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Key points of this past exam are: Binomial Coefficient, Fifth Order, Maclaurin Polynomial, Function, Rational Number, Approximates, Expansion
Typology: Exercises
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Math 106-C (Salomone) March 6, 2009 Show all your work!
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Score (25 points possible):
Problem 1. (15 points) Determine the fifth-order Maclaurin polynomial for the function
F(x) =
1 + x
and use it to find a rational number which approximates
Note: The coefficient of xk^ in this expansion is called the ”binomial coefficient” (^1 / k^2 ).
Problem 2. (8 points) Given that the Nth-order Maclaurin polynomial for f (x) = ex^ is
e x^ ≈ 1 + x +
x 2 +
x 3 + · · · +
x N^ ,
find the following. (Each answer will depend on the previous answer.)
(a) The third-order Maclaurin polynomial for e x
(b) The sixth-order Maclaurin polynomial for e−x 2
(c) The seventh-order Maclaurin polynomial for the ”mysterious” antiderivative
e −x 2 dx
Problem 3. (2 points) The function f (x) =
e −^1 /x 2 x! 0 0 x = 0
is called ”C∞^ -flat” at the origin because its derivatives satisfy
f (n)^ (0) = 0 for all n ≥ 0.
Find the Maclaurin polynomial of order 2,009 for this function, and de- scribe ”what this means for Taylor polynomials everywhere.”
