MAT 213

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• MAT 213
• Brief Calculus

Section 2.1Change, Percent Change, and Average
Rates of Change
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Change

• A main function of calculus is describing change
• While it is useful to know what something is, it
  can be more useful to know how it is changing
• Examples
• Change in home sales
• Change in profit/revenue/costs per unit produced
• Change in number of students attending SCC per
  semester

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Two ways of measuring change
Change New Value Old Value, sometime referred
to as absolute or total change Units output
units Percent Change Units percent We
calculated percent change back in 1.3 with
exponential functions
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Example After researching several companies, you
and your cousin decided to purchase your first
shares of stock. You bought 200 worth of stock
in one company Your cousin invested 400 in
another. At the end of a year, your stock was
worth 280, whereas your cousins stock was
valued at 500. Who earned more money? Who did
better?
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• We have another way of expressing change which
  involves spreading the change over an interval
• This is called the average rate of change
• We divide the change (or total change) by the
  length of the interval
• Example
• If you drove 100 miles in 2 hours, how fast were
  you going?
• Were you going the same speed the whole time?

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Change New Value Old Value Units output
unitsPercent Change Units
percent Average Rate of Change Units units
of output per unit input
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Average Rate of Change Suppose a large grapefruit
is thrown straight up in the air at t 0
seconds The grapefruit leaves the throwers hand
at high speed, slows down until it reaches its
maximum height, then speeds up in the downward
direction and finally, SPLAT! During the
first second, the grapefruit moves 22 feet
During the next second, the grapefruit moves 12
feet
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Average Rate of Change Use the table to
compute the average rate of change of the
grapefruit over the following intervals 4 t
5 1 t 3 0 t 5
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Average Rate of Change The grapefruits journey
can be modeled by this quadratic function Use
this function to compute the average rate of
change over the following intervals 4 t 5 1
t 3 0 t 5
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• Average Rate of Change
• Graphically, the average rate of change over the
  interval a x b is the slope of line
  connecting f(a) with f(b) We refer to this as
  the secant lineUse the graph of H(t) to compute
  the average rate of change of the grapefruit over
  the following intervals
• 4 t 5
• 1 t 3
• 0 t 5
• Notice that secant lines are linear functions
  that have the average rate of change between a
  and b as a slope

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A couple of formulas for compounding interest

• If an annual interest r is compounded n times per
  year, then the balance, B, on an initial deposit
  P after t years is
• If money is invested at a nominal rate of r and
  compounded continuously, we have the formula

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• The nominal rate or annual percentage rate (APR)
  is the percentage 100r
• The percentage change of the amount accumulated
  over one compounding period is
• The effective rate or annual percentage yield
  (APY) is the percentage change in the amount
  accumulated over one year

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• Suppose you have money to place into an interest
  bearing account. You have the following options
• 5.9 compounded quarterly
• 5.8 compounded continuously
• 6 compounded annually
• Which option would you choose?

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In groups lets try the following from the book

• 3, 13, 23, 27