assignment0

# Assignment0 - f x =    x sin 1 x if x 6 = 0 if x = 0 5 Find the following limits(a lim x → ∞ ln(1 x √ x(b lim x → ± 1 √ x-1 sin

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MATH1211/10-11(2))/A0/NKT THE UNIVERSITY OF HONG KONG DEPARTMENT OF MATHEMATICS MATH1211 Multivariable Calculus 2010-11 2nd Semester: Assignment 0 This assignment is for your revision of single-variable calculus. No need to hand in solution. 1. Find the derivative of the following functions. (a) f ( x ) = 2 x 3 - 3 x + x - 4 - 12 (b) g ( u ) = u sin u 5 + cos u (c) h ( t ) = e 2 t - ln( t 2 + 1) (d) k ( s ) = tan - 1 s 2 2. Let f be a function deﬁned on the closed interval [ - 3 , 3] by f ( x ) = 2 x 3 + 3 x 2 - 12 x + 1. (a) Find all relative and absolute extremum values attained by f on [ - 3 , 3]. (b) Find all inﬂection point(s) of f , and the sub-intervals on which f is concave up or concave down within the interval [ - 3 , 3]. (c) Sketch the graph of y = f ( x ) where x is within [ - 3 , 3]. 3. Find all the points on the circle x 2 + y 2 = 1 at which the slope of the tangent line to the circle is 3. 4. Determine the continuity and diﬀerentiability of the function f ( x ) at x = 0, where
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Unformatted text preview: f ( x ) =    x sin 1 x if x 6 = 0 , if x = 0. 5. Find the following limits. (a) lim x → + ∞ ln(1 + x ) √ x (b) lim x → + ± 1 √ x-1 sin x ² 6. Evaluate the following deﬁnite or indeﬁnite integrals. (a) Z 2 x 4-3 x 3 + 4 x-5 6 x 2 dx (b) Z 4 (1 + √ x ) 4 √ x dx (c) Z 2-1 | t | dt 7. Find the volume of the solid that remains after you bore a circular hole of radius a through the center of a solid sphere of radius b > a . 8. Show that Z 5-5 ³ 3 x 2-x 10 sin x + x 5 √ 1 + x 4 ´ dx = 2 Z 5 3 x 2 dx. Hint: some functions within the integral sign are odd functions while some are even . 9. Find h ( x ), given that h ( x ) = Z x 2-1 t 3 sin tdt. Hint: Let y = h ( x ) and u = x 2 , and apply the Fundamental Theorem of Calculus. 1...
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## This note was uploaded on 05/04/2011 for the course MATH 1211 taught by Professor Wang during the Spring '11 term at HKU.

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