Question 1 Let g(t) be the population (in thousands of people) of Calculus City, as a function the time t (in years), where t = 0 corresponds to the year 2020. In 2020, the population was 10000, and each year the population increases by 30%. (a) Model the population by nding a formula for g(t), using the appropriate choice of a linear, polynomial, power, trigonometric, or exponential function, or a piecewise combination thereof. (b) Evaluate the expression 9(6), and write a sentence explaining what this means. (c) Solve the equation g(t) = 16.9, and write a sentence explaining what this means. Question 2 Answer the following parts. (a) Write down a formula for the amount of money you would have in your account after 15 years if it started at $1000 and the account pays 2.5% annual interest compounded monthly. (b) If you buy a car for $20,000 and it loses half its value every year when will the car be worth $3,000? (c) A population is known to grow exponentially. If it starts at 100 and is 150 after 3 months when will the population be 300? Question 3 Newton's Law of Cooling (which applies to warming as well) says that the temperature difference between an object and its surroundings is an exponentially decaying function of time, provided that surrounding temperature remains constant. Suppose that the surrounding temperature does not depend on time, and denote this temperature T3. Let T(t) be the temperature of an object at time t. Translating \"the temperature difference between an object and its surroundings is an exponentially decaying function of time\" into an equation yields T(t) T,3 = ae'k' where a. and k are constants. (More specically we know is: > 0, otherwise the object temperature wouldn't approach the surrounding temperature in the long run.) Without too much trouble it can be deduced that Tit) _ Ts = (TO _ Ts)ekt1 where T0 is the temperature of the object at t = 0. A 98C hard-boiled egg is put into a big pot of 18C water at t = 0, where t is measured in minutes. After 5 minutes the temperature of the egg drops to 38C. (a) Use the data above to solve for T(t), the temperature of the egg at time t. The only variable in your answer should be the input, t. (b) What is a realistic domain of the temperature function? (c) Draw a rough sketch of the function and label any intercepts and asymptotes. (d) Assuming the water has not warmed appreciably, how long does it take the egg to cool to a temperature of 20 C"? Question 4 Average vs. instantaneous rates of change. Frank was monitoring the population of fruit ies as part of his research toward his honors biology project. He monitored the population over a 50 day period, counting at regular intervals. Below is a table of some of the data he collected, indicating the number of flies on certain days. Day 3 5 14 44 |Population of fruit ies | 150 180 225 345 (a) Use the information in the table to determine the average rate of change on the intervals i. day 3 to day 5. ii. day 5 to day 14. iii. day 14 to day 44' Include units. Math 140 Written Homework 2: 1.6-2.3 Page 2 of 2 (b) Does the data above support the sentence \"As time passes, the population of fruit ies increases faster and faster?\" Explain. (c) Estimate the instantaneous rate of change at t = 5. Give units. Question 5 For each part below, calculate the limit or determine that it does not exist. If a limit is 00 or 00, say so. Show all necessary steps. . 53236 (a) all133 w+6 5936 1' (b) 313}; (ccm2 2_ (c) limm 36 z>6 586 Question 6 Assume that lim f(x) = 2. For each part below, calculate the limit or determine that it does not exist. If a limit is oo or -oo, say so. Show all necessary steps. (a) lim (f(x)) 4 1 (b) lim x-+2 (f (2 ) ) 2 x-+2 (c) lim (3x + x2 f(x) ) (d) lim f(ac) x-2+x- 2 (e) lim f (ac) x-2-x - 2 (f) lim f (20) x-2x - 2