Plot the graph of f as
. Use a friendly window with
an x-range of about to
as a grid point. Have the grid off. Sketch the graph.
because of division by zero. Trace
to on your
graph. What feature does the graph
have at this point? By tracing closer and closer to
, find the
seems to be approaching as
x approaches 1. Is the limit as x approaches 1 from
the left side the same as when x approaches 1 from
the right side?
What relationship do you notice between the
equations for f (x) and g(x)?
Plot the graph of
9. The graphs of f and g from Problem 5 should look like this . On this figure, sketch the graph of h.
10. Simplify the equation for
synthetic substitution. Write the result in mixed-number
11. How is the polynomial part of
in Problem 10
related to the graph of
12. Why is it important for your work to be 100% correct
in problems like these that are sequential?
13. Write a paragraph summarizing the things you
learned about rational functions, removable
discontinuities, asymptotes, etc. as a result of doing
Exploration 15-5a: Rate of Change Date:
of a Polynomial Function
Objective: Find the instantaneous rate of change of
a polynomial function at a given
value of x.
Astronaut Spencer Spacey takes off from the planet
Alderaan. He starts his rocket countdown at time
minutes. Shortly thereafter his
spaceship takes off.
It rises for a while, then drops while his second-stage
rocket engine is starting up, and then rises again.
Spencer’s computer finds that his distance,
above the surface is given by
1. How far is Spencer from the surface at
min? How far did he go in this time
What was his average velocity for this time interval?
2. What is Spencer’s average velocity for the time
interval from 2 minutes to x = 2.001 minutes?
3. Spencer’s instantaneous velocity at 2 minutes is the limit of his average velocity between 2 and x minutes
as x approaches 2. From your answers to Problems 1
and 2, what do you conjecture that his instantaneous
velocity equals at 2 minutes?
time 2 and x minutes, Spencer travels
miles. So his average rate
and simplify the
numerator. Then use synthetic substitution to do the
division, thus removing the discontinuity at
Write a simplified equation for .
5. Use this simplified equation for
to find the limit
as x approaches 2. Does this confirm the
instantaneous velocity you found in Problem 3?
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