- Look at p. 578 #10, but only find the degree 2 polynomial using the
book's hint by
a) write the degree 2 polynomial of cos(x) since it is a known series
b) sub in 2x for x in the known Taylor series for cosine
c) using the trig identity and sub in your response to b) in it.
So you will find the
degree 2 polynomial rather than the first four nonzero terms that the book asks for. Show work.
- Show computation work for a direct computation of
the degree 2 Taylor polynomial of the original function cos2(x) about 0 (i.e. NOT using the trig identity---and directly by the table for each of the three
rows or similar),
using the second last bullet point on the series theorems sheet.
Show work like the power rule
and chain rule to compute the first derivative of cos2(x).
- Show that these methods give the same degree 2 polynomial by reducing algebraically.
- What does the sign of each term in the degree 2 polynomial tell you about the geometry/graph of the original
- If we wanted to find the radius of convergence of the Taylor series would we use geometric or ratio
test? (don't apply the test but do specify which would naturally apply)
- Give the Taylor/Lagrange error bound in approximating the function cos2(x)
with the degree 2 polynomial from the center 0 to 0.5.
Fill in the error formula on the series theorems sheet
in with numbers and show your reasoning for M, but do not simplify.