(b) Using the Taylor expansion of arctan x, 1
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Do the Part B in 10 minutes and Get the Thumbs up please show neat and clean work
![Question 4
(a) Calculate the following:
i. the area of the region in the first quadrant bounded by the curves of
f(x)=√x and g(x)=x-2, and the x-axis. Show your working.
ii. the area of the region bounded by the curves of f(x) = sin and
g(x) = cosa in the interval [4, 5]. Show your workin
(b) Using the Taylor expansion of arctan x,
1<x< 1, calculate the sum
1
Σ(-1)", 3n (2n +1)
n=0
I
(c) Let f: [0, +∞o) → R be continuously differentiable. Show that there exist
continuous, positive and increasing functions g, h: [0, +∞) → R such
that f=g-h.
Hint: You may use the fact that for continuous functions g, h: [0, +∞) →
R, it holds that max{g, h}
g+h+g-hand min{g, h}
g+h-lg-hare
also continuous.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fa2e672a1-54a8-4743-aad6-f3d1c4a0c4ab%2F0fafd1e3-38bc-4cac-8dad-5de7b23b559d%2F1edwvdi_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Question 4
(a) Calculate the following:
i. the area of the region in the first quadrant bounded by the curves of
f(x)=√x and g(x)=x-2, and the x-axis. Show your working.
ii. the area of the region bounded by the curves of f(x) = sin and
g(x) = cosa in the interval [4, 5]. Show your workin
(b) Using the Taylor expansion of arctan x,
1<x< 1, calculate the sum
1
Σ(-1)", 3n (2n +1)
n=0
I
(c) Let f: [0, +∞o) → R be continuously differentiable. Show that there exist
continuous, positive and increasing functions g, h: [0, +∞) → R such
that f=g-h.
Hint: You may use the fact that for continuous functions g, h: [0, +∞) →
R, it holds that max{g, h}
g+h+g-hand min{g, h}
g+h-lg-hare
also continuous.
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