Consider the function f x = x 2 - x - ln x Find a the...

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Tutorial 9

1 .   Consider the function f ( x ) = x 2 - x - ln x . Find ( a ) the intervals on which f is increasing or decreasing . ( b ) the local maximum and minimum values of f . ( c ) the intervals of concavity and the inflection points . 2 . Prove that ( a ) sin x < x , for 0 < x < 2 π . ( b ) tan x > x , for 0 < x < π 2 . ( c ) e x > 1 + x > ln ( 1 + x ) , for x > 0 . 3 . The function f ( x ) = x 3 + ax 2 + bx has the local minimum value f ( 1 √ 2 √ 3 9 . What are the values of a and b ? 3 ) = - 4 . Evaluate the limits . Use L’Hospital ’s Rul where applicable . If L’Hospital ’s Rule does not apply , explain why . x 7 - 1
( a ) lim x → 1 x 4 - 1 . e x - 1 - x sin x 2 . ( b ) lim x → 0 1 1 ( c ) lim x → 0 + x - e x - 1 . 3 e - x 2 . ( d ) lim x → ∞ x √ x ( e ) lim x → 0 + x . ( f ) lim x → 1 + x 1 1 - x . 1 ( g ) lim x x . → ∞ x
MATH 137 Tutorial 9 Instructor: Liu, J. March 14, 2017 Sections 4.3–4.5: I/D, Derivative, and Concavity Tests; Indeterminate Forms; Curve Sketching 1. Consider the function f ( x ) = x 2 - x - ln x . Find (a) the intervals on which f is increasing or decreasing. (b) the local maximum and minimum values of f . (c) the intervals of concavity and the inflection points. 2. Prove that (a) sin x < x , for 0 < x < 2 π . (b) tan x > x , for 0 < x < π 2 . (c) e x > 1 + x > ln ( 1 + x ) , for x > 0. 3. The function f ( x ) = x 3 + ax 2 + bx has the local minimum value

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