NOIC

Submit homework solutions and class exercises by hard copies to me or soft copies to tonychanyt@gmail.com.

Course Assessment

Download Xcas, a free computer algebra system.

After installing it, start the English version of the software, xcasEn.

In the Euclidean plane, if p = (p₁, p₂) and q = (q₁, q₂) then the distance is given by

Class Ex.

p. 175, Q10

C(t):=(30*t)/(200000+t)

plot(C(t),t=1E5..1E7,xstep=1E3)

Plot the interval before 1E5. What happens when t is small?

Plot the interval after 1E7. What happens when t is large?

CEMC - Euclid - Mathematics Competitions - University of Waterloo

Homework #6 due Oct 27

Oct 28, Class Exercise #8, textbook p. 219, Q. 20

Use the following Xcas commands to help you.

LS(x,y):=2*sin(x)*cos(y)

evalf(LS(5*pi/4,3*pi/4))

To help you to do it by hand, use this unit circle.

Tips: Sine corresponds to the vertical length which is the y-coordinate,

and cosine corresponds to the horizontal length which is the x-coordinate.

You only need to focus on the first quadrant.

Homework #7 due Oct 29 9am

Textbook, p. 217, Q. 9 & 10

Oct 29, Class Exercise #10

Enter the following Xcas commands in the same 2d figure under Geo.

t1(x):=1/sin(x)

plot(t1(x),color=blue,legend="t1")

t2(x):=2*cos(x)/(2*sin(x)*cos(x))

plot(t2(x),color=161,legend=" t2")

Explain in English what each of the command does.

Prove t1(x)=t2(x) algebraically.

Check out the this webpage: Sine Cosine Functions.

Follow its instructions.

Homework #8 due Nov 3 9am

Textbook, p. 260, Q. 19 & 22

Nov 3

Enter the following Xcas commands in the same 2d figure under Geo.

plot(sin(x),x=-pi..pi)

plot(csc(x),color=blue)

Explain in English what each of the command does.

Use these and other plots to perform the investigation on p. 261, Q. 1–10.

Chapters 4 & 5, trigonometry test will be on Nov 11. See an attachment below for the cover page of this test.

Some of the questions will come the textbook on pp. 244-247 & 300-305.

You should also know all the formulas on the last page of the book.

Compare the result of the following with Example 1 on pp. 292-3 of the textbook.

diff(20*sin(pi*x/60)+25

evalf(20*cos(pi*10/60)*pi/60

What exactly is the instantaneous rate of change at t=10?

A multiplicative inverse or reciprocal for a non-zero number x, denoted by 1/x or x⁻¹, is a number which when multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a.

Negative integer exponents, by definition, raising a nonzero number to the −1 power produces its reciprocal:

One also defines

for any nonzero a and any positive integer n.

If ƒ is a function from a set A to a set B, then an inverse function (or reverse function) for ƒ is a function from B to A. Informally, x becomes y and y becomes x. A round trip (a composition) from A to B to A (or from B to A to B) returns each element of the initial set to itself.

If ƒ is an invertible function with domain X and range Y, then

It is important to realize that ƒ⁻¹(x) is not the same as ƒ(x)⁻¹. In ƒ⁻¹(x), the superscript "−1" is not an exponent.

In trigonometry, for historical reasons, sin²(x) usually does mean the square of sin(x), i.e., 2 is an exponent.

However, the expression sin⁻¹(x) does not represent the multiplicative inverse to sin(x), i.e., -1 is not an exponent.

The function (sin x)^–1 is the multiplicative inverse to the sine, and is called the cosecant. It is usually denoted csc x:

Polynomials test on Chapters 1, 2, and 3 will be on Monday, Nov 15.

Class exercise: textbook, p. 330, Q. 19

Start a new spreadsheet from Xcas.

Enter the data into it.

Select (highlight) the data.

Click Maths --> 2d stats --> Scatterplot

Click OK

The same plot can be achieved also from a regular CAS command as follows:

scatterplot([1,10,100,1000],[0,1,2,3])

Class exercises:

1) Start with a $1 investment in year 1. How much money would you have 40 years later if the return is 10% compound annually? How about 20%, 30%, ..., 100%? Use Xcas spreadsheet to perform the calculation.

2) Start Geo --> New figure 2d

line(point(0,0), point(9,12)

O:=point(0,0)

Draw the unit circle on this figure.

Class exercise: textbook, p. 394, Example 1

Start a new spreadsheet from Xcas.

Enter the data into it.

Select (highlight) the data.

Click Maths --> 2d stats --> Scatterplot

Click OK

Click Maths --> Regressions --> Linear

Etc. ...