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| Up to Main Lab Page | Next Lesson - Trig Functions | Previous Lesson - Program Control |
| Topics for this Lab: | Groupings [...] {...} | Colors | Polynomials | Rational Functions | |
|---|---|---|---|---|---|
| Products | Algebraic Functions | Sums of Functions | Transcendental Functions | Absolute Value |
In this lesson we will use Maple to investigate the behaviour of some standard Calculus functions.
Thus {f, g} represents the unordered pair of functions f and g.
So
> f := x-> x;
> g := x -> x^2;
> plot({f(x), g(x)}, x = -1..1.5);
will plot both f and g on the same graph. We don't care wich order it does them
in so we can give f and g as an unordered pair.
Suppose now though that we wish to assign specific colors to f and g on the plot.
We can do this by using the optional color=name
argument to the plot function.
> plot({f(x), g(x)}, x = -1..1.5, color=magenta);
If we want to assign different colors to f and g we could try
> plot({f(x), g(x)}, x = -1..1.5, color={magenta, blue}); #WRONG
But this doesn't work since plot needs to know which order to
apply the colors in. Thus we should use the following.
> plot({f(x), g(x)}, x = -1..1.5, color=[magenta, blue]);
Strictly speaking, since we now care which order we want f and g to be plotted
(and hence which color each gets), we should put them as an ordered list as well.
> plot([f(x), g(x)], x = -1..1.5, color=[magenta, blue]);
This tells Maple unequiviqually that the first function, f, gets magenta while
the second function, g, gets blue.
| aquamarine | black | blue | navy | coral | cyan | brown | gold | green | gray |
| grey | khaki | magenta | maroon | orange | pink | plum | red | sienna | tan |
| turquoise | violet | wheat | white | yellow |
Color values can also be set by giving the RGB (Red, Green, Blue) values for
the color, through the COLOR function. For Example
> plot(g(x), x = -1..1.5, color=COLOR(RGB, .5607, .7372, .5607));
We presume you are familiar with the linear polynomial f(x) = mx + b,
which has slope m and intercept b. We can actually use the iteration operator
($) to generate a sequence of functions to
plot.
> f := x -> m*x;
> functions := [f(x) $ m = -5 .. 5];
> plot(functions, x = -5..5);
The second line creates an ordered list of functions, one for each m value.
On the same graph plot x + b for b from -3 to 3, in the range -5 to 5.
If f(x) is a quadratic, f(x) = ax2 + bx + c, we can complete the square to get f(x) = a((x + b/(2a))2 + (c/a - (b/2a)2). Since a, b and c are constants this has the form (x + d)2 + r = g(x + d) + r, where g(x) = x2. We now investigate the effect of the transformations x -> x + d and y -> y + r on the graph of f.
If we make the transformation x -> x + d in a graph, the effect is to shift the
graph to the left by d units. This is because what was the origin is now the
point -d.
> plot([(x + 3)^2, x^2], x=-5..5, color=[red, blue]);
Of course if d is negative the graph shifts to the right.
If we make the transformation y -> y + r, (i.e. f(x) -> f(x) + r) this has the
effect of shifting the graph up by r units, since if y = f(x), then y + r =
f(x) + r.
> plot([x^2 + 20, x^2], x=-5..5, color=[red, blue]);
Of course if r is negative the graph shifts down.
These two rules are generally true, that is:
f(x) -> f(x + a) shifts the graph of f left by a.
f(x) -> f(x) + a shifts the graph of f up by a.
From this we see that any quadratic must look essentially like y = x2, possibly shifted.
If we compare the graphs of x2 and x4:
> plot(x^2, x=-10..10);
> plot(x^4, x=-10..10);
we see that they are very similar. x4 has a slightly flatter bottom,
and rises more quickly, but the basic form is the same. We can see the
differences more clearly by plotting both functions on the same graph:
> plot([x^2, x^4], x=-1.5..1.5, color=[red,blue]);
For a generalised polynomial , as x tends to either + or - infinity the highest power of x will take over. For even degree polynomials this means that f(x) -> infinity as x -> +- infinity.
To see what other terms do consider the following:
> f := x -> x^4 + 20*a*x^2;
> fns := [f(x) $ a = -3..3];
> plot(fns, x = -9..9);
The x2 term induces a kink if a is negative. This is because for
small x the x2 term becomes significant and bows the shape of the
graph near 0. Away from 0 the x4 term takes over.
We can see this more clearly by plotting:
> plot(x^4-60*x^2+200,x=-10..10);
From this we can see how a quartic could have four roots. Notice that the
derivative of a quartic is a cubic and has three roots, the three turning
points on the graph.
In general an n degree polynomial may have up to n real roots and n-1 turning points.
To investigate the effect of an x3 term consider
> plot(x^4 + 2*x^3 - 60*x^2 , x = -9 .. 9);
Notice how the right hand minima is raised, and the left hand one is lowered.
On the same graph plot x^4 + a*x^3 - 60*x^2, for a from -2 to 3, on the range - 9 to 9.
Care must be taken when using this interpretation. Consider
(x + 1)4, in particular try
> expand((x+1)^4);
This shows that the existance of odd powers, x3 and x in a quartic, may just be the result of a linear shift x -> x + c.
Use the methods above to investigate sextics (degree six polynomials).
As x -> infinity x3 goes to infinity, as x -> -infinity, x3 goes to -infinity. This will be the limiting behaviour of any cubic.
We now consider the effect of introducing an x term. Consider
x3 + ax = x(x2 + a).
> f := x -> x^3 - a*x;
> fns := [ f(x) $ a=-2..2];
> plot(fns, x = -2..2);
Try plotting a couple of these individually to get a better idea of the effect of
each a.
If a is positive, x2 + a never goes negative and this can only add to the cubic. If a is negative x2 + a goes negative and introduces a kink in the cubic, giving the classic three roots.
Now consider the effect of an x2 term.
x3 - ax2 = x2(x - a) which has two roots, one at
x = 0 and one at x = a. This suggests a skewed graph.
> f := x -> x^3 - a*x^2;
> fns := [ f(x) $ a=0..3];
> plot(fns, x = -1..3);
Putting these together would indicate that x3 - ax2 - bx
has a skewed cubic look.
> f:=x->x^3-a*x^2-5*x;
> fns := [ f(x) $ a=0..3];
> plot(fns, x=-3..3);
Use the methods above to investigate quintics (degree five polynomials).
The order of a rational function is (the degree of p) - (the degree of
q).
The order of a rational function determines its behaviour as x tends to +-
infinity.
In general f will behave like xn, where n is the order of f
(possibly negative), as x tends to +-infinity.
We start by considering 1/x.
> plot(1/x, x=-5..5);
Oops, since 1/x has a discontinuity at x = 0 we must limit the y range.
> plot(1/x, x=-5..5, -10..10, discont=true, color=red);
As x -> infinity 1/x -> 0+, as x -> -infinity 1/x -> 0-.
We say that 1/x has a horizontal asymptote of 0.
Now consider 1/x2.
> plot(1/x^2, x=-5..5, -10..10, discont=true, color=red);
This has a very similar form to 1/x, except that when x is negative
1/x2 is positive, thus as x -> -infinity 1/x2 -> 0+.
The graphs of 1/xn look similar to one of these two, depending
whether x is odd or even.
Now consider f(x) = 1/((x-1)(x+1)).
> plot(1/((x-1)*(x+1)), x=-5..5, -10..10, discont=true, color=red);
Since the order of f is -2 f(x) has the behaviour of 1/x2 as x tends to
infinity. It also has two asymptotes, one at -1 and one at 1.
We can understand the behaviour of the function near these asymptotes by considering the
signs of (x-1) and (x+1) near the asymptotes.
| x < -1 | -1 < x < 1 | x > 1 | |
|---|---|---|---|
| x - 1 | - | + | + |
| x + 1 | - | - | + |
| f(x) | + | - | + |
Consider:
> f:=(x-a)/((x-1)*(x+1));
> fns := [f $ a = -2..2];
> plot(fns, x=-4..4, -10..10, discont=true);
The order of all of these functions is 1, and so thay behave like 1/x as x
tends to +-infinity.
See if you can explain why each of the lines goes where it does by considering the signs of the various factors. It may help to plot the functions on seperate graphs.
The form of these graphs can be understood in terms of multiples of functions, f(x)g(x). In this case f(x) is a linear function, so we have h(x) = (x - a)g(x).
If x > a then x - a > 0 and the sign of h will be the same as that of g.
If x = a then x - a = 0 and the h will have a root at a.
If x < a then x - a < 0 and the sign of h will be opposite to that of g.
Another common form is cf(x), where c is a constant. This stretches the graph
of f(x) vertically by a factor of c.
> f:= x^2;
> plot([f(x),2*f(x), 3*f(x)], x = -3..3, -10..10, color=[red, blue, cyan]);
Try the above plot with f(x) = 1/((x-1)(x+1))
Thus x1/2, x-1/3, x + x-1 - 3x-3/2, (x2 + 1)1/2 are all algebraic functions.
> plot(x^(1/2), x = -5..5);
Since x1/2 is not defined for x < 0 we don't need the negative
range.
> plot(x^(1/2), x = -1..10);
Note that x1/n is not defined if n is even and x is negative,
whereas if n is odd it is defined everywhere.
Plot the following:
We have already effectively considered sums when considering polynomials.
Go back to the polynomial section and consider the problem as an algebraic sum.
Sums of Functions
One of the important relations for constructing algebraic functions is taking
the sum of two functions.
When we add functions together they add pointwise. Thus if both f and g are
positive f + g will grow, whereas if one is negative it will pull f + g down.
Consider the following algebraic sum:
> f := x -> x^2;
> g := x -> 1/x;
> plot([f(x), g(x), f(x) + g(x)], x = -2 .. 2, -10..10, color=[red,blue,cyan]);
> f := x -> x^4;
> g := x -> x^3;
> plot([f(x), g(x), f(x) + g(x)], x = -2 .. 2);
Transcendental Functions
The transcendental functions are basically all functions which are not
algebraic. This includes the trigonometric functions, which we will consider in
the next lab, exponential and logarithmic functions, which we will also look at
in detail later.
One simple type of function which is not algebraic are the piecewise functions. Of particular interest is if absolute value function, abs(x) or |x|.
| abs(x) = | x | if x >= 0 |
| -x | if x < 0 |
| x < -1 | -1 < x < 2 | x > 2 | |
|---|---|---|---|
| x - 2 | - | + | + |
| x + 1 | - | - | + |
Thus
| abs(x + 1) + abs(x - 2) = | -(x + 1) -(x - 2) | if x < -1 | = | -2x + 1 | if x < -1 |
| (x + 1) - (x - 2) | if -1 < x < 2 | 3 | if -1 < x < 2 | ||
| (x + 1) + (x + 2) | if x > 2 | 2x + 3 | if x > 2 |
Plot The following:
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