University Problem Set #1 ECE 313
of Illinois Page 1 of 2 Spring 1999
Assigned: Wednesday, January 20
Due: Wednesday, January 27
Reading: Ross, Chapter 1.1–1.5, Chapter 2.1–2.5 and 2.7
Noncredit Exercises: (Do not turn these in) Ross, p. 16: 1–5, 7, 9;
p. 59-60: 3, 4, 9, 10, 11-14; pp. 61-64: 1–3, 6, 7, 10, 11, 12, 16.
Problems: These problems are based entirely on material covered in the prerequisites to this
course. You should have mastered this stuff already, but may need to review the material one
more time before starting the course. The problems below are assigned to help your review, and
also as a self–diagnosis aid. If you cannot solve all these problems correctly, you will have
difficulty in comprehending the material in the latter half of this course. It is not in your best
interest to discover after the drop date that you really don’t understand calculus as well as you
thought you did, and that consequently you are in some danger of failing this course.
Do not use Mathematica or Matlab or a calculator etc. to do these problems except
when you are specifically asked to do so.
1.(a) Does the commutative law of addition: a + b = b + a imply that – 4 + 1 equals 1 – 4?
(b) Determine whether –22+1 equals 1–22 and –23+2 equals 2–23 using
(i) ordinary grade–school arithmetic.
(ii) your calculator.
On an algebraic-entry calculator (e.g. TI, Casio, Sharp, most “desk accessory”
calculators on PCs and Macintoshes etc), what happens when you press the keys
marked – 2 x2 + 1 =?
(iii) the Microsoft spreadsheet program Excel. (Enter the four formulas = –2^2+1,
= 1–2^2, = –2^3+2, and = 2–2^3 into different cells in the spreadsheet)
(c) If you agree with Excel that –22+1 ≠ 1–22, while –23+2 = 2–23, find the error in the
following “proof” via the distributive and cancellation laws that –22+1 does in fact equal
1–22: –23+2 = 2–23 ⇒ (–22+1)×2 = (1–22)×2 ⇒ –22+1 = 1–22 (since 2 ≠ 0)
(d) True or False? The solutions to the quadratic equation ax2 + bx + c = 0 are given by
2c
–b ± b2 – 4ac
2. The angles in this problem are expressed in degrees and not in the radians more
commonly used in mathematical circles.
(a) Use your calculator to evaluate cot(10˚)cot(30˚)cot(50˚)cot(70˚) without writing down
intermediate results such as the values of cot(10˚), cot(30˚), etc and re-entering the
numbers into your calculator. If your calculator cannot be used in this fashion, you are
urged to replace it with a more sophisticated machine.
(b) If your calculator’s arithmetic unit is designed in accordance with the IEEE Standard for
floating-point arithmetic, you should have obtained exactly 3 as the answer to part (a).
Does it surprise you that cot(10˚)cot(30˚)cot(50˚)cot(70˚) = 3? Find four other integers a,
b, c, and d such that 0 < a < b < c < d < 90 and cot(a˚)cot(b˚)cot(c˚)cot(d˚) is an integer.
You will get 100% extra credit on this problem if cot(a˚)cot(b˚)cot(c˚)cot(d˚) = 2.
(c) Find the value of 1
2 + 1 + 1
3 + 2 + 1
4 + 3 + ... + 1
100 + 99
3. In this problem, all angles are expressed in radians.
(a) Use your calculator to evaluate 52 cos(3–1arctan(18 3/35)).
