1. Hierarchy of arithmetic operations: Use your own examples (at least three examples and execute them in maple) to prove which arithmetic operations are carried out first in Maple and if they are of equal priority, in which directions they are carried?

1.
> 11+20/3-5;

-here, division is performed first, then followed by addition and subtraction.
2.
3/5*7/8(5)+200;

-here, calculation involving parentheses is performed first, then followed by division, multiplication and addition.
3.
> 666-23+98/3*5^10;

-here,  exponentiation is executed first, followed by division, multiplication, addition and subtraction.
4.
987^5(9)*23-10;

-here,  calculation involving parentheses is executed first, followed by exponentiation, multiplication and subtraction.

5.
234+76-90*9.23;

-here, multiplication is performed first, followed by addition and subtraction.

2. Building fractions: Write an equation of your own, in the fractional form as shown in the class.  Use minimum number of parenthesis only.

> 1*1/2-5/6+(2*4/5);

3. Using your own examples (one example each and execute them in maple) clearly bring out all the differences between the following maple symbols and commands:
	
a) ; and :

' ; ' is used to execute the equation with the result displayed.

> 3+5;

' : ' is used to execute the equation without displaying the result.

> 3+5:

b) = and :=

' = ' is a temporary assigning symbol. It only works in a specific line. After moving to another line, the command will not be valid and computed. 
> myDist= 4*ft + 3*inches;

> convert(myDist, system, FPS);

' := ' is a permanent assigning symbol. It will work in the whole worksheet, until the value is changed or 'restart' is executed.
> restart;
> myDist:= 4*ft + 3*inches;

> convert(myDist, system, FPS);

c) ? and ???

' ? ' is used to expand all parts (i.e. example, description) of a certain topic when commanding help of Maple.
> ?plot;

 ' ??? ' is used to expand the example part of a certain topic with other parts (i.e. description) of the topic contracted when commanding help of Maple.

> ???plot[options];

d) expression and function

' expression' when used will execute the equation with no arrows being displayed.
> f:=x^2-5*x;

'function' when used will execute the equation with arrows being displayed. It is closer to mathematical representation.
> f:=x->x^2-5*x;

e) sum and add

'sum' is used to compute a formula for an indefinite or definite sum, whereby it is a "symbolic" summation.

> sum(x,x=5..n-1);

'add' is used to compute a formula when an explicit sum is needed.

> add(x,x=5..10);

4.	Use of Help facilities in Maple:  Use the help commands in Maple and explain the use of three Maple commands (not discussed in the class) with your own examples.

1. convert
> my_eq := (a*x^3+b)/(x*(-10*x^5/2-x+44));

> convert( my_expr, parfrac, x );

2. solving inequalities
> ineq := 3*x^2 + 5*x < 20;

> solve(ineq, {x} );

3. finite sum
> sum( (3+5*i)/(1/6+6*i^2.4), i=1..10 );

5. Use of units and Scientific Constants: Select any three problems involving different types of units from your textbooks and carry out the complete calculations including the units using Maple.  State the problem fully and comment whether Maple gives the correct final answer along with appropriate units.  (Refer the examples discussed in the class)

1. The red light associated with the aurora borealis is emitted by excited (high energy) oxygen atoms at 630.0 nm. What is the frequency,(lambda)  of the light? c=2.998 x 10^8 m/s.lambda must be in meters.

> restart;

> with(Units):
> f := GetSystems():
> lambda := 630.0*n*m * 1*m /(10^9*n*m);

> speed := (2.998*10^8)/(6.300*10^(-7))*Hz;

2. What is the energy of 4.57 electrons?
> restart;
> with (ScientificConstants):
> GetConstants():
> mass := 4.57 * 9.11e-31 * kg;

> c := 3.00e8 * m/s;

> E := mass * c^2 ;

3. How many molecules of acetone are in a 5 gram sample?
The chemical formula for acetone is C3H6O.

> number_of_moles_of_acetone := 3*Element(C, atomicweight) + 6*Element(H, atomicweight) + Element(O, atomicweight);

> evalf(number_of_moles_of_acetone);

> convert(%, units, kg, amu);

> 5./%;

> %*evalf( Constant( N['A'] ) );

6. Find out the dimensions of three different physical properties using Maple.

> restart;
> with(Units):
> GetDimensions():
> GetDimension(momentum);

> GetDimension(specific_heat_capacity);

> GetDimension(pressure);

7. Find out the values and units of any three different scientific constants using Maple?  Indicate the errors associated with these scientific constants.

> restart;
> with( Units):
> with( ScientificConstants):
> GetConstants();

> GetConstant(M[Earth]);

-The error with this scientific constant is that it has a big value of uncertainty.

> GetConstant(M[Sun]);

-The error with this scientific constant is that it has a big value of uncertainty.

> GetConstant(R[Earth]);

> 
-The error with this scientific constant is that it has an uncertainty.

8. Calculate the volume occupied by 1 kg of mercury at 25 C.

> restart;
> with(ScientificConstants):

> GetElement(Hg,density);

> density := 13.5336*(g/cm^3);

> mass := 1000*g;

> volume := mass/density;

9. Use Maple to calculate the number of water molecules in 1 g of liquid water.

> restart;
> with(ScientificConstants):
> number_of_mole_of_water := 2*Element(H, atomicweight)+Element (O, atomicweight);

> evalf (number_of_mole_of_water);

> convert(%, units, kg, amu);

> 1/%;

> number_of_molecules := % * evalf( Constant(N['A']));

10. What is computer algebra system and what are its advantages?

Computer Algebra System or Symbolic Computing is a type of computation with symbols representing mathematical objects including :
-integers (real & complex)
-polynomials
-derivatives & integrals
-systems of equations
-series expansions of functions
Its objective is to obtain closed form and exact solutions.
It contains thousands of built in functions and also many options for simplifying expressions.

Its advantages are:
-it contains thousands of built in functions with many options for simplifying expressions.
-it is free from representation or truncation and propagation error (errors accumulated arising from a sequence of calculations.
-it shows the exact results, rather than approximate values.

11. Determine whether (y + 8) / (x - 2) = x + 6   is linear or not? (Hint: solve for y, and comment)

> restart;
> solve ((y+8)/(x-2)=x+6,y);

-This is not a linear equation. It is a quadratic equation as the unknown x has a degree of 2.

12. Show that z=1, y=2, x=3 are the solutions for x+2y-3z = 4.  (Hint: use subs)

> restart;
> a := x+2*y-3*z=4;

> subs(z=1,y=2,x=3,a);

Since 4 = 4, thus z = 1, y = 2 and x = 3 are the solutions for x + 2y - 3z = 4.

13. a) Solve the equations:  x - y = -3 and x + 2y = 3.  Plot these equations in the same graph.  Do these graphs cross each other?  What is the meaning of the point of intersection? 

> restart;
> b := x - y = -3;

> c := x + 2*y = 3;

> solve({b}, {y});

> solve({c}, {y});

> plot({x + 3,(-x/2) + (3/2)}, x = -5..5);

The graphs do cross each other. The point of intersection represents the solution for both of the equations. 

b) Plot the equations: y = -x - 3 and y = -x + 2.  From the graphs what can you say about the existence of solutions for this set of equation?

> restart;
> eq1 := y = -x - 3;

> eq2 := y = -x + 2;

> solve ({eq1} , {y});

> solve ({eq2} , {y});

> plot ({-x - 3 , -x + 2} , x = -3..3);

There is no point of intersection. Thus, there is no solution for these equations.

c) Plot the equations: x + y = 1 and 2x + 2y = 2.  From the graphs what can you say about the existence of solutions for this set of equation?

> restart;
> alpha := x + y = 1;

> beta := 2*x + 2*y = 2;

> solve ({alpha} , {y});

> solve ({beta} , {y});

> plot ({-x + 1 , -x + 1} , x = -4..4);

The two lines overlap with each other. Thus, all the points on the line(s) are the solutions for both the equations.

14. Solve the equations: 2x + y - 2z = 8, 3x + 2y - 4z = 15 and 5x + 4y - z = 1.

> eqA := 2*x + y - 2*z = 8;

> eqB := 3*x + 2*y - 4*z = 15;

> eqC := 5*x + 4*y - z = 1;

> solve ({eqA , eqB , eqC} , {x,y,z});

15. Write any equation of your own and expand it using Maple.  Factorize and simplify the result and show that it gives back the starting equation.

> restart;
> myEq := (x + y) * (3*x^2 - 5*y);

> expand (%);

> factor (%);

> simplify (%);

16. Write any equation with three variables (x, y and z) and solve it using Maple.

> myEq1 := 3*x + 5*y^3 + 10*z = 10;

> myEq2 := 2*x^1/3 + y - 3/5*z = 4;

> myEq3 := 9*x - 15*y^1/2 - 4/3*z = 6/7;

> solve ({myEq1 , myEq2 , myEq3} , {x,y,z});