1. (Euler's Formula: Auxiliary Limits). Use suitable Rules of de L'Hôpital to evaluate the limits: (0) lim limtaretan ( Please give a complete solution to the problem.
1. (Euler's Formula: Auxiliary Limits). Use suitable Rules of de L'Hôpital to evaluate the limits: (0) lim limtaretan ( Please give a complete solution to the problem.
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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Question
![1. (Euler's Formula: Auziliary Limits). Use suitable Rules of de L'Hôpital to evaluate the limits:
().
(i)
lim
(ii)
lim tarctan
Please give a complete solution to the problem.
2. (Compiez Numbers: Eatraction of Cubic Roots; Scientific Caleulators). Consider the complex number
w= -117 -44i.
(i) Write a program for a scientific calculator to determine the polar form r(cos0 +i sin 0) of w and the 'first' root
of degree three of w. Execute the program and fix the results; along with the exact values, please provide also decimal
approximations of the module r, the argument 0, and the polar form of w to five decimal places.
(ii) Find then two other roots of degree three of w by evaluating the products
21 = Row and a = w",
where
is a primitive root of unity of degree three; as before, please also round the exact values of zi and 23 to five decimal places.
(i) Show the points 20, 21, 22 in the complex plane.
Present your answers to the problem in a table similar to the following table:
Subproblem Answers
w = 2+1li, roots of degree 3 of w = 20. z1, ?
(i)
r = w| = v2 +1l = 5/5 11.18034;
()-
e 1.39094 (since the number is in the first quadrant,
0 = arg(w) = arctan
the corresponding correction is equal to 0);
w s 11.18034 - (cos(1.39094) +i sin(1.39094)]
20=2+i
(ii)
21- i -1-
+
R -1.86603+1.23206;
29 = oi= -1+
+
2
V3-
-0.13397 - 2.23206
te
(ii)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Faaa34c5d-c989-40c5-b697-ae4d9381e0d4%2F2efcc8f6-c92d-4198-8499-892588070208%2Fko3cm5u_processed.jpeg&w=3840&q=75)
Transcribed Image Text:1. (Euler's Formula: Auziliary Limits). Use suitable Rules of de L'Hôpital to evaluate the limits:
().
(i)
lim
(ii)
lim tarctan
Please give a complete solution to the problem.
2. (Compiez Numbers: Eatraction of Cubic Roots; Scientific Caleulators). Consider the complex number
w= -117 -44i.
(i) Write a program for a scientific calculator to determine the polar form r(cos0 +i sin 0) of w and the 'first' root
of degree three of w. Execute the program and fix the results; along with the exact values, please provide also decimal
approximations of the module r, the argument 0, and the polar form of w to five decimal places.
(ii) Find then two other roots of degree three of w by evaluating the products
21 = Row and a = w",
where
is a primitive root of unity of degree three; as before, please also round the exact values of zi and 23 to five decimal places.
(i) Show the points 20, 21, 22 in the complex plane.
Present your answers to the problem in a table similar to the following table:
Subproblem Answers
w = 2+1li, roots of degree 3 of w = 20. z1, ?
(i)
r = w| = v2 +1l = 5/5 11.18034;
()-
e 1.39094 (since the number is in the first quadrant,
0 = arg(w) = arctan
the corresponding correction is equal to 0);
w s 11.18034 - (cos(1.39094) +i sin(1.39094)]
20=2+i
(ii)
21- i -1-
+
R -1.86603+1.23206;
29 = oi= -1+
+
2
V3-
-0.13397 - 2.23206
te
(ii)
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