PART I : polynomial functions
1)
The general formula for polynomial function can be written as:
p(x) = anxn + an-1xn-1 + ... + a1x + ao
example g(x) = 3x2 + 4x5 + x - x3 + 1
A) It is customary It is customary to write a polynomial with the powers in decreasing order from left to right. g(x) = ___________________
this is called the standard form of a polynomial.
B) an is the leading coefficient and n is the degree of the polynomials.
g(x) (see above, standard form) is a polynomial of degree _____.
The leading coefficient is ______.
C) also anxn are called terms. list all the terms of g(x) including aox0
__________________________________________________
D) the constants an, an-1, ..., ao are called the coefficients.
The values of g's coefficients are a5 = ___, a3 = ____, a2 = ____, a1 = ____.
D) ao is called the constant term. In the above example, the constant is ______
2)
graphs of polynomials tend to be curvy. they have no asymptotes. THe
maximum of x-intercepts =zeros of the polynomial = degree
of the polynomials.
For example g(x) can have _____ x-intercepts or ___ or ___ ....
The zeros are the values of x that will cause p(x) to be zero. p(x) = 0
3) consider the polynomial f(x) = x3 - x2 - 6x
The
leading term is ______, for very large value of x (take 1000 or -1000),
it will over take over. It will tells us the long-run behavior of the
function. So f(x), for large x (negative or positive) will looks like :
What happens to the long-run behavior if we study f(x) = - x3 - x2 - 6x instead ?
4) g(x) = x4 - x3 - 6x2
Which term shows the lead for big x values ? ___________
trace the long-run behavior of g(x)
what about - g(x) = - x4 - x3 - 6x2
5) g(x) = 3x6 - 2x5 + 4x2 -1
show long-run behavior.
MORE : see other examples presented by your instructor.
6) For each of the polynomials below, answer the questions:
1) degree ______, leading term _________, leading coefficient _______, trace the long-run,
max
number of x-intercepts ______ find the x-intercepts using your TI (use
zoom6 and calc/zeros), using the TABLE key of your TI find f(xi)s (xi
is a x-intercept )
f(x) = 2x^2 - 2x -4
f(x) = x^3 -7x^2 + 4x + 12
f(x) = x^3 - x + 5x^2 -5
f(x) = x^4 +2x^2 -x -0.5
-----------------------------------------------------------------------------------------------------
7) (14 p. 388) Let's answer together this question:
compare the graphs of f(x) = x3 + 5x2-x-5 and g(x) = -2x3-10x2+2x+10 on a window that shows all intercepts. (use zoom6 and then Zbox). How are the graphs similar? different ?
A)
enter f(x) and g(x) in your TI. (key y = , top left). Use zoom 6
(standard) to trace it. (key zoom at the top then 6). trace the graph.
Then use the zoom 1 key to have a better look at the x-intercepts
(spots at which the curve crosses the x-axis). That is frame the area
of interest using the zbox option. conclusion? compare the graphs.
B) Find the x-intercepts using the TI. Use the key 2nd calc (top) zeros.
x1 =______ x2 = ________ x3 = __________
Note: that the degree of the polynomials is _______ and there are ______ xintercepts.
Note:
The x-intercepts of the graph of f(x) = zeros of the polynomial f(x)
(that is the values of x that cause the polynomials to be 0) =
solutions of f(x) = y = 0 = roots of y=0.
The x-intercepts are solutions of x3 + 5x2-x-5 = 0
-------------------------------------------------------------------------------------------------------
8) (15 p. 388)
What is the x-intercept of a graph ?If you are asked to find the x-intercept, set y= ________
What is the y-intercept of a graph ? If you are asked to find the y-intercept, set x = ________
Find the equation of the line going through the y-intercept of
y = x4 - 3x5 -1 + x2 and the x-intercept of y = 2x -4
A) Find the y-intercept of y = x4 - 3x5 -1 + x2 (f(x) = x4 - 3x5 -1 + x2)
That is find the value y for which x = 0. The y-intercept is yin (0, ___)
B) Find the x-intercept of y = 2x -4. That is find the value of x for which y = 0 . (solution of y=0)
x-intercept is xint ( ___, 0).
C) Can you now find the equation of the line through both points ? (the equation of a line is y= ax+b,
solve for a and b)
----------------------------------------------------------------------------------------------
9) 18 p. 388
question: if f(x) = x2 and g(x) = (x+2)(x-1)(x-3), find all x for which f(x) < g(x).
Follow the steps:
A) graph the functions using your TI. use zoom6.
B) Find the intersections of the 2 graphs using your TI. Use 2nd calc intersect.
x1 = ________ and x2 = _______
C) for which interval(s) f(x) < g(x) that is the graph of f(x) is below g(x) ?
---------------------------------------------------------------------------------------------------
10)
p.388 19. Let V represent the volume in liters of air in the
lungs during a 5 second respiratory cycle. If t is time in seconds. V
is given by :
V= 0.1729t + 0.1522t2 - 0.0374t3
A) write V(x) in standard form V(x) = __________________
The leading term is _________, the leading coefficient is __________, the degree of plolynomials
__________, the maximum x-intercepts ________, sketch the long run below.
C)
trace the graph using zoom6. Understand that only 0<t<5 is of
interest for us. The values of V before or after have no
interest. (there is not negative time and you look at 5s cycle only)
D) Find the maximum value of V. Use your TI: 2nd/calc/max
E) What is the practical significance of the maximum value ?
11) 1 to 6 very easy. and 9,10,11
12) 20 p. 388 comparing the revenue function R(x) and the cost C(x)
hints. R(x) is the revenue function. It is the revenue of the firm in million of dollars.
A) C(x) is the cost in million dollars and x is the number of thousands units produced.
C(x) = (x-1)3 + 1
if x=3, y = 9 that means : it cost 9 million dollars to produce 3,000 units.
if x =2 , y = _____, it means _______________________________________
B) R(x) = x. what does that tell you about the price of each unit?
(take an example: if x = 3 (3000 units sold), y =3 (3 millions of $ benefit)
C)
graph both R(x) and C(x) . Use first zoom fit (6). then use Zbox to frame the
intersection area. FInd the intersections. (use 2ndcalc/intersect). For
what values of x the firm break even ? (R(x) = C(x)
make a profit ? (that is R(x) > C(x), graph of R above C) lose money ?
(R(x) < C(x) , graph of R below C)
------------------------------------------------------------------------------------------------------
13) 21 p. 388
The town of Smasville was founded in 1900. Its population y (in hundreds) is given by the equation:
f(x) = y = - 0.1x4 + 1.7 x3 - 9x2 + 14.4 x + 5 where x is the number of years after 1900.
if x= 3, y = 5: that means, the year is 1903 and the population is 500 people.
A) Find the population in 1900. (that is x=0)
B) Find the population when the town becomes a ghost town ? (that is y=0)
Find the year AND THE MONTH.
Find the solution of y = 0 or zeros of f(x) = 0 or x-intercepts of the graph of y.
(intersection of the curve with the x-axis). Graph the function . Use the window: 0<x<10 and -2<y<13.
You can move the cursor until you find the answer. There are 12 months. 1/12 = end of January.
2/12 = end of February 3/12 = end of March ...
C) What was the largest population of Smallsville after 1905 ? When did the town reach that maximum? year and MONTH ?
Find the maximum of the function. You can move the cursor until reaching the top. Find (x,ymax).
x is the number of years after 1900 and y the number of people in hundreds.
--------------------------------------------------------------------------------------------------
14) 24 p. 388
A
function that is not a polynomial can often be approximated by a
polynomial. For example, for certain x-values, the function f(x) = ex
can be approximated by the fifth-degree polynomial:
p(x) = 1 + x + x2/2 + x3/6 + x4/24 + x5/120
A) show that p(1) is about f(1). How good is the approximation.
Enter p(x) in the TI and compute p(1) using 2nd /table.
B) calculate p(5). How well does p(5) approximate f(5) ?
C)
USe the table option of the TI to compare p(x) and f(x). what range of
values of x does p(x) gives a good estimate for f(x) ? graph them together on the same set of axes.
--------------------------------------------------------------------------------------------------
Assignments, try without hints
15) p. 388 #25 A , B, C (try without the steps below )
Let f(x) = x - x^3/6 + x^5/120
A) i) First change your mode to radian on your TI. (use the key MODE and switch th radian)
ii) Then enter f(x) in Y11 and sin(x) in Y2.
iii)Then set the window (key window at the top) : -2п<x<2п (use the key п )
and -3 <y <3
iV) Use your TI trace to graph the 2 functions. With your cursor move left to right/right to left to see
how x and y varies as you ride the graphs. The up and down cursor is to switch from Y1 to Y2
B)
Observe the 2 graphs. (TRACE key) For a range of values of x, the graphs overlap.
The graph of f resembles the graph of sin(x) on a small interval. Give
the approximate interval. Use your cursor side way (in TRACE) to find
the interval.
C) use the key TABLE (2nd GRAPH) to compute f(п/8) and sin(п/8). How reasonable an approximation does f give for sin (п/8) ? (that is in the table, compare Y1 and Y2)
-------------------------------------------------------------------------------------------------------------------
16) 22 p. 388
The
volume , V, in millimeters, of 1kg of water as a function of
temperature T is given, for 0<T<30C by: V= 999.87 - 0.06426T
+ 0.0085143T2 - 0.0000679T3
A) leading term =________, leading coefficient = ____________, sketch the long run below
B) enter the function in Y1 (TI). (make sure the mode is back to degree). don't trace it yet.
C)
Using the TABLE key, Find V(0). What does that represent ?: If the
temperature of the_______ is ____ then the ______ is equal to ______ml.
B) Find V(10) (use the TABLE key). THat means :
______________________
D) You know 0<T<30 (that is 0<x<30). This is the window for the x values.
If T = 0, V = _________ and if T = 30, V == ____________
this means that ___ < V < ____ . It gives you the window for the y values.
Use
these 2 windows (for x values and y values) to set the proper ranges
using the key WINDOW in your TI. Then trace the function.
E)
DEscribe the shape of your graph. Does V increase or decrease as T
increases ? Does the curve upward or downward ? What does the graph
tell us about how the volume varies with temperature?
F) At what temperature does water have the maximum density ?
hints:
To answer the question, you have to know that density = Mass/Volume.
THe mass is a constant (1kg). So if the density is a maximum, the
volume is a ____________. This happens when T=____.
© haplosciences 2000-2007 -
vente par correspondance jeux éducatifs,
jeux scientifiques clement
o
ni , jeux educatifs
Amina, Wanikou, mael
r
oi du mail , laboratoire chimie, fiches
e
xperiences, thomas au temps des chevaliers, Pilou
la classe,
fichiers animation les francas, jeux engrenages
singes, engre
nages juniors, engr
e
nages train safari, prismes
microscopes, p
rismes, jeux gyzmo sous la mer, jeux engrenages, c
o
loriage, travaux manuels, animaux, sciences, vente prismes
tub es a essai, apprendre l
a
nglais , coloriae ,
anglais maternelle, jeux clémentoni, cahiers maternelle,
anglais enfants, anglais petits