cahier physique - physics workbook

PART I lesson, inquiry mode
If a function r can be written as the ratio of polynomials p(x) and q(x), that is, if

r(x) = p(x) / q(x)

Then r is called a rational function. (We assume that q(x) is not the constant polynomial q(x) =0 )

1) (p. 400 1 to 6)
Are the functions rational functions? If so, write them in the form p(x)/q(x), the ration of polynomials.
A) f(x) = x²/2 + 1/x B) (√ (x) +1)/(x+1) C) f(x)= (4^x + 3)/(3^x - 1)
D) f(x) = (x² +4 ) / e^x E) f(x) = (9x - 1 )/ (4√x + 7 ) + (5x³)/ (x²-1)
F) f(x) = x²/(x-3) - 5/(x-3)

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2) Consider the function f(x)= (x+2) / (x-1)
Plug this function in your TI (in Y1 ).
use zoom 6 and trace . Describe the graph from very large negative value of x
to very large positive vales of x.

A)Use your cursor (side way, left/right and right /left) to find the behavior of f(x) when x becomes
really large. (negative and positive). When x is really large, (big values) y approaches ____. It gets closer and closer to y= _____ as x becomes really big (in absolute value)
. y=1 is called an horizontal asymptote.
We can find this horizontal asymptote by looking at the expression of f(x):
For large x : f(x)= (big number + 2 ) / (big number -1 ) = (X +2 ) / (X - 1) = x/x =1

So the long run of rational functions will give you the horizontal asymptotes.

B) Move now your cursor from left to right toward x = 1. What happens before x reaches 1 ?
What happens after x has reached 1 ?
x = 1 is called a vertical asymptote
The vertical asymptote (if any) can be found by finding the values that will cause the denominator to be zero.
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3) f(x) = (2x² +5x -3 ) / (x² -2x - 8) = r(x)/p(x)
A) To find the horizontal asymptote, find the long-run behavior.
For big values of X,f(x) becomes (2X² +5x -3 ) / (X² -2x - 8) = _______ (cross out the X²)
so y = _____ is the horizontal asymptote

B) f(x) = r(x) / p(x) so p(x) = ______________
Let's find the values of x that cause p(x) =0
We need to find the roots of x² -2x - 8 (or solutions). Factor P(x) = ( ) ( )
Therefore, x= _____ and x= ______ are the vertical asymptotes.

C) on a graph paper trace the 3 asymptotes. Let's try to trace the function. You need to build
a table. Pick x values in the areas of interest (x<-2, -2<x<4, x>4).
Enter the function in your calculator and use the TABLE function:
moving from left to right try: x= -5, x= -3, x= -2.5, x= -2.1 then x = -1.8, x=-1.5, x=0, x=1, x=2, x=3, x=3.5 then x=4.1, x=4.5, x=5 .. You got the idea/
Trace your function using the TI to check your work. use zoom 6
Are you close ?

D) Using your calc/zeros function find the 2 x-intercepts of the function.
x1= _________ and x2 = ___________.
r(x1) = ________ r(x2) = __________
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4) f(x) = 3 / x+2
A) r(x) = _____, p(x) = _________
What is the value that will cause p(x) to be zero ? x = ____
So x = ___ is a vertical asymptote.

B) For really big values of x , y = 3/ (X + 2) = 3/X and 3/(big value) is really small.
THat is y is getting close to 0. So y= ____ is a horizontal asymptote.

C) On a graph paper, trace the 2 asymptotes and build a table to sketch the function.
Move from left to right. (x= -5, x= -3, x=-2.5, x= -2.1, x= -1.8, x= - 1.5, x= ......
Check your sketch using zoom6/trace.
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f(x) = 1/(x² +3)
A) p(x) = ___________ q(x) = _______________
Can you find a value of x (real one) that will cause q(x) to be zero ?
So there is no ___________ asymptote.

B) For really big value of x, f(x) = 1 / (X² + 3) = 1/X² so y gets close to ________
so y = ____ is a horizontal asymptote.

C) Try to find the graph. Use a table of values

D) trace it using your TI. Use zoom6 and zbox to see the bump. Any x-intercept ? FInd the maximum of the function.
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II - FROM the books. assignments

1) 7 p. 400
Compare the long-run behavior of:
f(x) = (x²+1)/(x²+5) g(x) = (x³ + 1)/(x² +5) h(x) = (x+1)/(x² +5)

2) 9,10,11 p.400
Find the horizontal asymptote if it exists, of the functions:
f(x)= 1/(1 + 1/x)
g(x) = (1-x)(2+3x)/(2x² + 1)
h(x) = 3 - 1/x + x/(x+1)

2) 12 p. 400
Let t be the time in weeks. At time = 0 (meaning there is no t negative), organic waste is dumped into a pond. He oxygen level in the pond at time t :
f(t) = (t²-t +1) / (t² + 1)
A) use TABLEset to chose auto and tbl 1. Use TABLE to answer:
f(0) = ___, f(2) = ___, f(3)= ____, f(4) = ___, f(5) = ____, f(6) = ___, f(10) = ____
Using these values, can you think of a good window to trace the graph ?
___ < x < ___ and ___ < y < ____

B) Using a good window (see A) graph the function

C) Describe the shape of the graph. What is the significance of the minimum for the pond ?

D) what eventually happens to the oxygen level (look at the y values over time)

E) Approximately how many weeks must pass before the oxygen level returns to 75% of its normal level?

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14 p.400
Bronze is an alloy, or mixture, of copper an tin. The alloy initially contains 3kg copper and 9kg tin. You add x kg of copper to this 12kg of alloy. THe concentration in the alloy is a function of x:

f(x) = concentration of copper = (total amount of copper) / (total amount of alloy)

A) Find a formula or f in terms of x, the amount of copper added

B) Evaluate the following expressions and explain their significance for the alloy:
(use TABLEset)
f( 1/2) ; f(0) ; f(-1)

C) Graph f(x) for -5<x<5 and -0.25<y<0.5 (use the window key)
Interpret the y-intercept in the context of the alloy

D) graph f(x) for -3<x<100 and 0<y<1. Describe the appearance of your graph for large x-values. Does the appearance agree with what you expect to happen when large amounts of copper are added to the alloy?
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