Homework 5

• Finding a polynomial that has specified values
  • Find a quadratic polynomial f(x) such that f(0)=2, f(1)=4, and f(2)=12.
  • (for extra credit) Prove that for any distinct x₁, x₂, x₃, and any y₁, y₂, y₃, such that the points (x₁, y₁), (x₂, y₂), and (x₃, y₃) do not lie on one line, there exists a quadratic polynomial f(x) such that f(x_i)=y_i for i=1,2,3.
    
• Binomial coefficients and binomial theorem: see e.g. wikipedia
  • Find the second and third terms in the binomial expansion of (x-2)¹⁰.
  • Use the first few terms of the binomial expansion of (1+0.002)¹⁰⁰ to estimate the value of 1.002¹⁰⁰ correct to two decimal places.
    Hint: start with just one term and try using more and more terms. As soon as the answers for k terms and for k+1 terms agree to three decimal places, you have the desired approximation.