Weekly Tutorial 3 for Math. 69.107A,
Elementary Calculus I

1. A potato at room temperature (20 ^oC) is put in an oven at 200^oC and heats up according to Newton's Law, that is, the temperature T(t), at time t, in minutes, satisfies the differential equation: T'(t) = -k(T(t) - 200), where k > 0.
Find the value of "k" if after 30 mins. the potato's temperature is 120^oC.

a) 0.523
b) 0.027
c) 0.811
d) None of these

2. Differentiate the following function: f(t) = tan { t²/sqrt(t² + 4) } and find the value of the derivative at t = 0. a) 0
b) -1
c) 1
d) sqrt(2)

3. Let Q(t) denote the total quantity at time t of a pollutant in a lake. Assume that this pollutant, when left alone, mixes evenly with the water in the lake and eventually becomes harmless. The model for the concentration of Q(t) in the lake is given by Q'(t) = -0.38 Q(t). How long will it take for 99% of the pollution to be removed from the lake?

a) About 6 years
b) About 9 years
c) About 12 years
d) About 15 years

4. The decline of the species Amospitza Maritima Nigrescens, known as the Seaside Dusky Sparrow can now be modeled, in hindsight, by the function P(t) where

so that, in 1955, there were approx. 3000 Duskies. Assume that t is measured in segments of 20 years. Thus t = 1 means 20 years; t = 2 means 40 years, etc. At which point in time, reckoned from 1955, was it to be expected that the Dusky sparrow would become extinct?
Hint: You need to solve the equation P(t) = 0.99 for t. Use the initial value x_o = 3

a) t = 1.2 (or 24 years later)
b) t = 1.5 (or 30 years later)
c) t = 1.68 (or 33.6 years later)
d) t = 3.53 (or 70.6 years later)

Solutions are given within the box: 1. b ... This the reverse of the "coffee problem" 2. a ... Chain Rule, etc. 3. c ... Similar to a decay problem 4. d ... Did this in class. Use Newton's method...