Worked Sample Problems - Math 103

CHAPTER P - Preliminaries

Section P.1, page 7

Problem 7

Maple can solve many inequalities, such as the one in this problem:

>	solve(5*x-3<=7-3*x);

This indicates that the solution is . Note what happens when the inequality is strict:

>	solve(5*x-3<7-3*x);

Of course, this indicates that the solution is .

Problem 25

>	solve(abs(z/5-1)<=1);

Problem 44

>	solve(abs(x-1)=1-x);

Section P.2, page 15

Problems 19-20

We can write a little Maple program that finds the line through two points as follows:

>	linethrough:=(p,q)->y=simplify(q[2]+(q[2]-p[2])/(q[1]-p[1])*(x-q[1]));

In Maple, points are indicated with square brackets, so problem 19 is to find the line through the points [3,4] and [-2,5]:

>	linethrough([3,4],[-2,5]);

>	linethrough([-8,0],[-1,3]);

Problem 45

Here is a plot of  and C=F:

>	plot({5/9*(F-32),F},F=-50..50,color=blue,thickness=2,labels=["F","C"]);

To find where the lines cross, we solve:

>	solve(F=5/9*(F-32),F);

Section P.3, page 25

Problem 32

To see whether a function is even or odd, we can add and subtract f(x) and f(-x) as follows:

>	g:=x->x^4+3*x^2-1;

>	simplify(g(x)+g(-x));

>	simplify(g(x)-g(-x));

Since g(x)-g(-x) is zero, the function g is even.

Problem 45 d-f

>	u:=x->4*x-5; v:=x->x^2; f:=x->1/x;

>	v(f(u(x)));

>	f(u(v(x)));

>	f(v(u(x)));

Section P.4, page 32

Problems 17, 19, 28

Of course, Maple makes these easy - here are all the answers on one graph:

>	plot({sqrt(x+4),abs(x-2),(x+2)^(3/2)+1},x=-5..5,-1..10,color=blue,thickness=2);

Problem 47

Wc can plot the circle using "implicitplot" -- be sure and use "scaling=constrained" so you don't get an ellipse:

>	circ:=x^2+y^2-3*y-4=0:

>	with(plots,implicitplot):

>	implicitplot(circ,x=-3..3,y=-1..4,thickness=2,color=blue);

To find intercepts:

>	solve(subs(y=0,circ),x);

So x-intercepts are [2,0] and [-2,0].

>	solve(subs(x=0,circ),y);

and y-intercepts are [0,4] and [0,-1].

The x and y coordinates of the center must be at the averages of the x and y intercepts. (Why?) - So:

>	center:=[(2-2)/2,(4-1)/2];

Problem 83

First we define f(x):

>	f:=x->5*x/(x^2+4);

>	with(plots,display):

>	A:=plot(f(x),x=-10..10,color=blue,thickness=4):

We made a thick copy of the graph of f to compare with the others we will make:

(a) We'll also plot f(ax) for a=2,3,10 and display them all together:

>	B:=plot({f(2*x),f(3*x),f(10*x)},x=-10..10,color=red,thickness=2):

>	display({A,B});

Apparently the bend in the middle of the graph gets sharper and bunches in toward the y-axis as a gets larger.

(b) Now we'll plot f(ax) for a=-2,-3 against the original f:

>	C:=plot({f(-2*x),f(-3*x)},x=-10..10,color=red,thickness=2):

>	display({A,C});

So the graph gets flipped (around either of the x or y axes), and bunches in as a gets more negative.

(c) Now for a=1/2, 1/3 and 1/4:

>	E:=plot({f(x/2),f(x/3),f(x/4)},x=-10..10,color=red,thickness=2):

>	display({A,E});

Now it appears that the graphs are getting "wider" - the bumps in the graph move away from the y axis (if we used small negative values of a, the graph would flip as well).

Section P.5, page 43

Problem 20

Again, plotting is easy -- just make sure to make the domain big enough to include a few periods (the period of this function is , of course.

>	plot(sin(x+Pi/2),x=-4*Pi..4*Pi,color=blue,thickness=2);

Problem 50

Maple doesn't mind trig functions, so to get it to evaluate  you have to tell it to convert it to "radical" form:

>	convert((sin(Pi/8))^2,radical);