HSC Mathematics. Workshop 4

Transcription

1 HSC Mathematics Workshop 4 Presented by Richard D. Kenderdine BSc, GradDipAppSc(IndMaths), SurvCert, MAppStat, GStat School of Mathematics and Applied Statistics University of Wollongong Moss Vale Centre October 2009

2 HSC Mathematics - Workshop 4 Richard D Kenderdine University of Wollongong October Applications of Calculus to the Physical World General Rates of Change For most of this topic we differentiate with respect to time. Derivatives are rates of change. So if a question asks for a rate of change it means find the derivative. If a questions gives the rate of change then the integral is probably required. We can interpret an integral as an accumulator - it adds up all the changes given by the rate of change. 1. Exercise: Over a 10-year period commencing on 1 January 1995 the population P (t) in a certain town was modelled by the equation P (t) = t where t is time in years, 0 t 10. Find: (a) the population on 1 January 1995 (b) the population on 31 December 2004 (c) the rate of change in the population on 1 January Exercise: After a thunderstorm the rate of increase in water stored in a dam is given by R(t) = 500 t+1 litres/day where t is in days. (a) If the dam contained 2000 litres before the storm calculate the amount of water in the dam when t = 6. (b) What is the maximum rate of flow and for what value of t does this occur? 1 rdk301@uow.edu.au 1

3 3. Exercise: (1985 HSC) At time t minutes after a jet engine starts the rate of fuel burn, R kg/min, is given by R = t (a) Sketch a graph of R as a function of t. (b) What is the rate of burn after 7 minutes? (c) What value does R approach as t becomes very large? (d) Calculate the total amount of fuel burned in the first 7 minutes. Exponential growth and decay There are numerous situations in which the rate of change in a quantity is proportional to the quantity itself. A discrete example is compound interest where the amount of interest in dollars is proportional to the amount invested. Now thinking of the continuous case we derive the appropriate function. We want to find a function Q(t) such that dq(t) t = kq(t) for some constant k. 4. Exercise: The rate of growth in the number of bacteria is proportional to the number present. If the initial population of 100 grew to 200 after 3 hours find: (a) the population after 5 hours (b) the population after 9 hours (c) the time, to the nearest minute, for the population to reach 1000 (d) the rate of growth in the population after 5 hours 5. Exercise: The half-life of a radioactive material is 50 years. How long will it take until only 10% of the original material is present, assuming exponential decay? 6. Exercise: The rate of growth in the number of locusts in a colony is proportional to the population present. If there were 5000 locusts 2 days after observations 2

4 began and after 5 days, calculate (a) the number of locusts in the population when observations began (b) the number of locusts after 10 days (c) how long it will take for the population to reach (d) whether the model used to answer the previous questions is applicable over an extended period of time. Motion of a Particle The motion of a particle considers three aspects of its movement in a line: displacement (x) from a fixed origin (+ve or -ve) velocity, the rate of change in displacement (v = dx dt = ẋ) acceleration, the rate of change in velocity (a = dv dt = d2 x dt 2 = ẍ) Note that velocity and acceleration are vectors, that is they have magnitude and direction. Speed only has magnitude so two particles can have the same speed but different velocities since they are travelling in different directions. 7. Exercise: The displacement of a particle in metres at time t seconds is given by x = ln(t 2 3t + 4). Find: (a) the initial displacement (b) the initial velocity (c) the direction in which the particle is initially moving (d) whether the particle comes to rest and if so, its location (e) the distance travelled in the first 2 seconds (f) the acceleration at t = 1 (g) a description of the velocity as time increases. It is useful to remember that the distance travelled between two times is the area under the velocity-time graph between those times. 8. Exercise: Find the distance travelled by a particle between t = π 6 velocity is given by v = sin 4t cm/s. 5π and 12 if the 3

5 9. Exercise: The acceleration of a particle is given by ẍ = 6 cos 2t m/s 2. If the particle momentarily stopped at x = π metres when t = π 4 the exact displacement at t = π sec. sec, find 10. Exercise: The displacement of a particle in metres is given by x = 4t 3 sin 2t. Find the maximum velocity of the particle. 11. Exercise: The velocity, in metres/sec, of a body that is initially at the origin is given by Find: ẋ = e2t 9 + e 2t (a) an expression for acceleration (b) an expression for displacement (c) the initial velocity (d) a full description the motion as time increases (e) graph velocity against time Probability The main things to remember with Probability questions are: Multi-stage events: tree diagrams. Multiply probabilities going out the tree, add going down. Complementary event: P (E) = 1 P (Ē). Use with the words at least one. Venn diagrams. 12. Exercise: There are 22 employees in a small business. Eight of those employees did not work elsewhere before joining the business and do not have a tertiary education. Of the remainder, 10 worked elsewhere and 6 have a tertiary education. If an employee is selected at random, what is the probability that the employee both worked elsewhere and has a tertiary education? 4

6 13. Exercise: Two teams play a series of 3 matches. At the outset the probability that Team A will win the first match is 0.7. The probability of winning the second match increases/decreases by 0.05 according to whether the team won/lost the first match. The same process applies to the third match so Team A would have a probability of 0.6 of winning the third match if it lost the first two matches. Find the probability that (i) Team A wins the series (ie more matches that Team B). (ii) Team B wins at least one match. 14. Exercise: There are 8 pairs of identical twins in a school. A random sample of 3 students is to be taken from the 16 twins and asked questions about their experiences as a twin. What is the probability that there are no matching twins in the sample? 15. Exercise: A game is played with tetrahedral (four sided, with equilateral traingles as faces) dice. One of the die has faces marked (2,4,6,8) and the other (1,3,5,7). The score after each toss is obtained by adding the number on each die. What is the probability of a total score greater than 27 after two tosses? 5