Section 3. Functions as Transformations
Functions play a key role in many applications of mathematics. The arrow notation is used to make functions and the unapply procedure can turn any expression into a function. The Matrix procedure can be used to make tables of data.
0. Pull down the Help menu, choose New Users/Full Tour, then click on the third item: Algebraic Calculations and work through the examples in that worksheet; pay special attention to function notation and solving equations.
Open a new Maple worksheet and do the following
1. Use arrow notation to define the function f(x)=cos(x) - x tan(x). Then use the entry
g := D(f);
to obtain g as the derivative function.
2. Continuing 1. Plot f over the interval x = -1 to x = 1 using plot( f(x), x=-1..1).
Then find the area of the region below the graph of f and above the x axis.
Hint. This will require the values b where f(b) = 0. Find the positive value using
b := fsolve(f(x)=0,x,0..1);
Check that f(-b) = 0 also. Once b is found, the area is the integral of f(x) from -b to b.
3. Continuing 2. Plot the graph of f from -1 to 1 and the tangent line segment to the graph at the point x = 0.5, y = f(0.5) over the interval x = 0 to x = 1.
Hint. You already have the derivative function g. Use it to define the tangent line function using
T := x -> f(0.5) + g(0.5)*(x - 0.5);
Then execute
plot( [f(x), [(t,T(t),t=0..1] ], x=-1..1);
You will see that Maple's default colors for two curves are red and green. The second curve is easier to see with the optional equation color=[red,blue]. Try it, you will like it.
4. Continuing 3. Find the length of the curve plotted in Exercise 2.
Hint. Do the integration numerically using evalf( Int( .... ) ).
5.* An animation.
The tangent line plot in Exercise 3 can easily be animated as follows.
First define the function T(a,x) whose value at (a,x) is the formula for the tangent line to the graph of f at (a,f(a)).
T := (a,x) -> f(a) + g(a)*(x - a);
Then use
plots[animate](plot,[ [f(x), [t,T(a,t),t=a-0.5..a+0.5] ],x=-1..1,-1..2,color=[red,blue]],a = -1..1);
Once the plot appears, click on it with the mouse and the context bar becomes a row of video controls; enjoy.
Read the Help page for plots[animate].
6. Solving a differential equation.
The dsolve procedure solves differential equations. The syntax is simply
dsolve( deqn )
where deqn is a differential equation or the name of one. Do a restart and define a simple first order differential equation as follows
restart; deqn := diff(y(x),x) + x*y(x) = x;
Get the general solution with the entry
dsolve( deqn );
Now obtain the solution satisfying y(0) = 0 using
soln := dsolve( {deqn, y(0)=0} );
7. Plot the last solution using
plot( rhs(soln), x=-2..2);
8.* Using unapply and Matrix.
Convert the second solution in Exercise 6 into a function f using
f := unapply(rhs(soln),x);
Evaluate the solution at x = 0, 0.2, 0.4, ... , 1.0 to 4 digits as follows
evalf[4]( f(0.2*k) $ k=0..5 );
Make a table of function values with the following Matrix entry
Matrix( [ [x, 0.2*k $ k=0..5] , [ 'f(x)', % ] ] );