DYNAMICS 1 Basic Definitions. Newtons Second Law.
1. A particle, of mass m, has position vector
r(t) = (x(t), y(t) = (3 sin 2t + 4 cos 2t, 3t + 2) .
at time t.
(i) Determine the particles momentum m(dr/dt) at time t.
2
at time t.
(ii) Determine the part
COMPLEX NUMBERS EXERCISES
1. By writing = a + ib (where a and b are real), solve the equation
2 = 5 12i.
Hence find the two roots of the quadratic equation
z 2 (4 + i) z + (5 + 5i) = 0.
2. By substituting z = x + iy or z = rei into the following equation
INDUCTION EXERCISES 2.
1. Show that n lines in the plane, no two of which are parallel and no three meeting in a point, divide the plane into
n2 + n + 2
2
regions.
2. Prove for every positive integer n, that
33n2 + 23n+1
is divisible by 19.
3. (a) Show th
CALCULUS EXERCISES 2 Numerical Methods and Estimation
1. Use calculus, or trigonometric identities, to prove the following inequalities for in the range 0 < <
2:
sin < ;
< tan ;
cos 2 < cos2 .
Hence, without directly calculating the following integrals
INDUCTION EXERCISES 1
1. Factorials are defined inductively by the rule
0! = 1 and (n + 1)! = n! (n + 1).
Then binomial coecients are defined for 0 k n by
n!
n
=
.
k
k!(n k)!
Prove from these definitions that
n
n
n+1
+
=
,
k
k+1
k+1
and deduce the B
ALGEBRA EXERCISES 1
1. (a) Find the remainder when n2 + 4 is divided by 7 for 0 n < 7.
Deduce that n2 + 4 is not divisible by 7, for every positive integer n. [Hint: write n = 7k + r where 0 r < 7.]
(b) Now k is an integer such that n3 + k is not divisibl
CALCULUS EXERCISES 3 Techniques of Integration
1. Evaluate
Z
Z
ln x
dx,
x
Z
x sec2 x dx,
3
dx
,
(x 1)(x 2)
Z
1
tan1 x dx,
0
Z
1
0
dx
.
ex + 1
2. Evaluate, using trigonometric and/or hyperbolic substitutions,
Z
dx
,
x2 + 1
Z
2
1
Z
dx
,
x2 1
Z
dx
,
4 x2
2
d
CALCULUS EXERCISES 1 Curve Sketching
1. Sketch the graph of the curve
y=
x2 + 1
(x 1) (x 2)
carefully labelling any turning points and asymptotes.
2. The parabola x = y 2 + ay + b crosses the parabola y = x2 at (1, 1) making right angles.
Calculate the va
ALGEBRA EXERCISES 2
1. Under what conditions on the real numbers a, b, c, d, e, f do the simultaneous equations
ax + by = e and cx + dy = f
have (a) a unique solution, (b) no solution, (c) infinitely many solutions in x and y.
Select values of a, b, c, d,
CALCULUS EXERCISES 5 Further Dierential Equations
1. Find all solutions of the following separable dierential equations:
dy
dx
dy
dx
d2 y
dx2
y xy
,
xy x
sin1 x
,
y (0) = 0.
=
y 2 1 x2
dy 2
1
= 1 + 3x2
where y (1) = 0 and y 0 (1) =
.
dx
2
=
2. Use the
Section 2.6 Class Notes Mathematical Models: Constructing Functions Guiding Problem 1 The price p, in dollars, and the quantity x sold of a certain product obey the demand equation 1 p = - x + 150, 0 x 500 . 4 (a) Find the revenue function. (b) What is th
Section 4.4 Class Notes Polynomial and Rational Inequalities Guiding Problem 1 Solve each polynomial inequality. 2 ( a ) ( x + 3) ( x - 4 ) > 0 ( b ) ( x + 1) ( x - 5) ( x + 3)
0
( c)
x 6 > 16 x 4
(d)
x3 + 2 x 2 - 8x 0
Guiding Problem 2 Solve each rationa
San Jos State University Mathematics Department Math 12, Number Systems, Sections 1 & 2, Fall 2007 Instructor: Office location: Telephone: Email: Office hours: Class days/time: School Holidays Classroom: Prerequisites: Dr. Trisha Bergthold MacQuarrie Hall
Prerequisite Mathematical Skills for Entering Precalculus Spring 2008 In order to successfully begin your Precalculus course, you must have a sound knowledge of certain mathematical skills and concepts. I have identified the items in the following list as
Section 4.5 Class Notes The Real Zeros of a Polynomial Function
Example 1: Find the real zeros of the polynomial function How many zeros could this polynomial have? Number of Real Zeros Theorem A polynomial function cannot have more real zeros than its de
Math 19 Grade Progress It is completely straightforward to keep track of your grade-to-date in this class. Every point is weighted exactly the same. All you need to do is add up the points you have earned so far, and divide by the total points it was poss
Homework Assignments in MyMathLab General Information
Before you start, you will need Course ID: Student Access Code: Valid Email address that you check regularly: System Requirements Operating Systems: Windows 2000, Windows XP, and Windows Vista TM (Macs
San Jos State University Mathematics Department Math 19, Precalculus, Sections 1 & 2, Spring 2008 Instructor: Office location: Telephone: Email: Office hours: Class days/time: School Holidays Classroom: Prerequisites: Dr. Trisha Bergthold MacQuarrie Hall
Math 19 Homework Assignments Unit 2: Functions and Their Graphs Each homework assignment below should be completed to the best of your ability by the due date. The given reading, notetaking, exercises, and problems for each homework assignment represent t
Section 2.1 Class Notes Guiding Problem #1 Information often comes in related pairs: name-birthdate, name-telephone number, food item-calorie count, calorie count-grams of fat, natural numbers-their opposites. Sometimes these relationships are unambiguous
Section 2.5 Class Notes Graphing Techniques: Transformations Guiding Problem 1 On the same set of axes, draw each of the following functions: f ( x ) = x 2 , g ( x ) = x 2 + 3, h( x ) = ( x - 3) , m( x) = 3 x 2 , n( x) = ( 3x )
2 2
Describe how each of th
Math 19 Precalculus Revised Course Schedule (Note: subject to change with fair notice.) Wk Date Work Due 1 2 W 1/23 F 1/25 M 1/28 W 1/30 3 F 2/1 M 2/4 W 2/6 4 F 2/8 M 2/11 W 2/13 5 F 2/15 M 2/18 W 2/20 6 F 2/22 M 2/25 W 2/27 7 F 2/29 M 3/1 W 3/5 8 F 3/7 M
Summary of Mathematical Processes (Learning Outcome 9) One of the ways in which college level mathematics courses differ from high school mathematics courses is in the expectation that students will grow and develop in their mathematical sophistication. I
Due Monday May 12, 2008 (will not be accepted late) Points Earned: _ out of 5 points
Math 19 Final Reflective Paper Turn in 2 copies of your paper. Attach this page to the first copy. No late submissions accepted Now that you are nearing completion of thi
Section 5.1 Class Notes
Composite Functions Guiding Question 1
2 For f ( x ) = 2 x + 3 and g ( x ) = x - 2 x , find:
( a) (
f o g ) ( 2)
( b) ( g o f ) ( 2)
( c) (
f o f ) ( -3 )
( d ) ( g o g ) ( -1)
Guiding Question 2 Find the domain of ( f o g ) ( x )
Math 19 Homework Assignments Unit 4: Polynomial and Rational Functions Each homework assignment below should be completed to the best of your ability by the due date. The given reading, notetaking, exercises, and problems for each homework assignment repr
Section 3.1 Class Notes Linear Functions and Their Properties Framing Thoughts for this Section We've already studied equations of lines and graphs of lines in chapter 1, so much of this material will seem familiar. The main difference is that we will re-
Section 4.1 Class Notes Polynomial Functions and Models Guiding Question 1 Determine which functions are polynomials and state the degree. If not, state why. 3+ x ( a ) f ( x ) = 5 x3 - 3x + 4 ( b ) f ( x ) = 3 x 2 ( c ) f ( x ) = ( d ) f ( x) = x - 3 x-3
Chapter 1 Review Class Notes
Formulas to Review Write the following formulas. (1) distance formula (2) midpoint formula (3) slope formula (4) general form for equation of a line (5) point-slope form for equation of a line (6) slope-intercept form for equa