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CALCULUS 1501 WINTER 2010
HOMEWORK ASSIGNMENT 10.
Due April 5.
10.1. Find the Maclaurin series for
f
(
x
). Find the radius of convergence of the series, and show,
using Lagrange’s remainder theorem that the series converges to
f
(
x
).
(i)
f
(
x
) =
xe
2
x
.
(ii)
f
(
x
) = sin
x
2
.
10.2. Find the ﬁrst 3 terms of the Taylor series of
f
(
x
) centred at
a
:
(i)
f
(
x
) = sec
x
at
a
=
π
4
.
(ii)
f
(
x
) = tan

1
x
at
a
= 1.
10.3. Find the Maclaurin series for
f
(
x
) =
x
2
√
1 +
x
.
10.4. Approximate the function
f
(
x
) =
x
x
at
a
= 1 by a Taylor polynomial of degree 2. How
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Unformatted text preview: good is your approximation at x = 1 . 1? 10.5. Find the Cartesian equation of the curve which is given by a parametric equation x = 2 cos θ, y = 3 sin θ, θ ∈ (π,π ) . Sketch the curve indicating with an arrow the direction in which the curve is traced as the parameter increases. 10.6. Evaluate the integral Z 1 ln(1x ) x dx. Hint: Use Taylor series expansion and the identity ∑ ∞ n =1 1 n 2 = π 2 6 . 1...
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This note was uploaded on 07/15/2010 for the course MATH 1501 taught by Professor Shafikov during the Winter '10 term at UWO.
 Winter '10
 Shafikov
 Remainder Theorem, Remainder, Maclaurin Series

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