Some students had a test problem somewhat like the one below, where they were given a graph of y = f(x) and asked to find the value of f(x) when x = 4, and also the value of x for which f(x) = 12.

• x = 4 occurs on the x axis.
• The corresponding graph point lies directly above the x = 4 point on the x axis.
• To find f(x), we have to understand that the graph represents f(x) vs. x, so that the y coordinate of the graph point is the value of f(x).
• We estimate the y coordinate of this point by moving to the left until we reach the y axis.
• The y coordinate at this point appears to be approximately 20.

Another question asked for the average rate at which a function y = skinThickness(volume) for the skin thickness of a balloon changes between two values of the volume.

• This problem was designed to test your understanding of function notation and the definition of average rate of change.
• In the figure below we have graphed skinThickness (volume) vs. volume, attempting to more or less mimic the way the skin to so they balloon would change as its volume increases.
• We see that as the volume increases from from volume1 to volume2, skin thickness changes from skinThickness (volume1) to skinThickness (volume2).

• Note that we have abbreviated skinThickness as skT and volume as vol in some of labeling below), so that

• the change in skin thickness is rise = [skinThickness (volume2) -skinThickness (volume1)] while
• the change in volume is run = [volume2 - volume1].

The graph of the exponential function y = 2 ^ x can be characterized by its y intercept (0,1), obtained by letting x = 0, and its x = 1 point (1,2).

• We note that the 'doubling time' of this function is `dx = 1, so that every time we move a distance 1 to the right of function will double, and every time removed a unit to the left the function value will become half as great.

We compare this with the graph of y = 1.5 ^ x, which has basic points (0,1) and (1,1.5).

• The graph of y = 1.5 ^ x (note that the red dotted line arrow pointing to the red dotted graph should not be there; the arrow should point from the red y = 2^x to the red dotted graph and there should be a blue arrow from the blue y = 1.5^x to the blue dotted graph, which is the graph of y = 1.5 ^ x) increases by a factor of 1.5 between x = 0 and x = 1.
• As you can verify, either algebraically or numerically, any time x increases by 1, the value of y and hence the height of the graph above the x axis will change by factor 1.5.
• If x decreases by 1,the value of y and hence the height of the graph will change by factor 1 / 1.5 = 2/3.
• The doubling time for the graph will be somewhat less than `dx = 2, since two subsequent increases by factor 1.5 will result in an increase by factor 2.25, more than the doubling.

As noted above we show the graph of y = 2 ^ x in red for comparison.

The first graph below shows the graph of y = 2 (1.5 ^ x).

• Its x intercept is (0,2), and its x = 1 point is (1,3).
• Note that the ratio of y values between these points is again equal to the growth factor 1.5, and any increase in x of 1 unit will result in an increase by factor 1.5 in the y coordinate.

Video File #02

http://youtu.be/q-XW4jUdVSI

As seen in the notes for the preceding class, the recurrence relation for an exponential function is P(n+1) = (1 + r) P(n).

• Recall the example of the preceding class where the growth rate was r = .12 and the initial quantity was P(0) = $10,000.
• The values of P(1) and P(2) are calculated below using these parameters.

Video File #03

http://youtu.be/GebZ_RxC8dM

We next look at the behavior of the temperature of approximately 1 ounce of water in a Styrofoam cup.

• In class we actually observed the temperatures at intervals of one minute, obtaining Celsius temperatures of 61.5, 58.1, 55.2, 52.6, 50.4, 48.9, 46.1, 44.9, 43.4 and 42.3 degrees.
• After each reading was taken every individual was to predict the next reading.

• Most people started out by assuming that if the temperature drops from 61.5 to 58.1 Celsius in a minute, a drop of 3.4 Celsius, then in the next minute the temperature should again drop by 3.4 Celsius to something in the neighborhood of 54.7.
• When the temperature dropped only to 55.2, some people were satisfied (after all, that was only half a degree different in the prediction, and everyone knows that the real world doesn't behave in a completely predictable fashion) and predicted another drop in the neighborhood of 3 Celsius degrees to 52.2.
• Again the temperature changed by less than most people predicted, and it began to dawn on just about everyone that as the temperature dropped closer in closer to room temperature it would change more and more slowly; predictions began to reflect this.

In hindsight we can see that if we write the temperatures as a sequence S = 61.5, 58.1, 55.2, . . ., we might observe a pattern in the difference sequence S' = -3.4, -2.9, -2.6, -2.2, -2.0, -1.8, -1.7, -1.5, -1.1.

• We see from S' that the temperature is indeed changing more and more slowly, and we even obtain a pretty good means of predicting these changes, except that the last value seems a bit too low.

• The graph of temperature T vs. t shows how the temperature changes more and more slowly as we approach the room temperature of 23 Celsius.

Video File #04

http://youtu.be/fJxVom9D_P8

It turns out that for an exponential function y = A b^t, the average rate at which the function changes is in fact proportional to the value y of the function.

• For example, for the 'temperature difference' function Td = T - Tr, where Tr is room temperature and Td is therefore the difference between room temperature and the observed temperature of the water in the cup, we could calculate the average rate at which temperature difference changes for two different times t.
• Obviously when the temperature difference Td is great we expect the temperature to change more quickly then when Td is small.

• The turns out in fact that for most solid objects, the average rates are in the same ratio as the temperature differences Td: that is, Td2 / Td1 = (ave. rate 2) / (ave. rate 1).
• This is just like a statement that (y2 / y1) = (x2 / x1), which expresses the proportionality y = kx.
• We therefore conclude that the rate which temperature changes is proportional to the temperature difference:

• Rate of change of Td (temperature difference) = k Td.

The figure below represents the average rates at two different clock times as graph slopes.

• We see that for the greater temperature difference the slope is greater.  
• Whether the proportionalities on this specific graph are consistent with the    rate = k Td  hypothesis  is left as an open question.

We could attempt to verify this proportionality for the temperature data we obtained for the water in the cup.

• However, we won't get very good results because water is liquid, and the water temperature changes in 'layers', which sometimes invert themselves due to convection; water also cools by evaporation, which doesn't follow this sort of pattern very well either.
• Better results would be obtained by heating a cube of potato or some fruit which has a cellular structure that prevents convection.

Video File #05

http://youtu.be/cKqxgELPvtk