AMTH122 Survey of Calculus with Applications

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AMTH122Survey of Calculus with Applications

• Linear and Absolute Function
• Nonlinear Functions

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Homework Due Today Review

• Topic 1 Exercise 9, 13, 16
• What is a function?
• How to evaluate the value of a function?

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Linear function

• y mx b.
• m --- slope
• b ---- y-intercept

Y-intercept
X-intercept
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Slope

• The slope of the line through the distinct points
  (x1,y1) and (x2,y2) is
• where x2 x1 ? 0.
• Note, we let m denote slope.

12/12/2009
Section 7.2
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Sign of m v.s Graph of the line
mgt0
mlt0
m0
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Understand the slope and y-intercept in
application

• 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.
• Y-intercept can be interpreted as original value

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Example

• A cleaning service charges a flat fee of 50 plus
  10 per hour. Write a linear function describing
  their total charges for any number of hours, h.
  If Malcolm uses the service for 6 hours, what is
  his charge?
• Steps of Problem Solving
• Understand the problem
• flat fee 50 , hourly rate 10
• charges v.s number of hours
• Set up the problem
• Let x represent number of hours and y the total
  charge.
• m ? b ?
• Solve the problem and check the answer

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Lets try it together!

• 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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Business Application

• Rate of depreciation
• A small business buys a color laser printer for
  1500 and estimate its lifespan to be 8 years,
  after which it will be worth 50.
• Find a formula for the value of the printer after
  t years, and state a reasonable domain.
• What is the value of the printer after 5 years?

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Business Application

• Fixed Costs - start-up, or overhead eg. rent and
  utitlities
•  Variable Costs - extra expenditures eg supplies,
  travel, custodial, etc.
• Total Cost fixed cost variable cost
• Revenue ( income) qty. sold price
• Profit revenue - total cost
• Break-even point when revenue equals cost
  (profit equals 0)

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Examples Break even

• You and several of your friends decide to
  mass-produce I love calculus and you should
  too! T-shirts. Each shirt will cost you 2.50 to
  produce. Additional expense include the rental of
  a downtown building for a flat fee of 675 per
  month, utilities estimated at 100 each month,
  and leased equipment costing 150 per month. You
  will be able to sell the T-shirts at the premium
  price of 14.50 because they will be in such
  great demand.
• Give the equations for monthly revenue and
  monthly cost as functions of the number of
  T-shirts sold.
• How many shirts do you have to sell each month to
  break even?

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Supply and demandEquilibrium point S(p) D(p)

• The supply and demand functions for oranges(in
  thousands of bushels) are given by
  S(p)1500p-15000 and D(p) 17500-1000p, where p
  is the price per bushel. Determine the
  equilibrium point and explain what is represented
  by that point.
• S(p)D(p)
• Solve the equation
• p 13
• Price is 13 per bushel.

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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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Quadratic Function

• General form
• Axis of symmetry
• vertex
• Vertex form
• The graph of this function is a parabola with its
  vertex ( turning points) at (h,k)

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Graph
agt0 , open upward
alt0 , open downward
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Lets try it

• Sketch the graph of
• Graph of y (x 4)2
• Graph of y 4 x2
• Graph of y (x 3)2 5
• Graph of y x2 4x 3

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Cubic function

• Cubic function y ax3 bx2 cx d
• Other form f(x) a(x h)3 k
• Point of inflection (h,k)
• S- shape graph
• ExampleGraph of y 4 (x2)3

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Radical function

• General form
• Domain is restricted
• radicand must be non-negative.
• Range is restricted
• principal square root is always positive.

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Before next class

• Read topic 2
• And finish homework
• Topic 1 Exercise 33, 49, 53,61, 69, 73
• Topic 2 Exercise 1-11(odd), 29, 33, 47, 57