class 0304

In our last class we found that the demand function demand(price) = 200 - 3 * price gives us the revenue function

revenue(price) =

price ( 200 - 3 * price) =

-3 * price^2 + 200 * price. 

This is a quadratic function of price, and it was analyzed using the standard methods for analyzing a quadratic.

A graph of the function is a parabola opening downward, with vertex (33.33, 3333) and zeros at (0, 0) and (66.67, 0).

For the revenue vs. price function of the preceding example,

• What is the rise of the graph between the points where price = 30 and price = 40 and what does the rise mean?

If price = 30 we have revenue(price) = revenue(30) = -3 * 30^2 + 200 * 30 = 3300, meaning a revenue of $3300.

If price = 40 we have revenue(price) = revenue(40) = -3 * 40^2 + 200 * 40 = 3200, meaning a revenue of $3200.

The rise of the graph is the change in the y = revenue coordinate, which is 3200 - 3300 = -100, indicating the $100 drop in the price.

• What is the run of the graph between the points where price = 30 and price = 40 and what does the run mean?

The run of the graph is from x = price = 30 to x = price = 40, a run of 10 units.

This represents the $10 change in price of $30 to $40.

• What is the average slope of the graph between the points where price = 30 and price = 40 and what does the slope mean?

The average slope of the graph between these points is

• ave slope = rise / run = -100 / 10 = -10,

meaning -100 dollars / (10 dollars) = -10 dollars / dollar = -10, or -10 dollars of revenue change per dollar of price increase.

This tells us that in this price range, every dollar of price increase corresponds to an average of a 10 dollar loss of revenue, on the average.

• What is the equation of the straight line through the points where price = 30 and price = 40?

The two points are (30, 3300) and (40, 3200).  The slope = slope formulation tells us that

(y - 3300) / (x - 30) = (3200 - 3300) / (40 - 30) so that

(y - 3300) / (x - 30) = -100 / 10 or

(y - 3300) / (x - 30) = -10.  Solving for y we first get

y - 3300 = -10 ( x - 30) so that

y = -10 ( x - 30) + 3300 and

y = -10 x + 3600.

The straight line connecting these two points is therefore

revenue = -10 * price + 3600.

DERIVE syntax:

declare price as a variable with the line price:=

do the fit using FIT([price, m·price + b], [30, 3300; 40, 3200])

simplify, approximate

you get 3600 - 10·price

If the length of a spring is 30 cm when no weight is suspended and the length increases by 5 cm for every additional pound of suspended weight then

Immediate response before even reading the rest is to construct a graph of length vs. weight and specify everything we can.

We use the same guidelines as in the preceding class:

• Recognize that this is a linear function.

• Find the intercepts.

• Find the basic points.

• Find the slope.

• Sketch the graph.

Recognize that this is a linear function:

Length increases by the same amount for every additional pound, so a graph of length vs. weight will be linear.

Find the intercepts.

The length is 30 cm when no weight is suspended, which corresponds to the point (0, 30) on a length vs. weight graph.

Find the basic points.

One basic point is the y intercept (0, 30).

Another is the point 1 unit to the right of the y axis, which corresponds to x = weight = 1.  In moving from the x = weight = 0 point (0, 30) to the x = weight = 1 point the weight changes by 1 lb, which according to the given information increases the length by 5 cm.  So the new point will be (0 + 1, 30 + 5) = (1, 35).

This shows us that the graph is increasing as we move to the right, and increasing by 5 units in the y direction for every unit in the x direction.

Find the slope.

The graph increases by 5 units in the y direction for every unit in the x direction, so the slope is rise / run = 5 / 1 = 5.

Sketch the graph.

A sketch will start at (0, 30) and will have a slope of +5, passing through the point (1, 35).

• What function length(weight) describes the length of the spring in terms of the weight?

The y = weight intercept is 30 and the slope is 5 so the equation is y = m x + b = 5 x + 30, i.e.,

length = 5 * weight + 30.

• What is the meaning of the slope of this graph?

The slope indicates change in length / change in weight, which is the rate at which length changes with respect to weight.

• What is the meaning of the y intercept of this graph?

The y intercept is the length when the weight is zero, i.e., the unstretched length of the spring.

For the number sequence 11, 14, 17, 20, 23, ...

• What is the pattern of the sequence?

The pattern is that the sequence starts at 11 and every new number is 3 greater than the preceding number.

• What function f(n) can be used to give the numbers in the sequence?

This sequence could be modeled by a linear function with y-intercept 11 and slope +3:

f(n) = 3 * n + 11.

We might also specify that n = 0, 1, 2, ... .

Plugging in n = 0, 1, 2, ... gives us f(n) values 11, 14, 17, ... .

If a(n) = a(n-1) + 5, with a(0) = 12, then what are the terms a(1), a(2), a(3) and a(4)?

Substituting n = 1 we get

• a(1) = a(1-1) + 5, or

• a(1) = a(0) + 5, which since a(0) = 12 gives us

• a(1) = 12 + 5 = 17.

Substituting n = 2 we then get

• a(2) = a(2-1) + 5, or

• a(2) = a(1) + 5, which since a(1) = 17 gives us

• a(2) = 17 + 5 = 22.

Substituting n = 3 we get

• a(3) = a(3-1) + 5, or

• a(3) = a(2) + 5, which since a(0) = 12 gives us

• a(3) = 22 + 5 = 27.

If we substitute n = 4 we get a(4) = ... = 32.

The sequence is 12, 17, 22, 27, 32, ... .

The process for manually fitting a straight line to data points is as follows: 

• sketch a graph of the data points

• sketch the straight line you think comes as close as possible, on the average, to the data points

• Pick two points on your straight line.  Don't pick two of the original data points.

• Find the equation of the straight line through the two points you picked.

Your equation should approximate that of the best-fit straight line through the points. 

The actual best-fit line is found by minimizing the squares of the deviations between the predictions of the straight line and the data points. 

There is a formula for doing this, and graphing calculators or computer algebra programs also have commands for doing so (e.g., in DERIVE the command is fit([x, mx+b], ##) where ## is the data set.

We began by observing two spherical containers full of water.  After holding and comparing their 'feel' everyone was asked the following question:

• If you were dying of thirst and someone offered to trade you a bunch of the smaller containers for six of the larger, what is the smallest number of smaller containers you would accept? 

Answers varied from 20 to 30, which as we will see turns out to be pretty good.

The other question was

• How many times the diameter of the smaller is the larger?

Most people said about twice the diameter, and very few were able to make a decision between more than twice and less than twice the diameter.

So the consensus was that the big sphere contains between 3 and 4 times as much water and is close to twice the diameter of the smaller.

Is this consistent?

No.  As we've seen already if one solid has twice the linear dimension of another geometrically similar solid it has 2*2*2 = 2^3 = 8 times the volume.  We're all very sure that if we were dying of thirst we wouldn't need 48 of the smaller sphere to make up for 6 of the larger.

What is our conclusion about how many times the diameter of the small we have in the large sphere?

If we have less than 8 times the volume, which seems very clear after handling and thinkin' about drinkin' the things, then we conclude that the big sphere is less than twice the diameter of the small.

Measurement shows that the smaller contains 100 ml of water while the larger contains 263 ml of water.