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

• 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?

• 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

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

Calculus Week 1 Video Clip 04 graph

• 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

• 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 slope, interpret

• 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

• 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

• 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

• 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.

• 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

• 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

• 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

• 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

• 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

• 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?

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

• 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

• 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

• 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

• 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

• 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

• 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.

• 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)

• 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

• 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.

• 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?

Lecture #4:  The Concepts of Differentiation and Integration in the Context of Rate Functions

Lecture #5:  :  Growth of an Exponential Function; Trapezoidal Representation of Approximate Derivatives and Integrals

Lecture #09:  Text problems; Introduction to Natural Logarithms and Composite Functions; Numerical Solution of Differential Equations

Lecture #10:  Predictor-corrector methods; Sine and Cosine functions

Lecture #11:  Trigonometric Functions; Brief Intro. to Polynomials

#20:   The Fundamental Theorem of Calculus; Derivatives of Polynomials

#27:   Implicit Differentiation; Tangent Line Approximation

#28:   Implicit Differentiation; Maxima, Minima and Inflection Points