Suppose
we have these pressure and temperature readings for a gas in a rigid container:
| pressure (kPa) | 0.913 | 0.959 | 1.006 | 1.051 | 1.099 | 1.140 |
| temperature (C) | 30.12 | 33.46 | 36.78 | 40.11 | 43.43 | 46.77 |
Exercises:
1. Sketch the graph of temperature as a function of pressure.
2. Estimate the slope of the graph when the pressure is 1 kPa.
3. What does this mean in practical terms? Your answer should look something like this: "When the pressure is approximately 1kPa, the temperature increases about ___ times as fast as the pressure increases."
4. Now think of the same table as expressing pressure as a function of temperature. What's the best way to express the information graphically?
5. Estimate the slope of this graph when the temperature is about 36C.
6. What does this mean in practical terms?
7. What is the connection between your answer to question #2 and your answer to question #5?
8.
Graph
the cubing function on your calculator.
Look closely at the point (1/2, 1/8).
If you know that the slope of the graph of the cubing function is 3/4 when
the input value is 1/2,
what can you deduce about the slope of the graph of the cube root function?
9.
Graph
the cube root function on your calculator.
Check whether your conclusion in exercise #8 looks correct.
If you know that the slope of the graph of the cube root function is 1/12
when the input value is 8,
what can you deduce about the slope of the graph of the cubing function?
10.
Suppose that f and g are functions which are inverses of
one another.
If f(3) = 5, what do you know about g?
If the output of f increases half as fast as the input when the
input is near 3
(this is information about the slope of the graph of f),
what can you say about the relationship between changes in the input of
g and the output of g?
What can you say in general about slopes of graphs of inverse functions?