diff --git a/source/proteus/proteus-3-2.xml b/source/proteus/proteus-3-2.xml index cef9ddd61..f1652a728 100644 --- a/source/proteus/proteus-3-2.xml +++ b/source/proteus/proteus-3-2.xml @@ -6,7 +6,7 @@

- In the Desmos window below, there is a graph of a function that you can experiment with: + In the Desmos window below, there is a graph of a function that you can experiment with by adjusting the slider for the value of the parameter b:

@@ -14,26 +14,32 @@

  1. - Estimate the location of the relative minimum. + With b = 1, estimate the numerical coordinates of the relative minimum.

  2. - Estimate the location of the inflection point. + With b = 1, estimate the numerical coordinates of the inflection point.

  3. - Play with the slider for a. What happens to the location of the relative minimum? of the inflection point? + Play with the slider for b. What are two specific things you notice about how the graph of the function changes?

  4. - Play with the slider for b. What happens to the location of the relative minimum? of the inflection point? + As you increase the value of b, what happens to the coordinates of the local minimum?

    5. +
  5. +

    + As you increase the value of b, what happens to the coordinates of the inflection point? +

    +
+ diff --git a/source/proteus/proteus-3-3.xml b/source/proteus/proteus-3-3.xml index 758952251..a98da55d5 100644 --- a/source/proteus/proteus-3-3.xml +++ b/source/proteus/proteus-3-3.xml @@ -4,37 +4,39 @@ PROTEUS exercises -

- Sketch a graph of a function with the following combinations of properties, or explain why such a combination is impossible. -

-
    -
  1. -

    - One relative maximum and one relative minimum -

    -
    2. -
  2. -

    - One relative maximum but no global maximum -

    -
    3. -
  3. -

    - One global minimum but no relative minimum -

    -
    4. -
  4. -

    - Two relative maxima but no local minimum -

    -
    5. -
  5. -

    - Two relative maxima but no global minimum -

    -
    6. -
+ +

+ For each of the following prompts, sketch a graph of a function that has the stated combination of properties, or explain why such a combination is impossible. +

+
    +
  1. +

    + One relative maximum and one relative minimum +

    +
    2. +
  2. +

    + One relative maximum but no global maximum +

    +
    3. +
  3. +

    + One global minimum but no relative minimum +

    +
    4. +
  4. +

    + Two relative maxima but no local minimum +

    +
    5. +
  5. +

    + Two relative maxima but no global minimum +

    +
    6. +
+ + - diff --git a/source/proteus/proteus-3-4.xml b/source/proteus/proteus-3-4.xml index 1ab0cbcae..cb5333ecc 100644 --- a/source/proteus/proteus-3-4.xml +++ b/source/proteus/proteus-3-4.xml @@ -4,38 +4,41 @@ PROTEUS exercises -

- Suppose you are building a fence around a rectangular field, and the field needs to have an area of 9000 square meters. There are many rectangles of different dimensions that you could consider. -

-

-

    -
  1. -

    - One option you have considered is to make the field be a square. Draw a picture of what the field would look like in this case. -

    -
    2. -
  2. -

    - Another option you have considered is to make the field look like a piece of notebook paper. Draw a picture of what the field would look like in this case, labeling the measurements of your shape. Remember, the field still needs to have an area of 9000 square meters. -

    -
    3. -
  3. -

    - Draw at least one more picture of a rectangular field that is different than what you’ve drawn before and would still have an area of 9000 square meters. -

    -
    4. -
  4. -

    - Which measurements are different among your three fields? Which measurements are the same? -

    -
    5. -
  5. -

    - Choose letters to represent the key measurements of the field. What are the lowest and highest values of those measurements? -

    -
    6. -
-

+ +

+ Suppose you are building a fence around a rectangular field, and the field needs to have an area of 9000 square meters. There are many rectangles of different dimensions that you could consider. +

+

+

    +
  1. +

    + One option you have considered is to make the field be a square. Draw a labeled picture of what the field would look like in this case. +

    +
    2. +
  2. +

    + Another option you have considered is to make the field look like a piece of notebook paper. Draw a picture of what the field would look like in this case, labeling the measurements of your shape. Remember, the field still needs to have an area of 9000 square meters. +

    +
    3. +
  3. +

    + Draw at least one more picture of a rectangular field that is different than what you've drawn before and would still have an area of 9000 square meters. +

    +
    4. +
  4. +

    + Which measurements are different among your three fields? Which measurements are the same? +

    +
    5. +
  5. +

    + Choose letters to represent the key measurements of the field. What are the lowest and highest values of those measurements? +

    +
    6. +
+

+ + diff --git a/source/proteus/proteus-3-5.xml b/source/proteus/proteus-3-5.xml index 7855a7e1b..c0a580fcb 100644 --- a/source/proteus/proteus-3-5.xml +++ b/source/proteus/proteus-3-5.xml @@ -4,38 +4,41 @@ PROTEUS exercises -

- A cylindrical water tank on top of a city water tower has a radius of 8 meters and a height of 15 meters. The amount of water in the tank changes as water is being pumped in or out, depending on the time of day and the local water usage; it is usually completely refilled by about 2am, and then starts to drain around 7am as people wake up and do morning routines (e.g., showering, cooking, etc.). -

-

-

    -
  1. -

    - What is the maximum amount of water the tank can hold? Note: the formula for the volume of a cylinder is V = \pi r^2 h. -

    -
    2. -
  2. -

    - Draw what you imagine the water in the tanks would look like at 2am, at 8am, and at 11am. -

    -
    3. -
  3. -

    - Which measurements of the water are different among your different pictures? Which measurements are the same? -

    -
    4. -
  4. -

    - Determine a formula for the volume of water in the tank as a function of only one variable. What is the varying quantity on which your volume formula depends? -

    -
    5. -
  5. -

    - Draw a graph relating the volume of water to the water level, and a second graph relating the volume of water to time. -

    -
    6. -
-

+ +

+ A cylindrical water tank on top of a city water tower has a radius of 8 meters and a height of 15 meters. The amount of water in the tank changes as water is being pumped in or out, depending on the time of day and the local water usage; it is usually completely refilled by about 2am, and then starts to drain around 7am as people wake up and do morning routines (e.g., showering, cooking, etc.). +

+

+

    +
  1. +

    + What is the maximum amount of water the tank can hold? Note: the formula for the volume of a cylinder is V = \pi r^2 h. +

    +
    2. +
  2. +

    + Draw what you imagine the water in the tanks would look like at 2am, at 8am, and at 11am. +

    +
    3. +
  3. +

    + Which measurements of the water are different among your different pictures? Which measurements are the same? +

    +
    4. +
  4. +

    + Determine a formula for the volume of water in the tank as a function of only one variable. What is the varying quantity on which your volume formula depends? +

    +
    5. +
  5. +

    + Draw a graph relating the volume of water to the water level, and a second graph relating the volume of water to time. +

    +
    6. +
+

+ + diff --git a/source/sec-2-6-inverse.xml b/source/sec-2-6-inverse.xml index 4f024c344..a101a717a 100644 --- a/source/sec-2-6-inverse.xml +++ b/source/sec-2-6-inverse.xml @@ -63,9 +63,11 @@

- + + + Basic facts about inverse functions