5-Minute Check on Activity 5-7

1
5-Minute Check on Activity 5-7

• Match the following interest types
• Compound Earning interest
  only on the principal
• Simple Earning
  interest on principal and interest
• If the effective yield bigger or smaller than the
  interest rate?
• How does the compounding period affect the
  effective yield?
• What is the formula for a continuously compounded
  account?
• How much money would you have at retirement, if a
  rich uncle deposited 5000 in a stock market fund
  that earned 10 interest compounded continuously
  the day that you were born?

always bigger
more compounding increases the effective yield
A Pert
A Pert 5000 e0.1(65) 5000 e6.5
3,325,708.17
Click the mouse button or press the Space Bar to
display the answers.
2
Lab 5 - 8

• Continuous Growth and Decay

Kathmandu, Nepal 11/05/2005
3
Objectives

• Discover the relationship between the equations
  of exponential functions defined by y abt and
  the equations of continuous growth and decay
  exponential functions defined by y aekt
• Solve problems involving continuous growth and
  decay models
• Graph base e exponential functions using
  transformations

4
Vocabulary

• None new

5
Activity

• The US Census Bureau reported that the US
  population on April 1, 2000 was 281,421,906. The
  US population on April 1, 2001 was 284,236,125.
  Assuming exponential growth, the US population y
  can be modeled by the equation y abt, where t
  is the number of years since April 1, 2000 (when
  t 0).
• What is the initial value, a?
• What is the annual growth factor, b?

a 281,421,906
b 284,236,125 ? 281,421,906 1.01
6
Activity cont

• Assuming exponential growth, the US population y
  can be modeled by the equation y abt.
• What is the annual growth rate?
• What is the equation for US population as a
  function of t?
• Use this to estimate the US population on 1 Apr
  2011.

r b 1 1.01 1 0.01
y(t) 281,421,906(1.01)t
y(11) 281,421,906(1.01)11 281,421,906(1.115668
347)
313,973,513
Estimate as of yesterday http//www.census.gov/m
ain/www/popclock.html
7
Activity cont

• Change the equation y abt, to a continuous
  growth form of y aekt. So bt ekt and ekt
  (ek)t
• How are b and ek related?
• Using our calculator, let Y1 ex and Y2 1.01
  and find their intersection (solution for b
  ek)?
• Rewrite the US population function in continuous
  growth format.

b ek or 1.01 ek
k 0.00995
y 281,421,906e0.00995t
8
Continuous Growth Reminder

• Continuous growth is modeled by the equation
• y aekt
• where a is the initial amount, k is the constant
  continuous growth rate and t is time

9
Continuous Growth Example

• A bacterial growth in a culture increases by 25
  every hour. If 10000 are present when the
  experiment starts
• Determine the constant, k, in continuous growth
  model
• Write the equation for the continuous model
• When will the sample double?

b 1 .25 1.25 b ek 1.25 ek
via graph k 0.2231
T A0ekt 10000e0.2231t
20000 10000e0.2231t t 3.11 hours
10
Continuous Decay Example

• Tylenol (acetaminophen) is metabolized in your
  body and eliminated at a rate of 24 per hour.
  You take two Tylenol tablets (1000 milligrams) at
  1200 noon.
• What is the initial value?
• Determine the decay factor, b.
• Find the constant continuous decay rate, k.
• Write the continuous decay function

1000 milligrams
b 1 - .24 0.76
b ek 0.76 ek via graph k
-0.27444
T A0ekt 1000e-0.27444t
11
Graph of ex function

• y ex
• Domain all real numbers
• Range y gt 0
• Increasing or Decreasing
• always increasing (positive slopes)
• y-intercept 1 no x-intercept
• y 0, x-axis, is a horizontal asymptote

12
ex Transformations

• Compared to y ex, describe the graphic
  relationship between its graph and the following
  graphs
• y - ex
• y ex2
• y ex 2
• y 2ex
• y e-x
• y 1 2ex

Outside Reflection across x-axis
Inside Shift left 2 units
Outside Shift up 2 units
Outside Vertical stretch by 2
Inside Reflection across y-axis
Outside Vertical stretch by 2 reflected across
x-axis and shifted up by 1
13
Summary and Homework

• Summary
• Quantities that increase or decrease continuously
  at a constant rate can be modeled by y aekt.
• Increasing k gt 0 k is continuous rate of
  increase
• Decreasing k lt 0 k is continuous rate of
  decrease
• The initial quantity at t0, a, may be written in
  other forms such as y0, P0, etc
• Remember the general shapes of the graphs
• Homework
• page 604-09 problems 2, 3, 8