THIS DOCUMENT IS CONTAINED IN THE SUMMARY DOCUMENT FOR THE FIRST TWO WEEKS OF CLASS, WHICH COVERS MOST OF ASSTS 1-3
Precalculus 1 Quiz 040115:
Sketch a smooth curve through your depth vs. clock time data points.
Estimate the coordinates of three points on your curve but not coinciding with actual data points.
From your curve estimate the clock time at which depth is 53, as well as the depth at clock time t = 10.
Suppose the points are (5, 70), (11,50) and (14, 44).
Plug the coordinates of each point into the form y = a t^2 + b t + c. You'll get three equations.
The equations are
25 a + 5 b + c = 70
121 a + 11 b + c = 50
196 a + 14 b + c = 44.
For the first two equations subtract the second from the first
25 a + 5 b + c = 70
- [ 121 a + 11 b + c = 50 ]
_____________________
-96 a - 6 b = 20
Subtracting the second equation from the third we get
75 a + 3 b = -6
This gives us the system
-96 a - 6 b = 20
75 a + 3 b = -6
Doubling the second equation gives us the system
-96 a - 6 b = 20
150 a + 6 b = -12
which we add to get
54 a = 8
We easily solve this to get
a = 8 / 54 = 4 / 27
We plug this into either of the two equations in a and b. Picking the first of these equations
-96 a - 6 b = 20
we obtain
-96 (4 / 27) - 6 b = 20.
Multiplying both sides by 27 to clear the fraction and doing the rest of the obvious stuff we get, if my arithmetic is correct,
b = - 154 / 27.
Plugging our values of a and b into the first of the original equations we get
25 (4 / 27) + 5 (-154 / 27) + c = 70
which we solve by familiar but irritating details to get
c = 2560 / 27.
Our values of a, b and c give us the model
y = 4 / 27 t^2 - 154 / 27 t + 2560 / 27.
We can approximate this equation by
y = 0.148·t^2 - 5.70·t + 94.8
The Excel best-fit model for our actual data is
y = 0.131 t^22 - 6.74 t + 95.4
which should be better than our hand-graphed result.
Class estimates of the clock time at which depth is 53, as well as the depth at clock time t = 10, give us respective averages of t = 7.4 and y = 46.
How do these estimates compare with the predictions of our function
y(t) = 0.148·t^2 - 5.70·t + 94.8?
y stands for depth, so if we plug in y(t) = 53 we obtain the equation
53 = 0.148·t^2 - 5.70·t + 94.8.
If we can solve this equation for t then we have a prediction based on the model.
We can solve this equation by putting it into the form a t^2 + b t + c = 0 and using the quadratic formula.
We end up with two solutions:
t = 9.86 or
t = 28.7.
The t = 28.7 doesn't correspond to anything that happens in the real world, but the t = 9.86 does. Our graph isn't overly accurate, and our t = 7.4 estimate doesn't compare badly with the t = 9.86 from the equation.
To get the model's prediction of the depth when t = 10 we simply plug t = 10 into
y(t) = 0.148·t^2 - 5.70·t + 94.8
to get
y(10) = 0.148·10^2 - 5.70·10 + 94.8 = 52.6.
Again this compares reasonably well with our sloppy-graph estimate of y = 46.
smooth curve thru all data points ...
Ask specific questions, get specific answers, email first and you need to do the qa; also ask questions at the beginning of class, not stick around after and ask, general phil being that I need to address your questions in a way that benefits everyone and not just you. You ask me a question, you write out a transcript and share with everyone, anonymously or otherwise, your choice.
another precalc question: why is the quad model not good for the freq vs. length data; make sure they graph data for all the first three situations.
note how grainy our depth data was but how smooth the model was and how well it approximated when we consider that any of the data points could have been half a unit to the right or left
take a look at the major quiz and see where we're heading
gotta move into trap graphs and interp etc.
challenge problem: if we know at each instant how quickly temp is changing how can be figure out the temp change over a given time interval?
maybe first bring up the concept of rate in terms of how fast temperature changes; hate to get into numerical examples but
algebra review topics need to be brought up on a regular and scheduled basic
we're not going to bother with lin eqns but gonna need quad form and applications, techniques for solving power fn models, right from the beginning and we should state in review examples of the kinds of things we can do to solve equations: adding, mult, taking powers of both sides etc.
if y = at^2 + bt + c represents depth vs. clock time (and incidentally note that I need to find the curve fit before the next class) then to find the clock time at which a certain depth occurs
plug depth in for t (maybe say x instead) and evaluate
plug in depth for y and rearrange the equation so the quad form applies then use the quadratic formula