Maple Lab #2-Due, Thursday Oct 23
Mat 200.1637 Rudy Meangru
1.
Define each of the following terms. Use examples or graphs to illustrate.
(a) Domain: the set of all possible values of the argument of a function.
Such as, y=x-6
Domain is real number.
(b) Range: the set of all possible values for the output of a function.
Such as, y=x-6
Range is real number
(c) X-intercept: the value of x at the point where a curve crosses the x-axis.
Such as, y=x-6, when y=o, x=6.
(d) Y-intercept: the value of y at the point where a curve crosses the y-axis.
Such as, y=x-6, when x=0, y=-6.
(e) Degree of a polynomial: is a highest possible exponent of algebra.
Such as, y=9 -4 +12x, the highest root is 3, there are 3 possible values for x, so the
degree is 3.
(f) Turning point: Points on the functions graph corresponding to relative extreme
values are turning points or points where the function changes from decreasing to increasing or
vice versa. Suppose the degree=n, the number of turning points are at most n-1.
a,b, and c are the turning points.
(g)
Even and odd multiplicity:
When the roots of a function touch the x-axis, is even multiplicity.
f(x)=
When the roots of a function across the x-axis is odd multiplicity
f(x)=
(h) Local minimum and maximum
A and C are local minimum, B is local maximum
2.
A function f is said to be increasing over some interval containing
and
if for
<
f(x )<f( ).
A function f is said to be decreasing over some interval containing
and
<
f( )<f( ).
Let the function f be defined by f(x)= -9 -4 +12x
f(x)= + -8 -12x
(a) Sketch the graph
(b)
Describe the domain and range
(c) Find the x-and y-intercept
(d)
Determine whether each x-intercept is of odd or even multiplicity.
(e) State the degree of the polynomial
(f) Determine the number of turning points.
(f) Determine the local maximum and local minimum, if any exist.
(h)
Over what intervals is f increasing?
(i) Over what intervals if f decreasing?
f(x)= -9 -4 +12x:
a) > plot(x^5-9*x^3-4*x^2+12*x,x=-10..10,y=-50..20);
if for
c)
b) Domain:(Range:
> solve(x^5-9*x^3-4*x^2+12*x=0);
0, 1, 3, -2, -2
c)
x-intercept: 0,1,3,-2;
y-intercept.is 0.
d)
When x=-2 intercept, it’s even multiplicity.
When x=0,1,3 intercepts are odd multiplicity.
e)
degree=5.
f)
There are 4 turning points
g)
Local minimum: -39.06648784, { [ { x2.376619459 }, -39.06648784 ] }
Local maximum: 3.943129547, { [ { x0.5467953101 }, 3.943129547 ] }
h)
the intervals of f increasing:
(-∞,-2), (, -0.9234147692 , 0.5467953101 ), ( 2.376619459 ,∞)
i) the intervals of f decreasing:
(-2, -0.9234147692 ), ( 0.5467953101 , 2.376619459 )
f(x)= + -8 -12x
a) > plot(x^4+x^3-8*x^2-12*x,x=-10..10,y=-50..20);
b) Domain:(-
Range:
x-intercept:
c) > solve(x^4+x^3-8*x^2-12*x=0);
0, 3, -2, -2
x-intercept: 3,-2,0. y-intercept is 0.
d) When x-intercept are 3,0,12,they are even multiplicity.
e) Degree=4
f) There are 3 turning points
g) Local maximum: 4.394531250, { [ { x-0.7500000000 }, 4.394531250 ] }
Local minimum: -32, { [ { x2 }, -32 ] }
h) The intervals of f increasing: (-2,-0.75),( 2,∞)
i)
The intervals of f decreasing: (-∞
,(-0.75,2)
3. A linear function is of the form f(x)= mx+b
For the linear functions f(x)=2x-3 and g(x)=-3x+2,
(a)
Compute the average rate of change over the intervals
[-3,-2],[-2,-1],[-1,0][0,1],[1,2]and [2,3]
(b) What observations can be made about the result in (a)?
(c)
Graph f(x) and g(x)
(d) What is the relation of the direction of each graph with its slope?
a)
f(x)=2x-3:
> f:=x->2*x-3;
f := x2 x3
> (f(-2)-f(-3))/((-2)-(-3));
2
> (f(-1)-f(-2))/((-1)-(-2));
2
> (f(0)-f(-1))/((0)-(-1));
2
> (f(1)-f(0))/((1)-(0));
2
> (f(2)-f(1))/((2)-(1));
2
> (f(3)-f(2))/((3)-(2));
2
So the average rate is 2
g(x)=-3x+2:
> f:=x->-3*x+2;
f := x3 x2
> (f(-2)-f(-3))/((-2)-(-3));
-3
> (f(-1)-f(-2))/((-1)-(-2));
-3
> (f(0)-f(-1))/((0)-(-1));
-3
> (f(1)-f(0))/((1)-(0));
-3
> (f(2)-f(1))/((2)-(1));
-3
> (f(3)-f(2))/((3)-(2));
-3
So the average rate is -3.
b) The rates of change over the intervals are same.
c) f(x):
g(x):
d)
If slope>0, the direction of the graph is increasing.
If slope<0, the direction of the graph is decreasing.
4.
A quadratic function is of the form f(x)=a +bx+c, where a
The graph of a quadratic
function is called a parabola. If a>0, the graph concaves upward, if a<0, the graph concave
downwards.
Let f(x)=
and g(x)=- +3x+28
(a)
Sketch the graphs of f(x) and g(x).
(b)
Describe the domain of each function
(c)
Sketch the graph of both functions
(d) locate the vertex of each.
(e)
Describe the range of each.
(f) Determine the interval where the function is increasing or decreasing.
:
a) > f:=x->x^2-2*x-15;
f := xx 22 x15
> plot(f);
b) Domain:
c) The same as a).
d) > minimize(x^2-2*x-15,x=-2..2,location);
-16, { [ { x1 }, -16 ] }
Vertex:(1,-16)
e) Range: [-16, ∞).
f) Increasing: [1, ∞); decreasing:(-∞,1)
g(x)=- +3x+28:
a) > f:=x->-x^2+3*x+28;
f := xx 23 x28
> plot(f);
b) Domain:
.
c) The same as a).
d) > maximize(-x^2+3*x+28,x=-2..2,location);
121 
3 121 
, {  { x },
}
4
2
4 

> evalf(%);
30.25000000, { [ { x1.500000000 }, 30.25000000 ] }
Vertex:(1.5, 30.25).
e) Range:(-∞,30.25).
f) Increasing:(-∞,1.5; decreasing: [1.5, ∞).
5.
Suppose that the function s(t)=3 +10 represent the position that a particle moves along a
straight line, where s is in feet and t is in seconds.
(a)
Compute the average rate of change of s(t) over the following time intervals.
[2,2.1],[2,2.01],[2,2.001] and [2,2.0001]. The average rage of change here is referred to as the
average velocity.
> s:=t->3*t^2+10;
s := t3 t 210
> (s(2.1)-s(2))/(2.1-2);
12.3
> (s(2.01)-s(2))/(2.01-2);
12.03
> (s(2.001)-s(2))/(2.001-2);
12.003
> (s(2.0001)-s(2))/(2.0001-2);
12.0003
(b)What value is the average velocity approaching as the interval gets smaller, that is, as t
gets closer to 2? This target value is called the instantaneous velocity.