< Previous Section | Home | Next Section > |

Rate of Change at a Moment

The examples from Section 4.3 and Exercise 4.2.5 both embody the general idea of rate of change at a moment. They illustrated the idea that a

“Moments” need not refer to moments in time. In Exercise 4.1.5, height was the independent variable. A moment while a function’s independent quantity varies is a tiny interval of values that includes a specific value of the independent quantity.

In a later module we will make the idea of rate of change at a moment much more precise, and we will develop techniques for determining whether a function has a rate of change at a moment. Our goal, for the present, is that you form a productive image of what “rate of change at a moment”

If

**Reflection 4.4.1.** Take a moment (pun intended) to restate the last sentence in the previous paragraph using the __meaning__ of “moment”. Do not use the __word__ “moment”.

**Reflection 4.4.2. **Use Figure 4.4.1 (right graph) to estimate the rate of change of *f* at the moment that $x = -1.36$.

It might appear that for values of

Our emphasize here is not so much that you can

- Zoom in (shift-click-drag) enough times around the point ($x_0, g(x_0)$) so that you are convinced that this part of the graph displays a rate of change that is essentially constant over that interval.

- Replace the value of d
*x*with an appropriate value to have GC calculate the constant rate of change over the interval you determined and to graph the linear function having that rate of change that passes through $(x_0, g(x_0))$. Set the value of d*x*to make the arrows appear within your graph window.

- Print your file. Write your answers to parts a-c below on the printout.

a) As you zoom in you eventually see two purple arrows. What do these arrows represent?

b) There is a command line in this file that calculates a value of*m*. Explain this calculation. Explain the value of *m*.

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

b) There is a command line in this file that calculates a value of

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

2. Download and open Exercise 4.4.2.

- Zoom in (shift-click-drag) enough times around the point($x_0, g(x_0)$) so that you are convinced that this part of the graph displays a rate of change that is essentially constant over that interval.

- Replace the value of d
*x*with an appropriate value to have GC calculate the constant rate of change over the interval you determined and to graph the linear function having that rate of change that passes through $(x_0, g(x_0))$. Set the value of d*x*to make the arrows appear within your graph window.

- Print your file. Write your answers to parts a-c below on the printout.

a) As you zoom in you eventually see two purple arrows. What do these arrows represent?

b) There is a command line in this file that calculates a value of*m*. Explain this calculation. Explain the value of *m*.

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

b) There is a command line in this file that calculates a value of

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

3. Download and open Exercise 4.4.3.

- Zoom in (shift-click-drag) enough times around the point ($x_0, g(x_0)$) so that you are convinced that this part of the graph displays a rate of change that is essentially constant over that interval.

- Replace the value of d
*x*with an appropriate value to have GC calculate the constant rate of change over the interval you determined and to graph the linear function having that rate of change that passes through $(x_0, g(x_0))$. Set the value of d*x*to make the arrows appear within your graph window.

- Print your file. Write your answers to parts a-c below on the printout.

a) As you zoom in you eventually see two purple arrows. What do these arrows represent?

b) There is a command line in this file that calculates a value of*m*. Explain this calculation. Explain the value of *m*.

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

b) There is a command line in this file that calculates a value of

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

4. Download and open Exercise 4.4.4.

- Zoom in (shift-click-drag) enough times around the point ($x_0, g(x_0)$) so that you are convinced that this part of the graph displays a rate of change that is essentially constant over that interval.

- Replace the value of d
*x*with an appropriate value to have GC calculate the constant rate of change over the interval you determined and to graph the linear function having that rate of change that passes through $(x_0, g(x_0))$. Set the value of d*x*to make the arrows appear within your graph window.

- Print your file. Write your answers to parts a-c below on the printout.

b) There is a command line in this file that calculates a value of

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

5. Download and open Exercise 4.4.5.

- Replace the value of d
*x*with an appropriate value to have GC calculate the constant rate of change over the interval you determined and to graph the linear function having that rate of change that passes through $(x_0, g(x_0))$. Set the value of d*x*to make the arrows appear within your graph window.

- Print your file. Write your answers to parts a-c below on the printout.

b) There is a command line in this file that calculates a value of

c) There is a command line in this file that graphs a linear function. Explain what this graph represents.

6. What did you learn from doing Exercises 4.4.1 - 4.4.5?

7. In Exercise 4.3.1, no exact value of elapsed time was given as “the moment” that the camera’s shutter opened. How might we interpret the statement, “The moment when the camera’s shutter opened” so that it is consistent with the definition given in this section of rate of change at a moment?