Q3 (a) State Rolle's Theorem. i) Verify Rolle's theorem for the function f(x) = (x – a)2020 (x – b)2021, where x € [a, b]. ii) Show that f(x) = x³ – 7x² + 25x + 8 has exactly one real root. %3D (b) Using L’Hospital's rule evaluate the following limits. lim (1+ sin 2 x)cotx x→0+ i) 4x ii) lim x-x 2x + sin x (c) x² x3 Using the differentiation of functions, prove that x – 2 < In(1+x) < x - 3(x+1) 1 +, where x > -1. Hence, prove that lim (In(1+ x) – x) = -. x→0 x2
Q3 (a) State Rolle's Theorem. i) Verify Rolle's theorem for the function f(x) = (x – a)2020 (x – b)2021, where x € [a, b]. ii) Show that f(x) = x³ – 7x² + 25x + 8 has exactly one real root. %3D (b) Using L’Hospital's rule evaluate the following limits. lim (1+ sin 2 x)cotx x→0+ i) 4x ii) lim x-x 2x + sin x (c) x² x3 Using the differentiation of functions, prove that x – 2 < In(1+x) < x - 3(x+1) 1 +, where x > -1. Hence, prove that lim (In(1+ x) – x) = -. x→0 x2
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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![Q3 (a)
State Rolle's Theorem.
i) Verify Rolle's theorem for the function f(x) = (x a)2020 (x – b)2021, where x E
[a, b].
ii) Show that f (x) = x³ – 7x2 + 25x + 8 has exactly one real root.
Using L'Hospital's rule evaluate the following limits.
(b)
i)
lim
x-x 2x + sin x
4x
ii)
lim (1+ sin 2 x)cot x
x-0+
(c)
x2
Using the differentiation of functions, prove that x -
x3
+
2
< In(1+x) < x –
3(x+1)
x2
x3
-, where x > -1. Hence, prove that lim (In(1+ x) – x) = -;.
2
3
x→0 x2](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F70d86970-59e6-4750-a0d7-1a72be8e9bdd%2F7d6bf4cf-6244-49e1-bc48-c5f467e35d19%2Fbyo38p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Q3 (a)
State Rolle's Theorem.
i) Verify Rolle's theorem for the function f(x) = (x a)2020 (x – b)2021, where x E
[a, b].
ii) Show that f (x) = x³ – 7x2 + 25x + 8 has exactly one real root.
Using L'Hospital's rule evaluate the following limits.
(b)
i)
lim
x-x 2x + sin x
4x
ii)
lim (1+ sin 2 x)cot x
x-0+
(c)
x2
Using the differentiation of functions, prove that x -
x3
+
2
< In(1+x) < x –
3(x+1)
x2
x3
-, where x > -1. Hence, prove that lim (In(1+ x) – x) = -;.
2
3
x→0 x2
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