(2) In each graph above, calculate the velocity in feet per second that is represented.

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Alexandrina Meryl Johnston
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1 Calculus Week 1 CHALLENGE 1-Velocity Exercise 1: Examine the two graphs below and answer the questions. Suppose each graph represents the position of Abby, our art teacher. (1) For both graphs above, come up with a scenario the graph represents (i.e. Abby walks away from the building at 2ft/sec). What does the negative vs. the positive mean? (2) In each graph above, calculate the velocity in feet per second that is represented. (3) For the first graph, How far away from the building is Abby after 2 seconds? After 5 seconds? After 10 seconds? (4) Come up with a general rule: If you have a horizontal line on a distance vs. time graph, how do you determine the person s velocity?

2 Exercise 2: Examine the two graphs below and answer the questions. Suppose each graph represents the position of Abby, our art teacher. (1) For both graphs above, come up with a scenario the graph represents (i.e. Abby walks away from the building at 2ft/sec). What does the negative vs. the positive mean? (2) In each graph above, calculate the velocity in feet per second that is represented. (3) For the first graph, How far away from the building is Abby after 2 seconds? After 5 seconds? After 10 seconds? (4) Come up with a general rule: If you have a line on a distance vs. time graph, how do you determine the person s velocity?

3 Exercise 3: Examine the two graphs below and answer the questions. Suppose each graph represents the position of Abby, our art teacher. (1) For both graphs above, come up with a scenario the graph represents (i.e. Abby walks away from the building at 2ft/sec). (2) In each graph above, why doesn t it make sense for me to ask you about velocity in general? (3) For the first graph, what s Abby s velocity at time t=1? What s her velocity at time t=2? What s her velocity at time t=3? (4) For the second graph what s her velocity at time t=0.8? What about at time t=2? (5) Come up with a general rule: under what circumstances is it problematic to try to find velocity from a graph? What needs to be true about a function for velocity to be findable?

4 Exercise 4: Abby finds a scooter out in front of the school and hops on. She kicks the ground pretty hard to get going fast, but unbeknownst to her someone attached a chain of rubber bands to the board of the scooter. Before she s more than a few feet from the school, her forward motion is slowed, then stopped, then the scooter, with her on it, is pulled back towards the school. Here s a graph of her distance. (1) How far from the school is she after 1 second? After 2? After 3? (2) What s her velocity after 0 seconds? (3) What s her velocity after 1 second? (4) What s her velocity after 2 seconds? (5) What s her velocity after 3 seconds? (6) Explain how you calculated the above velocities.

5 CHALLENGE 2- Areas under curves Exercise 1: Suppose Abby, the PCCS art teacher, is moving away from the school. The graphs below show her velocity (how fast she moves away from the school) as a function of time in two different scenarios (1) What s the difference between the scenarios shown in the two different graphs? (2) How far away from PCCS is Abby after two seconds, 4 seconds and 6 seconds in the first graph? (3) How far away from PCCS is Abby after two seconds, 4 seconds and 6 seconds in the second graph? (4) Find the AREA under the first graph between t=0 and t=4. (In other words, draw two vertical lines, one at t=0, the other at t=4. Then find the area of the rectangle you see sandwiched between the two vertical lines, the function, and the t-axis.) (5) Find the AREA under the second graph between t=0 and t=4. (6) What does the AREA under the graph represent if the graph is a velocity vs. time graph?

6 Exercise 2: Suppose Abby, the PCCS art teacher, is moving away from the school on a scooter. The graph below shows her velocity in ft/s as a function of time. Ignore the second blank graph till the bottom of the page. (1) How fast is Abby going after 1 second? After 2 seconds? After 3 seconds? After 6 seconds? (2) If her speed is constantly changing (increasing at a linear rate, in this case), do you think it s possible to find how far she has gone? Remember back to the previous exercise. (3) Find the AREA under the graph between t=0 and t=2. (In other words, draw two vertical lines, one at t=0, the other at t=2. Then find the area of the triangle you see sandwiched between the two vertical lines, the function, and the t-axis.). Keep track of units! What does this area represent? (4) Find the AREA under the graph between t=0 and t=4. What does this area represent? (5) Find the AREA under the graph between t=0 and t=6. What does this area represent? (6) Draw the function that represents Abby s distance from the school in feet as a function of time. Find the equation for that function.

7 Exercise 3: Suppose Abby is too exasperated by the students. She flees class, hops on a scooter and starts pushing hard against the ground to get the scooter going fast. A few seconds later, she realizes she has a job to do and lets the scooter coast to a stop. The graph below shows her velocity in ft/sec as a function of time. (1) What is her velocity after 2 seconds? After 3 seconds? After 5 seconds? (2) Calculate the area under the curve. Be careful about units. What does this area mean in the context of the problem? (3) How far away from the school is she after 3 seconds? (4) Can you figure out how far from the school she is after 5 seconds? Why are you having a hard time? (5) Graph her distance as a function of time on the blank grid as best you can.

8 Exercise 4: Abby again is riding away from the school on a scooter due to a need to escape. This time, it s raining so she first kicks the ground much faster than she did before to get to Whole Foods More quickly, and when she comes to her senses and realizes she has to return to work, the scooter slows to a stop more quickly too. The graph below shows her velocity in ft/sec as a function of time. (1) How fast is Abby going after 2 seconds? After 3 seconds? After 4 seconds? (2) Calculate the area under the curve (the area between enclosed by the parabola and the x-axis) as best you can. Use any method and show how you did it. What does this area represent in the scenario? (3) Graph her distance as a function of time on the blank grid as best you can.

9 CHALLENGE 3- Graphing functions Exercise 1: Suppose you know that a 4 th degree polynomial (so it s equation is something like: Y=ax^4+bx^3+cx^2+dx+e). You also have figured out two of its intercepts. They re at (-1,0) and (2,0). (1) Draw me 3 sketches of what this function could look like. (2) What additional information do you want to know to graph this function more completely? (3) I ll throw you another bone. Suppose it has a y-intercept of 2. Meaning it passes through the point (0,2). Draw me 3 sketches of what it could look like now. (4) Here s the equation for the polynomial: y=-3x^4+6x^3+2x^2-5x+2. Which of your sketches above best match the equation for this polynomial? How do you know?

10 Exercise 2: On graphs, there are points on a graph called critical points. There are five types of critical points. Not every graph has all five of them. The five types of critical points are absolute maximums, relative maximums, absolute minimums and relative minimums and inflection points. Not every function has all five. Parabolas, for example, only have either an absolute maximum or an absolute minimum. (1) Look at the graphs below. Put dots on the graph where you think critical points might be located. (Take your best guess at what they mean.) Use the abbreviations to make it easier: Absolute maximum (AMAX) Relative maximum (RMAX) Absolute minimum (AMIN) Relative minimum (RMIN) Inflection point (IP) (2) Now, thinking about the fact that we re trying to graph a complete picture of the function y=-3x^4+6x^3+2x^2-5x+2, what questions might you want to ask about it? Do you have any inklings of what is true about this functions critical points? How many does it have do you think? Where might they be located? (3) Suppose they told us another point the function passed through. We know it s intercepts and now they tell us that the function passes through the point (-0.8,3). We DO NOT know if this is a critical point. Draw 4 possibilities for what our function could look like. Would knowing a critical point be better?

11 Exercise 3: Consider the three graphs below. (1) Using words, how could you describe their differences? (2) Now, play a game with your partner. Look at the 8 graphs below. Choose 1 in your head and have your partner guess which graph you re thinking of by asking just yes or no questions. Try to practice the language outlined in the last exercise.

12 Exercise 4: Finally, consider the fact that we were trying to graph the function: y=-3x^4+6x^3+2x^2-5x+2 (1) I d like you to dredge up some of your pre-calc knowledge and show me how we d find the intercepts of this function. (2) Now suppose there was a magical process by which you could find the x-coordinates of critical points of the function. You do this magical process and it tells you there are exactly three critical points. One at x=0.541, one at x=0.5 and one at x= Use this information and previously revealed information to graph the complete function. How can we be sure there aren t more wiggly bits between the critical points? (3) Any ideas about what the magical process is that gives you the critical points? (If you ve done challenge 1, you might have an idea.)

13 SUMMARY- I expect your answers to these questions might be longer than the space provided. Please write up your answers on a separate sheet of paper. (1) There are three major problems that mathematicians and physicists kept encountering before the development of calculus. To solve these three problems, they invented calculus. What are the three problems? (2) In playing with these three problems, did you notice any commonalities or interrelationships between how you went about trying to solve them? (3) In all three problems, where does infinity come into play? (4) For all three problems, to solve them, you have to rely on the assumption that the function we re working with is something called continuous. What does this mean do you think? Can you give some examples of what might happen if you were trying to solve one of these problems on a function that isn t continuous?

The Jumping Dog Quadratic Activity

Standards: The Jumping Dog Quadratic Activity A2.4.1 Identify the family of function best suited for modeling a given real-world situation. A2.4.3 Using the adapted general symbolic form, draw reasonable