# CLEAR Calculus

## Lab 6: The Derivative at a point - part 2

You will work with people who worked in groups on the other two contexts assigned. Teach each other

1. The meaning of the important quantities in your context, especially the instantaneous rate of change that you approximated and the average rates that you used as approximations
2. How you computed your approximations

3. How you computed error bounds

Turn in a write-up in which you

1. Clearly show your work to approximate the instantaneous rate of change for ALL THREE contexts. Although you are now required to achieve a specified degree of accuracy, your work from Lab 5 in each context should guide your process. Explain how you know your approximation is accurate enough.

2. For the other two contexts, justify why your smaller approximations are in fact underestimates and your larger approximations are in fact overestimates. Note the following:

• You must frame your justifications in terms of what happens in the physical contexts.

• You cannot rely on the shapes of the graphs for your justifications (but you can use them

to subsequently verify your arguments).

• It is not sufficient to simply say one approximation is larger and one is smaller (you could

have two underestimates or two overestimates).

3. Explain the meaning of the average rate of change for your approximations. That is, your description should begin with something like, "This average rate of change [number][units]/[unit] is the constant rate that..." (replacing the words in brackets and completing the sentence).

In-Class Context: A bolt is fired from a crossbow straight up into the air with an initial velocity of 49 m/s. Accounting for wind resistance proportional to the speed of the bolt, its height above the ground is given by the equation $$h(t) = 7350 - 245t - 7350e^{-t/25}$$ meters (with $$t$$ measured in seconds). Approximate the speed when $$t =2$$ seconds accurate to within 0.1 m/s.

Context 1: Approximate the instantaneous rate of change of the volume of a sphere with respect to its radius when the radius is 5 cm. Make your approximation accurate to within 0.1 $$\text{cm}^3$$/cm.

Context 2: NASA has determined that asteroid 1999 RQ36 has a 1 in 1000 chance of colliding with Earth on September 24, 2182*. The force of gravity, $$g$$, in Newtons (N) between two objects is inversely proportional to the square of the distance, $$s$$ in meters (m), separating them. The constant of proportionality is $$GMm$$ where $$G$$ is the "universal constant of gravity" $$6.67\times10^{-11}$$ $$\text{Nm}^2/\text{kg}^2$$ while $$M = 5.97\times10^{24}$$ kg and $$m = 1.4\times10^{11}$$ kg are the masses of the earth and the asteroid, respectively. Approximate the instantaneous rate of change of the gravitational force between the Earth and 1999 RQ36 with respect to distance when the two objects are 10,000,000 m apart. Make your approximations accurate to within 6 N/m.

Context 3: The half-life of Iodine-123, used in medical radiation treatments, is about 13.2 hours. Approximate the instantaneous rate at which the Iodine-123 is decaying 5 hours after a dose of 6.4 $$\mu$$g is administered. Make your approximations accurate to within 0.0001 $$\mu$$g/hr.

*(Class is canceled on September 24, 2182.)