University Problem Set #1 ECE 313
of Illinois Page 1 of 2 Fall 2001
Assigned: Wednesday, August 22
Due: Wednesday, August 29
Reading: The ECE 313 FAQ at http://courses.ece.uiuc.edu/ece313/faq.html
Ross, Chapter 1.1–1.5, Chapter 2.1–2.5 and 2.7
Noncredit Exercises: (Do not turn these in)
Problems Theoretical Exercises Self-Test Problems
Chapter 1 1–5, 7, 9 4, 8, 13 1-15
Chapter 2 3, 4, 9, 10, 11-14 1–3, 6, 7, 10, 11, 12, 16, 19, 20 1-8
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. Think of this problem set as a diagnostic 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 = 5 ≠ 1–22 = –3, while –23+2 = 2–23 has value –6,
what happens when you divide both sides of the latter equation by 2?
(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 part (a) 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) Find the value of 1
2 + 1 + 1
3 + 2 + 1
4 + 3 + ... + 1
100 + 99 . You can
use your programmable calculator if you wish.
3. In this problem, all angles are expressed in radians.
(a) Use your calculator to evaluate 52 cos(3–1arctan(18 3/35)).
(b) Find the limit of 1
[sin x]2 – 1
x2 as x approaches 0. (Hint: the answer is not 0, or 1, or ∞)
(c) Find the maxima of f(x) = x25(1.00001)–x for x > 0. (If you have a graphing calculator,
try it on this problem; otherwise just use standard calculus methods)
