#---------------------------------------------------------------------------
# M1225		First Lab Assignment		Due Thursday, Jan 21, 1999

# Treibergs	Logs and exponentials
#---------------------------------------------------------------------------

# For those of you who are new to MAPLE, this lab assignment will give you some idea of what it 
# can do. We will explore graphical and numerical features of logarithms. Type in the 
# instructions as you see them on the assignment and see that you get the same answers that I did.

# When you are ready to prepare the lab questions, modify the examples or enter new code that 
# answers the question. Hand in only the solutions of the lab problems. Make your papers self 
# contained by entering text or writing by hand on the printout enough to describe the question 
# and the answer. You will have to label functions on graphs and points.

# Plotting functions

# To plot a single function

> plot(sin(t),t=-2*Pi..2*Pi);
> 

# To plot several several functions, list them in a set (surrounded by curly braces)


> plot({sin(x),sin(2*x),sin(3*x)},x=-2*Pi..2*Pi);
# Scaling=constrained gets the vertical scale compatible with the horizontal scale
> plot({sin(x), sin(2*x),sin(3*x)},x=-2*Pi..2*Pi,scaling=constrained);
# MAPLE can efficiently make a list of objects depending on a parameter via sequence

# and put them all in a set data structure by surrounding the sequence with braces. The set of 
# functions is given the name SetOfFns. This named set of functions may be plotted. Note that set 
# data structures don't keep the objects in order.


> SetOfFns:={seq(t^k,k=1..9)};
                                2   3   4   5   6   7   8   9
               SetOfFns := {t, t , t , t , t , t , t , t , t }


> plot(SetOfFns,t=-1.1..1.1,scaling=constrained);
# We can increase the set by union with another set containing the constant function
> plot(SetOfFns union {1},t=-1.1..1.1,scaling=constrained);
# 

# Implicit plotting

# The relationship between a function and its inverse is always if y = f(x) then x = g(y) where  
# g(y) is the inverse function of  f. Now the inverse relation  R={(x,y) : x=f(y)} 

# may not be a graph, but MAPLE can plot it anyway with implicitplot. You have to load in the 
# extra plotting routines first.

> with(plots):
> implicitplot(x^2 + y^2 = 4,x=-3..3,y=-3..3,scaling = constrained);
# Now lets plot y=sin(x) and the inverse relation R={(x,y):x=sin(y)}.
> implicitplot({y=sin(x),x=y,x=sin(y)},x=-7..7,y=-7..7,scaling=constrained);
# The wiggly curves are mirror images across the 45¡ line. The relation  R  is not a function 
# because the curve has several y values for each x. By restricting to those  x  where  y  is 
# monotone allows us to define the principal part of the inverse function. 

> implicitplot({y=sin(x),x=y,x=sin(y)},x=-Pi/2..Pi/2,y=-Pi/2..Pi/2, scaling=constrained);

# Of course MAPLE has the inverse sine which is a function for -1²x²1.


> plot({sin(x),x,arcsin(x)},x=-1..1,scaling=constrained);


# Exponentials and Logarithms

# The natural logarithm and exponential are written  ln(x)  and  exp(x).  We make a sequence of 
# lists (a list is a data structure containing an ordered sequences of objects.)


> seq([k,ln(k),exp(k)],k=1..10);
[1, 0, exp(1)], [2, ln(2), exp(2)], [3, ln(3), exp(3)], [4, ln(4), exp(4)],

    [5, ln(5), exp(5)], [6, ln(6), exp(6)], [7, ln(7), exp(7)],

    [8, ln(8), exp(8)], [9, ln(9), exp(9)], [10, ln(10), exp(10)]


# To get the decimal equivalents,
> seq([k,evalf(ln(k)),evalf(exp(k))],k=1..10);
     [1, 0, 2.718281828], [2, .6931471806, 7.389056099],

         [3, 1.098612289, 20.08553692], [4, 1.386294361, 54.59815003],

         [5, 1.609437912, 148.4131591], [6, 1.791759469, 403.4287935],

         [7, 1.945910149, 1096.633158], [8, 2.079441542, 2980.957987],

         [9, 2.197224577, 8103.083928], [10, 2.302585093, 22026.46579]


# To get logs and exponentials of other bases MAPLE provides the commands log[b](x), log10(x)  
# and  a^x

> seq([k,evalf(log[2](k)),evalf(2^k)],k=1..8);
 [1, 0, 2.], [2, 1., 4.], [3, 1.584962501, 8.], [4, 2.000000000, 16.],

     [5, 2.321928094, 32.], [6, 2.584962500, 64.], [7, 2.807354922, 128.],

     [8, 3.000000000, 256.]


> 


# Integral for logarithm

# Maple can do integrals, e.g, if f(t)=t^3 + sin(t) then  ºf(t) dt  from 0 to  x  is


> int(t^3 + sin(t),t=0..x);
                                  4
                             1/4 x  - cos(x) + 1


# It knows about  ln.


> int(1/t,t=1..x);
                                    ln(x)


# We replace 1/t by the function which is 1/2 for 1 < x ² 2, 1/3 for  2 < x ² 3 and so on

# We can define the staircase shaped function by the command


> f:= t -> 1/floor(1+t);
                                           1
                           f := t -> ------------
                                     floor(1 + t)


# where  floor means the greatest integer part, e.g. floor(2.4)=2, floor(3)=3,  floor(-2.3) = 
# Ð3. Now plot. discont=true tells MAPLE to expect a discontinuous function.


> plot({1/t,f(t)},t=1..8,scaling=constrained,discont=true);
> 

# The area under the staircase is less than the area under 1/t  (which is the logarithm)

# To get the area from 1 ² x ² 8 we may sum Sfn(8) = 1/2 + 1/3 + ... + 1/8 or in MAPLE


> sum(1/j,j=2..8);
                                     481
                                     ---
                                     280


# To see the decimal equivalent, use " to represent the previous quantity.
> evalf(");
                                 1.717857143


# We can make this a function
> Sfn:= m -> evalf(sum(1/j,j=2..m));
                                            m
                                          -----
                                           \
                        Sfn := m -> evalf(  )   1/j)
                                           /
                                          -----
                                          j = 2


> 

> Sfn(8);
                                 1.717857143


> 

# Compare to the integral from 1 to  10.  Ifn(8) = int(1/t,t=1..8)


> evalf(int(1/t,t=1..8));
                                 2.079441542


> 

# Laboratory Problems
# 1.  On the same graph, plot log[2](x), log[3](x),...,log[10](x) for 0.2 ² x ² 5. How do the 
# logarithms change depending on the base? Do the same for  log[1/2](x),...,log[1/10](x).

# 2.  On the same graph, plot 2^x,3^x,...,8^x for -5² x ² 1.2 How do the exponentials change 
# depending on the base? Do the same for (1/2)^x,...,(1/8)^x.

# 3. Does the function  f(x) = x + 2*x^3  have an inverse function? Find a subdomain of the 
# reals for which  f  has an inverse function.  Plot  y=f(x), and  y=f^(-1)(x)  (the inverse 
# function)  on the same graph. From your graph, approximate f^(-1)(5). (Read the mouse 
# coordinates.)

# 4.  Does the function  y = ln(x)  have an inverse for all x? For x>0? Plot the inverse relation 
# using implicitplot. What function is the inverse function? Verify by plotting. (for example plot 
# y=exp(x)) Answer the same question for y=log[3](x) and for  y=log[1/4](x).

# 5.  Compute Ifn(n), Sfn(n)  and  Ifn(n) - Sfn(n) for n=2,3,4,10,100,1000,10000.  This is 
# evidence for the fact that there is a number  c  such that   Ifn(n)-c ² Sfn(n) ² Ifn(n)  for all  
# n.  What is your guess as to the value of  c? (we will come to this eventually, page 535). This 
# shows that it takes a lot of terms in the sum to add up to some number. Using your  c, estimate 
# how many terms (what  n) is needed for 1/2 + 1/3 + ... + 1/n to  exceed  10?  Can you find 
# the least  n  by trial and error using the sum? (MAPLE is not actually summing!!!!)