
• Range - set of all possible outputs (dependent
  variable, observed variable)
• eg. 165, 163, 161, 159, 157, 155
• weight 155 lt weightlt 165
• Notation f(x) , W(t)

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Describe function y is a function of x

• Ordered Pairs
• Equation
• Graph
• Q what kind of correspondence between two sets
  represents a function?
• Rule of thumb one value of x --? exactly one
  value of y
• Vertical line test

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Recognizing Functions

• Not all graphs represent functions.
• Consider possible values in context, even
  from a table. Use definition to determine if you
  have a function.
• Vertical line test
• Given the graph of a relation, inputs on
  horizontal, and outputs on vertical. If at any
  point you can draw a vertical line that crosses
  the graph in two or more places, the graph does
  not represent a function.

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Determining the value of the function

• Use rule substitute
• W(4)
• From table
• locate input, read output
• From graph
• locate input on horizontal axis, move to
  
• graph, encounter output on vertical axis
•   units it is important to know and state the
  units of measure for each variable (axis)

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Skills Reviews

• Given
• Find
• Use the vertical line test to identify graphs in
  which y is a function of x.

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Linear function

• y mx b.
• m --- slope
• B ---- y-intercept
• Example Graph the linear function y ?x 3 by
  using the slope and y-intercept.

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Understand the slope

• Slope is defined as a ratio of a change in y to a
  corresponding change in x.
• Slope can be interpreted as a rate of change in
  an applied situation.

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Example

• Wal-Mart is offering one free DVD movie with the
  purchase of one or more DVD movies priced at
  19.99 each.
• Write the linear function that describe the total
  cost of DVD movie each transaction.
• Whats the cost of buying 5 DVD movies?
• How many DVDs could you buy in one transaction if
  you have 60 ?

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Absolute value function

• Graph of f(x) x
• Graph of f(x) x -3
• Graph of f(x) x-2
• Graph of f(x) -x

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Transformation

• Horizontal transformation
• f(x-h) shift h units to the right
• f(xh) shift h units to the left
• Vertical transformation
• f(x) k shift up k units
• f(x) k shift down k units
• Reflection
• - f(x) reflects the graph of f(x) about the x
  axis
• f(-x) reflects the graph of f(x) about the y axis

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Before next class

• Read topic 1
• And finish homework
• Topic 1 Exercise 9, 13, 16

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• Break-even point when revenue equals cost and
  (profit equals 0)
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• On graph
• x-intercept break-even point
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• y-intercept fixed cost (typically)