Applied Mathematics

Assignment 4 Due : 4pm Monday 15 October MAS 182 Applied Mathematics Semester 2, 2012

Mathematics and Statistics

Please show all your working. Where possible give the exact answer.

/1 .,1 1. Consider the integral -12x + 3 dx. Using the trapezoidal method with n = 4 -i_ and n = 8, estimate the integral numerically. Calculate the integral exactly (using calculus and showing your working) and compare this with your numerical results.

Find the following integrals. Show all working.

(i) I (x2 + 3x — 1)e-2x dx (ii) I X-2 ln(3x) dx

Find the average value of the function 1 mction f (s) = over s E [0, 2]. e3x

A solid gold tip of a tunnelling electron microscope has the shape given by rotating the graph of the function f (x) = 3 — about the x-axis for s E [0, 9]. Find the volume of gold required to build the tip. Solve the following differential equations. Note that the solution may be either a general solution or a particular solution. )

-.4 5

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(ii) 4c1-3.1.d + 2xy — 3x = 0 with y(0) = 2 (iii) X dy (TT:. — 2xy + 5 = 0 with y(1) 3

Notes:

• 10% of the marks for this assignment are reserved for presentation. • There are penalties for late assignments. You must contact your tutor before the due date if you have difficulties making the deadline.

Mathematical Applications

4th Assignment 4 p.m. deadline Monday, October 15th, 2012 MAS 182 Applied Mathematics Semester 2 Mathematics and Statistics

Please display all of your work. Give the exact answer wherever possible.

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1. Take a look at the integral -12x + 3 dx. Estimate the integral numerically using the trapezoidal method with n = 4 -i_ and n = 8. Calculate the integral exactly (showing your work using calculus) and compare it to your numerical answers.

Find the integrals below. Show all the things that are working.

(x2 + 3x — 1) I

I X-2 ln(3x) dx e-2x dx e-2x dx e-2x dx e-2x dx e-2x dx e-2x dx

Find the average value of the 1 mction f (s) = over s E [0, 2] function. e3x

A solid gold tip of a tunnelling electron microscope has the shape given by rotating the graph of the