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Mathematics Practices

Exploration 15-4b: Limits and Curved Asymptotes

Objective: Analyze graphs of rational algebraic functions with the help of synthetic
substitution and factoring .

1. Let

Plot the graph of f as . Use a friendly window with
an x-range of about to that includes
as a grid point. Have the grid off. Sketch the graph.

2. is undefined 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 limit 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?

3. Remove the discontinuity at algebraically by
the numerator and reducing the fraction .
Evaluate the resulting quotient polynomial at
Is the answer equal to the limit you found in
Problem 2?

4. How do you read What does this
equation mean?

5. Let

Plot the graph of g as using “thick” style. What
feature does the graph of g have at Sketch the
graphs of and here, showing their
relationship to each other.

6. Try to find by tracing to x- values closer and
closer to 1. Try x-values on both sides of 1. What
happens to the quotient as x approaches 1?

7. To understand why the graph of resembles
the graph of simplify the equation for
by synthetic substitution . Write the equation in
“mixed- number ” form as

What relationship do you notice between the
equations for f (x) and g(x)?


8. Let

Plot the graph of as .

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 by long division or
synthetic substitution. Write the result in mixed-number
form as

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 have
learned about rational functions, removable
discontinuities, asymptotes, etc. as a result of doing
this Exploration.








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, miles,
above the surface is given by

1. How far is Spencer from the surface at min? at
min? How far did he go in this time interval?
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?

4. Between time 2 and x minutes, Spencer travels
miles. So his average rate is

Substitute for and 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?

6. Plot a line through the point that has a slope
equal to the instantaneous velocity you found in
Problem 5. Take into account that the two axes have
different scales . How does the line relate to the


7. Find Spencer’s instantaneous velocity at time
3 minutes by starting with

Explain the fact that the answer is negative .

8. Let . Find and . How do
your answers compare with the instantaneous
velocities at times and

9. There is an algebraic way to find the instantaneous
rate of change of a polynomial function. Suppose
that If is the average rate of change
between the fixed time c and the variable time x,

Divide by means of synthetic substitution. Write a
polynomial equation for . (It will involve the
constant c.)

10. By appropriate calculations, show that

11. Explain how you can get the 5 and the 4 in from
the original polynomial

12. Suppose that What do you think would
be an equation for the function that gives the
instantaneous velocity of ?

13. Show that you can use the patterns you found in
Problems 11 and 12 on the function

to get the velocity function from Problem 8,

14. Look in Section 15-5 to find the special name given
to the velocity function.

15. What did you learn as a result of doing this
Exploration that you did not know before?




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