Tangents And Velocities

MAT 135: Calculus In-Class Group Work on Tangents and Velocities Instructions • You will be assigned arbitrarily into groups of 2 or 3. Each group will have the responsibility of writing up clean, correct work for each of the exercises below AND for presenting selected solutions at the board or document camera. Decide in advance who will be writing the final copy and who will present. • The written-up copy of your work is to be handed in. Please submit only one written-up copy per group. It will be assessed as part of your grade for In-Class Work. Each group member will get the same score. Your entire work during class -- including this assignment and any board presentations we do -- will be graded on a scale of 0 to 4 on the basis of completeness, effort, and a reasonable attempt at mathematical correctness. • Try to be right and definitely make significant headway on each task, but don’t be afraid to be wrong.

1. In an earlier example, we modeled the population of fleas in Prof. Talbert’s house during a recent (and thankfully past) infestation as a function of time using the exponential function

F(t) = 15 (1.1225 )

t

where t was measured in hours and F in number of fleas. This function tells us the size of the flea population at any given time. We want now to examine the rate at which this population is growing. (a) Using Winplot, make a graph of this model in an appropriate viewing window. Does the population grow at the same rate all the time? If not, describe how it does grow over time. (b) The average rate of change of the population over a time interval is defined to be the amount by which the population grew (or declined) over that time interval, divided by the length of the time interval. For example, the average rate of change of the flea population from t = 0 to t = 24 (that is, over the first day) is

(Population at t = 24) − (Population at t=0) F(24) − F(0) 240.194828 − 15 = = ≈ 9.383 24 − 0 24 − 0 24

(c) (d) (e) (f) (g)

Since we are dividing number of fleas by hours, the units of measurement on 9.383 are in “fleas per hour”. (Sounds like a speed.) Does this mean that the population is always gaining about 9 fleas each hour? Why not? Find the average rate of change in the flea population from t = 12 to t = 24; from t = 20 to t = 24; and from t=23 to t = 24. Excel would be a good tool to use, or you can just use a calculator. Find the average rate of change over the time intervals ending at t = 24 and starting at t = 23.5, t = 23.9, t = 23.99, t = 23.999, and t = 23.9999. Find these values using Excel by entering in data and dragging a formula. How long is the last time interval you used (from t = 23.9999 to t = 24), in seconds rather than hours? How fast is the population changing right at t = 24 hours? Justify your answer, using the work from parts (d) and (e). Since this rate is not being taken over an interval but at a single point, we call this the instantaneous rate of change in the population at t = 24. Explain how you would calculate the instantaneous rate of change in the flea population at t = 36, and describe how this quantity would compare to the instantaneous rate of change at t =24 (bigger, smaller, the same, etc.).

2. Bob pitches a baseball straight up into the air. The height (in feet) of the baseball above the ground is a function of time (in seconds) given by the graph below. (Bob catches the ball on its return trip, so its motion stops at t = 2.4 when it is roughly 4 feet above the ground.) 30 28 26 24

Height above ground (feet)

22 20 18 16 14 12 10 8 6 4 2 0 -2

0

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

2.8

3

Time (seconds)

(a) The average velocity of the ball over a time interval is the average rate of change in its position over that interval -- that is, it is the total change in position divided by the length of the time interval. For example, the average velocity of the ball from t=0 to t=2 seconds is

(Position at t=2) − (Position at t=0) 16 − 0 = =8 2−0 2−0

(b) (c) (d) (e)

And since we are dividing feet (position) by seconds (time), the units of measurement on this answer are feet per second. Does this mean the ball is always travelling 8 feet per second? Why not? Find the average velocities of the ball over the time interval from t = 1 second to t = 2 seconds. Draw the secant line through the points (1,24) and (2,16). What is the slope of this line? What does the slope of a secant line have to do with average velocity? Find the average velocity of the ball from t=1 second to t=1.5 seconds; from t=1 to t=1.2, and from t=1 to t=1.1. Draw the secant lines having these slopes. What is happening to the secant lines as the time interval shrinks? Draw the tangent line to the graph right at t=1 and use the grid markings to estimate its slope. What do you think this slope would tell you in terms of the baseball’s motion>