BC 2-3
Polynomial Approximations I
Name: ____________________
Interpolation
One way to approximate a function with a polynomial is to choose a polynomial that actually
equals the function at a few points, in the hopes that it will be a good approximation at other,
nearby points.
Consider the function f(x) = cos(x). Let’s find a polynomial that actually equals this function at a
few points.
1) Find a quadratic polynomial, q(x) = ax2 + bx + c, which has the same values as f(x) = cos(x) at


x = 0, x  , and x =  .
2
2
2) Sketch graphs of both f(x) and q(x) on both sets of axes below. Try to draw accurately,
labeling which graph is which!
2
-2
-2
IMSA
5
2
-5
-5
Poly Approx 1 p.1
5
Fall 12
3) Why is the second graph so much worse-looking an approximation than the first?
Interpolation is the term given to choosing a polynomial that exactly matches the function we are
interested in at a few selected points, hoping that it will make a good approximation at other,
nearby points. It does a pretty good job in general.
4) If we interpolate at x = -, x = 0, and x =  instead of -/2, 0, and /2, would our results get
better or worse? In what way?
5) If we wish to interpolate at n points, explain why we can always use a polynomial whose
degree is n – 1 (or less).
It can be quite a lot of work to find all the coefficients of the interpolating polynomial! We have
to plug in the points, get a bunch of equations, and solve for all the coefficients. Try to
interpolate cos(x) with a few more interpolation points (official term: “knots”):
6) Find the interpolating polynomial that matches cos(x) at x   , 


, and  . (Hint:
2
2
since it’s a fourth degree polynomial (n = 5 in question 5), we can sort of “factor” the polynomial


  



as a0  a1  x     a2  x     x    a3  x     x   x  a4  x     x   x  x   and
2
2
2 
2



then plug in various values of x and solve for the a’s. Then multiply it all out! You may use the
space on the back page of this packet for scratch work.)
IMSA
Poly Approx 1 p.2
, 0,
Fall 12
7) Plot cos(x) and your fourth-degree interpolating polynomial together. Choose a couple of
different ranges to get an idea of where the polynomial is a good approximation and where it
isn’t. Label your graphs!
8) Could the interpolating polynomial be used in calculus applications? Evaluate its derivative at
x = 1, and its integral from 0 to 1. Compare to the exact values you would obtain for cos(x).
IMSA
Poly Approx 1 p.3
Fall 12