Below is a list of Mathematica functions we have used. You should be familiar with each of these functions and know how to use them.
Abs Cos D Do Exp FindRoot If Interval Log Module N Plot Sin Solve Sqrt Sum Table Tan While
1. Use the Mathematica Sum function to define a function that computes the polynomial
Solution p[x_] := Sum[x^n/n,{n,1,21,2}]
2. A right triangle has a hypotenuse of length 1 and an angle of 
between the hypotenuse and one of its sides. Determine how the area of the triangle depends on 
and make a plot of the triangle's area versus 
for 
in the range from 0 to 
.
Solution: area[t_] := Cos[t] Sin[t]/2
Plot[area[t],{t,0,Pi/2}]
3. Construct a function that correctly computes the triangle area for 
in the range from 
to 
. The solution you developed for problem 2 above may produce a negative area for 
outside the range from 0 to 
, so you will have to use an If[] construct to ensure that function produces only positive areas throughout the entire range.
Solution: area[t_] := If[0<=t<=Pi/2,Cos[t] Sin[t]/2,-Cos[t] Sin[t]/2]
4. The following function can be used to compute simple interest on a principle of p with an annual rate of r and term of n years.
simple[p_,r_,n_] := p*(1+r)^n
You purchase a government bond with a face value of $1000 for $800. In five years you can redeem the bond for $1000. What rate of simple interest would a bank account have to earn to give you the same return on your $800 over five years?
Solution: Solve[simple[800., r, 5] == 1000., r]
where 
is the length of the segment from P to A and 
is the length of the segment from P to B.
Solution First we construct a function to compute the quantity to be minimized.
f[x_] := (x+1)^2+(x^2-1)^2 + (x-1)^2+(x^2-1)^2
We have to minimize this function, so we solve for critical points.
deriv = D[f[x],x]
Solve[deriv == 0.,x]
This produces three critical points. Substituting each of those points back into f[x] will show which of the critical points is a minimum.
6. You are standing an unknown distance away from the base of a tall building. From where you stand the angle of elevation to the top of the building is 
. You move forward x meters and discover that the angle of elevation has changed to 
. Construct a mathematical model for this situation and write down the sequence of Mathematica evaluations you would use to solve for the height y of the building.
Solution Let the distance from the second location to the base of the building be z. Using simple trigonometry we can write two equations for the two triangles involved.
This is two equations in two unknowns (y and z). We can reduce it to one equation in one unknown by using the second equation to solve for z.
From here we can let Mathematica take over:
z = y/Tan[phi] eqn = Tan[theta] == y/(x + z) Solve[eqn,y]
7. Develop a mathematical model and then use Mathematica to solve the following problem. Find the angle 
that makes the shaded region take up exactly 1/3 of the area of the circle.
Solution: The area of a sector with radius R and central angle 
is 
. The area of the isoceles triangle below the shaded region is 
. We want to know what value of 
causes the difference of these two areas to be 
. Since the solution is independent of the value of R, we can solve this problem with the following commands:
area[t_] := R^2 t/2 - R^2 Cos[Pi/2-t/2] Sin[Pi/2-t/2]
R = 1
FindRoot[area[t] == Pi R^2/3, {t, Pi/2}]