Challenge Problems


Trigonometry Problems

  1. ** Use trig identities to calculate cos(2pi/5) explicitly.
  2. * Use trig identities to calculate T3(x), T4(x), T5(x) and T6(x), where Tn(x) is the n'th Tchebycheff polynomial as introduced in class. [Reminder: Tn(x) is the unique polynomial such that Tn(cos x)=cos(nx).]
  3. * Do you see how the Tn(x) behave like prime numbers?


Derivatives of Polynomials

  1. * Consider the two parabolas y=2x2-1 and y=x2-4x+7. Find the common tangent lines to these two parabolas. That is, find the line(s) that are tangent to the two parabolas at the same time.
  2. * Consider the cubic y=x3-4x. Find the tangent line to y=f(x) at x=1. Find where this tangent line intersects the cubic (besides the trivial point x=1.)
  3. ** Repeat the previous problem but obtain general formulas using $y=ax3+bx2+cx+d and with x=x0.