Exponential Growth Models
This figure allows use of the Exponential Growth Models:
and
to create
graphs of exponential growth or decay model with base e or base b on the interval [0.01, 3]. The eight
features of this interactive figure are noted below.
1.
Click the bubble to select the model. Move the P0 slider to set the original value and
the k slider to set the rate. Adjust the ymax in the Window Setting if necessary to see the graph.
2.
Click the bubble to select the model. Move the P0 slider to set the original value and
the K slider to set the rate. Adjust the ymax in the Window Setting if necessary to see the graph.
Equation Check the box to show equation of the exponential growth model.
Grid Check the box to show the grid.
Window Setting Move the appropriate slider to set the window as needed. The limits are xmin
[-50, 0]; xmax [5, 1000]; and ymax [5, 50000].
x slider Check the box and move the slider to see the point and its coordinates on the graph.
Large Check the box to increase the screen graph to about 150% of original size.
Reset Click the button in the upper right corner to return figure to default settings.
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Exploratory Exercise: Click the Reset button
in the upper right corner. Click the bubble for
. Set the Window Setting so that xmin = 0, xmax = 5, and ymax = 10000. Check the
x slider box. Slowly move the P0 and k sliders while observing the changing graph and equation
on the grid. (Hold down the Alt key for finer slider control.) Set the k slider near 1. Move the x
slider and observe the red point on the graph. Move the P0 slider to 480. Set the Window
Setting so that xmin = 0, xmax = 5, and ymax = 500. Move the k slider between -2 and 0 and
observe the changing graph and equation.
Click the Reset button in the upper right corner. Click the bubble for
. If the graph
goes off the grid, adjust the Window Setting for xmin, xmax, and ymax as needed to view the
graph. Slowly move the P0 , b, and K sliders while observing the changing graph and equation
on the grid.
a) Click the Reset button in the upper right corner. Uncheck the Equation box. Move the P0
and k sliders so that P0 = 15 and k near 0.055. Write the equation of the function in the form
. Using the graph estimate the value of P when x = 5. Use your equation to
calculate the value of P when x = 5.
b) Check the Equation box and the x slider box. Move the x slider to 5. Compare your answers
from Part (a) with those shown on the grid.
c) Click the Reset button
in the upper right corner. Uncheck the Equation box. Check the
bubble for
. Move the P0 and K sliders so that P0 = 975 and K = 100. Set the
Window Setting so that xmin = 0, xmax = 100, and ymax = 1000. Move the b slider until
. What is the value for b? Write the equation of the function in the form
.
d) Check the Equation box and the x slider box. Move the x slider to get as close to the point
(40, 400) as you can. Compare your answers from Part (c) with those shown on the grid.