MATH 106B,C - CALCULUS II FALL 2011
QUIZ 6
NAME:
Show ALL your work CAREFULLY.
(a) Find the third-order Taylor polynomial P3(x) for the function g(x) = sin(2x) based at x0=π
2.
Since g(x) = sin(2x), we have g0(x) = 2cos(2x), g00(x) = −4 sin(2x)and g000 (x)−8 cos(2x).
It follows that g(π
2) = sin π= 0,g0(π
2) = 2 cos π=−2,g00(π
2) = −4 sin π= 0, and g000(π
2) =
−8 cos π= 8. Thus, the third-order Taylor polynomial for g(x)based at x0is given by
P3(x) = 0 + (−2) x−π
2+0
2! x−π
22
+8
3! x−π
23
= (−2) x−π
2+8
3! x−π
23
.
(b) Let f(x) = xex. Using Taylor’s theorem, what is the error committed by M2, the Maclaurin
polynomial of degree 2 for f(x) over the interval [−1,1]?
To estimate the error using M2, we need to first find the third derivative of f(x). Since
f(x) = xex, we have f0(x) = ex+xex=ex(1 + x),f00(x) = ex(1 + x) + ex(1) = ex(2 + x), and
f000(x) = ex(2 + x) + ex(1) = ex(3 + x). Over the interval [−1,1],|f000(x)|=ex|3 + x| ≤ 4e. We
now let K3= 4e. Then Taylor’s theorem asserts that error committed by M2over the
interval [−1,1] is less than or equal to
K3
3! · |x−0|3=4e
3! |x|3≤4e
3! =2e
3.
Date: October 28, 2011.
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