Assignment 01.doc

BACS 1263 Mathematics for Computer Science II Faculty of Information &Communication Technology

Assignment 1 FACULTY OF INFORMATION & COMMUNICATION TECHNOLOGY UNIVERSITI TEKNIKAL MALAYSIA MELAKA

Mathematics for Computer Science II BACS1263

SEMESTER 2

SESI 2008/2009

Due Date: 23 Feb 2009 before 4.00 p.m.

Question 1: A stone is dropped into a kale, creating a circular ripple that travels outward at a speed of 60cm/s.

a) Express the radius r of this circle as a function of the time t (in seconds). b) If A is the area of this circle as a function of the radius, find A ° r and interpret it.

Question 2: Sketch the region in the plane consisting of all points (x,y) such that x+y≤1

Question 3: In the theory of relativity, the mass of a particle with velocity v is m=m01-v2c2 where m0 the rest mass of the particle and c is the speed of light. What happens as v→c-.

Question 4: A telephone company wants to estimate the number of new residential phone lines that it will need to install during the upcoming month. At the beginning of January, 2009 the company had 100,000 subscribes, each of whom had 1.2 phone lines, on average. The 1

BACS 1263 Mathematics for Computer Science II Faculty of Information &Communication Technology

company estimated that its subscribership was increasing at the rate of 1000 monthly. By polling its existing subscribers, the company found that each intended to install an average of 0.01 new phone lines by the end of January.

a) Let s(t) be the number of subscribers and let n(t) be the number of phone lines per subscriber at time t, where t is measured in years and t=0 corresponds to the beginning of 2009. What are the values of s(0) and n(0)? What are the company’s estimates for s’(0) and n’(0)? b) Estimate the number of new lines the company will have to install in January, 2009, by using the Product Rule to calculate the rate of increase of lines at the beginning of the month. Question 5: If the equation of motion of a particle is given by s=Acos(ωt+δ), the particle is said to undergo simple harmonic motion.

a) Find the velocity of the particle at time t. b) When is the velocity 0?

Question 6: A waterskier skis over the ramp shown in figure below at a speed of 30ft/s. How fast is she rising as she leaves the ramp?

4ft 15ft

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BACS 1263 Mathematics for Computer Science II Faculty of Information &Communication Technology

Question 7: Under the certain circumstances a rumor spreads according to the equation pt=11+ae-kt where p(t) is the proportion of the population that knows the rumor at time t and a and k are positive constants.

a) Find limt→∞p(t). b) Find the rate of spread of the rumor. c) Graph p for the case a=10, k=0.5 with t measured in hours. Use the graph to estimate how long it will take for 80% of the population to hear the rumor.

Question 8: If the initial mass of a sample of 90Sr is 24mg, then the mass after t years is

mt=24∙e-In 2t/25

a) Find the mass remaining after 40 years. b) At what rate does the mass decay after 40 years? c) How long does it take for the mass to be reduced to 5mg?

Question 9: Find the value of k that makes the following antidifferentiation formula true (1-2x)3dx=k(1-2x)4+C

Question 10: 3

BACS 1263 Mathematics for Computer Science II Faculty of Information &Communication Technology

A rock dropped from a bridge has a velocity of -32t feet per second after t seconds. Find the average velocity of the rock during the first three seconds.

Question 11: If K(t) is the annual rate of income at time t, and if the income is to be received over the next N years, then the present value P of the stream of income at interest rate r is defined by the integral P=0NK(t)e-rtdt

A printing company estimates that the rate of revenue generated by one of its printing presses at time t will be 80-2ti thousand dollars per year. Find the present value of this continuous stream of income over the next 4 years at a 10% interest rate.

Question 12: Evaluate 1∞x2x3+8dx.

Question 13: Let fx,y,z=11xy+14yz+15xz, be the heat-loss function. Calculate and interpret ∂f∂x(10,7,5).

Question 14: Find all points (x,y) where fx,y=x3-3xy+12y2+8 has a possible relative maximum or minimum.

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