You should view the video clip relevant to your course.  Calculus and Physics students should also view the files referred to in the following.

Precalculus and Calculus video clips are concerned with the introductory flow model and its mathematical analysis.  Note that this model might seem like physics, but we aren't doing the physics, we are doing the mathematics.  We use a physics situation to provide us with an accessible and accurate modeling problem for which data can be obtained quickly and easily.  In the mathematics class we don't think much about the physics, any more than you think about the physics when you drive your car.

• Calculus students who don't completely understand the Calculus analysis are invited to view the precalculus clips, which go into details with which we assume calculus students are familiar.
• Calculus students should view the physics video clips to familiarize themselves with the ideas of velocity and acceleration, and the behavior of the graphs of these quantities.   Similar models are standard examples in any calculus course.

Physics students should view and understand the Precalculus and/or Calculus clips:

• General College Physics students should view and understand the Precalculus clips (recall that Precalculus is a corequisite/prerequisite for this course)..
• University Physics students should view and understand the Calculus clips (recall that Calculus is a corequisite/prerequisite for this course).

Precalculus Week 1 Video Clip 02 copy cal02

http://youtu.be/DUFJVmNbGZk

• Is the graph of depth vs. time increasing or decreasing?
• Is the depth changing and increasing, a decreasing or a constant rate, and how would each of these possibilities be seen in the shape of the graph?

Precalculus Week 1 Video Clip 03 describes depth vs. clock time data, asks about evidence for way rate changes, for graph shape from the table

http://youtu.be/EN1jinPpwzU

• How does the depth vs. clock time data give us evidence for the shape of the graph?

Precalculus Week 1 Video Clip 04 evidence for slowing of flow, idea of rate

http://youtu.be/8it5s9UYS4I

• What evidence is there in the table of data for the speeding up or slowing down of the flow?

Precalculus Week 1 Video Clip 05 the graph of the data, steepness indicates rate

http://youtu.be/5qxVLDlAD9Q

• How is the steepness of the graph related to the rate at which water flows from the cylinder? What does the steepness of the graph tell us about the way water flows from the cylinder?

Precalculus Week 1 Video Clip 06 columns for `dt, `dy, `dy/`dt; rate calculations

http://youtu.be/A7qogD-YUVs

• Given the table of y = depth vs. t = clock time data, how do we calculate the corresponding time and depth intervals `dt and `dy, and how we represent these quantities on the table?
• How do we calculate the average rates `dy / `dt? Why we call these average rates? What do they tell us about the water in the cylinder?

Precalculus Week 1 Video Clip 07 rate calculations and slope

http://youtu.be/nQy5A5eINwI

• How are our rate calculations represented graphically by the slope?

Precalculus Week 1 Video Clip 08 is there a mathematical rule, a function, that predicts these quantities? If so, call it a mathematical model.

http://youtu.be/LQyNvGzVGAA

• Why would we expect that there exists a mathematical rule or function that predicts the depth y vs. time t for the flow situation?

Precalculus Week 1 Video Clip 09 y = x^2 to general quadratic: stretching, shifting, always gives quadratic, every quadratic can be so obtained

http://youtu.be/NT7Zq5Pzld0

• Why does it seem reasonable that by stretching in shifting the y = t^2 graph we should obtain a graph that models be observed depth vs. time data?

Precalculus Week 1 Video Clip 10 choosing points and obtaining simul. eq. with 3 parameters.

http://youtu.be/_hfwbuF1OXQ

• How by choosing three points do we obtain three simultaneous equations for the three parameters of the quadratic model?
• What are the three parameters of the quadratic model?
• What are some of the things we should think about as we choose the points to put into our quadratic model?

Precalculus Week 1 Video Clip 11 eliminating c

http://youtu.be/IDMfsHpIX9E

• How do we eliminate a variable from a system of three simultaneous equations in three unknowns to obtain a system of two simultaneous equations in two unknowns?

Precalculus Week 1 Video Clip 12 eliminating b, obtaining a

http://youtu.be/yEtldcKMlrc

• Given a system of two simultaneous equations into unknowns, how do we eliminate one of the unknown and solve for the value of the other?

Precalculus Week 1 Video Clip 13 back substitution, plugging into form to get model

http://youtu.be/bl2JzTws-LM

• Once we have found the value of one of the variables, how do we use back substitution to find the values of the others?

Precalculus Week 1 Video Clip 14 comparing model with data: deviations and residuals

http://youtu.be/WyBVVWNvnHU

• How do we find the deviations and residuals of our model, and what do they tell us about the quality of our model?

Calculus Week 1 Video Clip 01 flow: is depth changing at increasing, decreasing or const rate? What about range?

http://youtu.be/E5wB-rL0A-U 

• How could we determine whether the depth of the water is changing at a constant, decreasing or increasing rate?
• How could we determine whether the range of the water stream is increasing a constant, decreasing or increasing rate?
• How is the range of the stream related to the rate of which the water level changes?
• Is the rate at which the water level changes related to the rate of which the range of the stream changes?

Calculus Week 1 Video Clip 02 which graph shape (if any) best represents the depth vs. clock time?

http://youtu.be/Qm03aMGGjNU

• Is the graph of depth vs. time increasing or decreasing?
• Is the depth changing and increasing, a decreasing or a constant rate, and how would each of these possibilities be seen in the shape of the graph?

Calculus Week 1 Video Clip 03 we observe data in long tube, 90 cm to 0 cm; data is average over class observations, not too reliable

http://youtu.be/kW5RB7uXe-o

• How does the data indicate whether the depth is changing a constant, an increasing or decreasing rate?

Calculus Week 1 Video Clip 04 graph

http://youtu.be/3cViWkLhHSk

• How does the shape of the graph indicate whether the water depth is changing at an increasing, they decreasing or a constant rate?

Calculus Week 1 Video Clip 05 calculate `dy / `dt values
http://youtu.be/XQNqYHI04mw

• How do we calculate the average rate at which depth changes over each of the time intervals represented on the table?

Calculus Week 1 Video Clip 06 calculate slop44e, interpret
http://youtu.be/qvs4xL0xeOY

• How do we use the graph to calculate the average rate of which the depth changes over given time interval?

Calculus Week 1 Video Clip 07 summarize from table to graph, show slopes between successive points
http://youtu.be/0MCweQvr41k

• How are the average rates of depth change for the various intervals depicted on the graph and on the table?

Calculus Week 1 Video Clip 08 Physics says that we can model data with a quadratic model

http://youtu.be/5IFLGswqZu0

• Physics uses the techniques of calculus to determine what kind of model will work for a given physical phenomenon. We are worried about the details of physics here, but physics tells us that the depth vs. time function for this situation is going to be a quadratic function.

Calculus Week 1 Video Clip 09 creating model: select three points, obtain system of 3 simul. lin. eq. for parameters

http://youtu.be/L532Ka8AWdk

• How and why do we plug into the form of a quadratic function y = a t^2 + b t + c the depth in time coordinates of three of our data points?

Calculus Week 1 Video Clip 10 solving the equations: reducing system to 2 eq.

http://youtu.be/ULfj8yr_7L8

• How do we reduce a system of three simultaneous linear equations in three unknowns to a system of two simultaneously linear equations into unknowns?

Calculus Week 1 Video Clip 11 reducing 2 eq. to 1, solving 1

http://youtu.be/yC0zk-JW0Yg

• How do we obtain a single equation in a single unknown from system of two equations into unknowns, and how does this give us the value of one of the parameters?

Calculus Week 1 Video Clip 12 back substitution

http://youtu.be/vjiSOaNINJ4

• How do we use back substitution to determine the values of all three parameters given the value of one?
• How do we obtain our mathematical model of depth vs. time from our solutions of the three simultaneous equations?

Calculus Week 1 Video Clip 13 questions to ask of model

http://youtu.be/Z-QRi475Jws

• How do we determine the depth predicted by the model for a given clock time?
• How do we determine the clock time in which a given depth is achieved, according to the model?

Calculus Week 1 Video Clip 14 answers to questions: graphical

http://youtu.be/MDoin27-Rm8

• How do we use a graph of our model to determine the depth data given clock time?
• How we use a graph of our model to determine the clock time corresponding to a given depth?

Calculus Week 1 Video Clip 15 when depth is 0

http://youtu.be/4MABYa6KPmo

• How do we determine when depth is 0?
• We know that the depth will eventually be 0. How can be that the depth vs. time function does not necessarily have a solution corresponding to a depth of 0?

Calculus Week 1 Video Clip 16 what is y when t=30, when is y=42?

http://youtu.be/lniUOSquL-s

Calculus Week 1 Video Clip 17 modeling rate of depth change by function: finding the two points

http://youtu.be/7hrcYHJdV4w

• If we have a model of depth vs. time, and wish to find the average rate at which depth changes between two given clock times, how do we find the two depths corresponding to the given clock times? Where the coordinates of the graph points corresponding to these two instants?

Calculus Week 1 Video Clip 18 calculating the slope, the ave. rate of change

http://youtu.be/KSDXXCqtR_4

• How do we calculate the average rate of change once we know the two positions and clock times required for the calculation?
• How do we calculate the average rate of change once we have found the coordinates of the two graph points?
• How is the average rate which depth changes represented on the graph of depth vs. time?

Calculus Week 1 Video Clip 19 goal: exact rate at t = 10. calculating rate over a shorter time interval

http://youtu.be/Xjm-StLIdAI

• How do we use the calculation of average rates over shorter and shorter time intervals to approach the instantaneous rate at a given clock time?

Calculus Week 1 Video Clip 20 symbolizing the shortening of the interval, calculating y in symbols

http://youtu.be/3mzkb1AgrLM

• What is the advantage of using symbols rather than specific numbers to represent our calculation of average rates over shorter and shorter time intervals?
• Given a quadratic model of depth vs. time, how would we obtain a symbolic expression for the depth corresponding to a clock time given by the symbolic expression t + `dt?

Calculus Week 1 Video Clip 21 calculating the slope in symbols

http://youtu.be/xSn9O7LadjA

• Once we have symbolic expressions for two clock times in the two corresponding depths, how do we arrive at a symbolic expression for the average rate which depth changes?

Calculus Week 1 Video Clip 22 taking the limit

http://youtu.be/VIbvEQFkX0E

• Once we have a symbolic expression for the average rate which depth changes, in terms of the length `dt of the time interval, how do we calculate the instantaneous rate which depth changes at the given clock time?

Calculus Week 1 Video Clip 23 arbitrary point t: finding y(t+`dt); student should complete.

http://youtu.be/hntRh6E5Mlk

• Given the function y(t), how do we find a symbolic expression for y(t+`dt)?

Calculus Week 1 Video Clip 24 another model: finding y(t+`dt) - y(t), cancellation of all of y(t)

http://youtu.be/HgHKo5dx6qQ

• For a given quadratic function y(t), how do we find the symbolic expression for `dy = y(t + `dt) - y(t)?

Calculus Week 1 Video Clip 25 completing `dy / `dt, taking limit

http://youtu.be/r4GK0um1Y-A

• For a given quadratic function y(t), how do we find the symbolic expression for `dy / `dt = (y(t + `dt) - y(t)) / `dt?

Calculus Week 1 Video Clip 26 the connection between the original model and the rate function, dy/dt and y'(t) notations.

http://youtu.be/8ykAv_ENaVk

• For a given depth vs. time function y(t), how do we find the rate of change function y'(t)?
• What is the meaning of the symbolic expression dy / dt?

Intro Force Exp:  Joining Rubber Bands            Introductory Force Experiment

Intro Force and Pendulum Exp:  Making the Pendulum             Pulling Back the Pendulum

Physics video clip 01 ball rolls down level ramp, questions: const vel? how to measure? shape of graph?

http://youtu.be/hWYse-JUAdU

• A ball rolls down a straight inclined ramp. It is the velocity the ball constant? Is the velocity increasing? Is the velocity decreasing?
• If the ball had a speedometer we could tell. What could we measure to determine whether the velocity of the ball is increase or decreasing?
• What is the shape of the velocity vs. clock time graph for the motion of the ball?

Physics video clip 02 ball rolls down curved-end ramp, same questions const vel? how to measure? shape of graph?

http://youtu.be/VbQDgXnPdSM 

• A ball rolls down ramp which curves upward at the starting end and otherwise rests on a level table. Answer the same questions is on the preceding video clip.

Physics video clip 03 v vs. t graph: certainly increasing, but at const, incr or decr rate?

http://youtu.be/PoSC4LHc6f8

• For the ball on the straight inclined, we would certainly agree that the ball's velocity is increasing. Is the velocity increasing the constant, and increasing, or decreasing rate? What is the graph of velocity vs. clock time look like?

Physics video clip 04 increase from graph: note `dt and `dv

http://youtu.be/EGsqMFwLL8g

• Given a graph of velocity vs. time how do we determine the rate which velocity increases?

Physics video clip 05 calculate ave. rate of change

http://youtu.be/TPi_j21-_TM

• If we know the velocities at two given clock times how do we calculate the average rate of which the velocities changing during that time interval?

Physics video clip 06 speedometer example: rate at which speedometer moves

http://youtu.be/MsZ2OBiw164

• Does the rate which is speedometer needle moves tell us how fast the vehicle is moving?
• Does a stationary speedometer needle implies stationary vehicle?
• Does quickly moving speedometer needle imply a quickly moving vehicle?
• What does it feel like insider car when the speedometer needle is moving fast?
• What does the speed of the speedometer needle tell us?

Physics video clip 07 slope triangle, instantaneous acceleration, tangent line

http://youtu.be/IGGvdewuysY

• How do we represent the calculation of the rate which velocity changes on graph of velocity vs. time?
• If we know the velocity at every clock time, that how we find the precise rate at which velocity changes adding given clock time?
• How do we depict the instantaneous rate of velocity change on a graph of velocity vs. time?

Physics video clip 08 finding displacement: define displacement, then calculate by common sense, by formula, by area

http://youtu.be/42KhTfAU2-c

• What is the difference between displacement and distance?
•  

In how many different ways can we represent the calculation of the displacement over a constant-acceleration time interval?

• How do we calculate displacement from velocity and time interval?  What is the common sense of this calculation?
• How do we use a graph to help visualize the calculation of displacement when velocity is constant?
• What aspect of a v vs. t graph tells us the displacement?

Physics video clip 09 displacement for linear v vs. t graph: common sense, formula, area

http://youtu.be/w18S9FJTGWc

• If we know the initial and final velocities over some time interval, and if the rate which velocity changes is constant, then how do we calculate the displacement over a the time interval?
• If we know the initial and final velocities over some time interval, and if rate which velocity changes is constant, then how do we calculate the displacement of the object during that time interval?

Physics video clip 10 continuing 09: calculation of area

http://youtu.be/WTxdmvZ9zkU

• How do we use a graph of v vs. t to depict the calculation of the displacement over a time interval?
• What aspect of the graph gives the displacement during the time interval?

Physics video clip 11 `ds on single time interval: ave. ht * width = vAve * `dt = `ds

http://youtu.be/ucgqZ6Q4Qek

• How do we calculate the displacement during a single specific time interval, given a linear v vs. t graph?
• If we know the initial and final velocities over some time interval, and if the rate at which velocity changes is constant, then how do we calculate the displacement of the object during that time interval?
• In how many different ways can we represent the calculation of the displacement over a constant-acceleration time interval?

Physics video clip 12 continuing ph11

http://youtu.be/vbvWSJH1Nf8

• How does the graphical calculation connect with our common sense about velocity, displacement and time?

Physics video clip 13 repeat for all trapezoids under curve / all short time intervals from t0 to tf

http://youtu.be/Bkt9gDc34HE

• If we know the initial and final velocities over some time interval, and if the rate which velocity changes is constant, then how do we calculate the displacements over a series of time subintervals?

Physics video clip 14 emphasize adding displacements, mention accumulated area

http://youtu.be/atA3hXJ_G4U 

• Having calculated displacements over all specified subintervals, how do we obtain the total distance moved from the initial clock time up through the end of each subinterval?
• How do we construct a position vs. time graph from a velocity vs. time graph?

Physics video clip 15 show generally what we get from a trapezoid, emphasize how trapezoid tells us when to add and

http://youtu.be/_QTyl-hIMoc

• How do we symbolically and graphically represent the calculation of the displacement corresponding to a given initial and final velocity over a given time interval?
• How do we know when to subtract and when to add the initial in final velocities in calculating average rates of velocity change and displacements from initial and final velocities and time interval?

Physics video clip 16 Experiment 1: Find the average velocity of a ball rolling from rest down a straight inclined ramp; infer initial velocity and final velocity, then infer acceleration. Use the program TIMER (on CD, or ask instructor to E-mail) and a measuring device (e.g., tape measure, ruler, meterstick).

http://youtu.be/naiDpyg5svE

• What do we measure to obtain average velocity?
• How do we obtain average velocity from our observations?
• How do we then infer initial and final velocities?
• How then do we infer acceleration?