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THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS AND STATISTICS Term 2 2019 MATH1131 MATHEMATICS 1A *. TIME ALLOWED – 2 hours 9. TOTAL NUMBER OF QUESTIONS – 3 B. ANSWER ALL QUESTIONS C. THE QUESTIONS ARE OF EQUAL VALUE F. ANSWER EACH QUESTION IN A SEPARATE BOOK J. THIS PAPER MAY BE RETAINED BY THE CANDIDATE L. ONLY CALCULATORS WITH AN AFFIXED ʼnUNSW APPROVEDŊ STICKER MAY BE USED N. A SHORT TABLE OF INTEGRALS IS APPENDED TO THE PAPER P. TO OBTAIN FULL MARKS YOUR ANSWERS MUST NOT ONLY BE CORRECT BUT ALSO ADEQUATELY EXPLAINED CLEARLY WRIT- TEN AND LOGICALLY SET OUT. All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing sketching or graphical work. *. i Use integration by parts to evaluate ∫ 0 √3 tan−1x dx. ii. a Write down the definitions of sinhx and coshx in terms of the exponential function. b Evaluate ∫ 0 ln3 coshx + 1 sinhx dx. iii. Let u = ⎜⎜⎜⎜⎝ ⎛ −1 −2 3 5 1 ⎟⎟⎟⎟⎠ ⎞ and v = ⎜⎜⎜⎜⎝ ⎛ −3 −6 β 9 3 ⎟⎟⎟⎟⎠ ⎞ be two vectors in R5. a. Find β if u and v are parallel. b. Find β if u and v are perpendicular. iv. Let a = ⎛⎝ 41 6 ⎞⎠ and b = ⎛⎝ −20 10 ⎞⎠ be two vectors in R3. a. Calculate Projba the projection of a onto b. b. Hence find a vector c not equal to a in the plane spanned by a and b such that |a| = |c| and Projba = Projbc. v. For the following system of equations use Gaussian elimination to deter- mine which values of λ if any will yield: a. no solution b. infinite solutions c. a unique solution. x. y + z = 4 x. λy + 2z = 5 2x+ λ+ 1y + λ2 − 1z = λ+ 7. vi. Let fx = 2 + 1 x and consider the uniform partition Pn of the interval r. [email protected] given by Pn = { 0 n 1 n 2 . . . n− n 1 n n } . a. Sketch a graph of f for 0 ≤ x ≤ 1 and indicate the rectangles defining the lower Riemann sum for f on P4. b. Write down an expression for the lower Riemann sum for f on Pn. c. Hence or otherwise show that n→∞ lim 2n+ 1 1 + 2n+ 1 2 + z z z+ 3n 1 = ln 3− ln 2. 2. i Let the region S be defined as S = {z Ʉ C : |z − 2− 2i| ≤ 2 and Rez > 1}. a. Sketch the region S on a carefully labelled Argand diagram. b. State in a+ ib form the complex number in S of maximum modulus. ii. A table sits on horizontal ground. In Figure 1 a bird stands on the table directly above a cat sitting on the ground and the distance between the tops of their heads is 105 cm as shown. In Figure 2 the cat and the bird have swapped places and the distance between the tops of their heads is now 145 cm. x Figure 1 105 cm y 145 cm z Figure 2 Let x be the height of the bird y be the height of the cat and z be the height of the table with all measurements made in cm. a. It follows from Figure 1 that x− y+ z = 105. By considering Figure 9. write down a second linear equation in x y and z. b. By reducing an appropriate augmented matrix to echelon form and back- substituting find the height z of the table and express the heights of the bird and cat in terms of a parameter. c. Given that the sum of the heights of the bird and cat is 112 cm find the height of the bird. iii Consider the following pentagon ABCDE with five equal-length sides. C B u A E Let u = −→AB and v = −−→ED. Observe that − −→BD is parallel to −→AE and hence D v BD −−→ = λ −→AE for some λ > 1. Similarly −→AC = λ ED −−→ = λv and −−→EC = λ −→AB = λu. a. By considering the triangle ACE prove that −→AE = λ v − u b. Find another expression for −→AE in terms of λ v and u and hence find the exact value of λ expressing your answer in surd form. iv Use the following Maple output to explain why A is invertible and to calculate A−1 for the given matrix A. Give reasons for your answer. > withLinearAlgebra: > A := <<2/3-1/3-2/3>| |> > DeterminantA > A^3-A A := ⎢⎢⎢⎢⎢⎢⎣ ⎡ − − 3 2 3 3 2 1 − − 3 2 3 3 2 1 − − − 3 3 3 2 2 1 ⎥⎥⎥⎥⎥⎥⎦ ⎤ −1 ⎡ ⎣0 0 00 0 0 0 0 0 ⎦ ⎤ v a By considering seventh roots of unity find a non-real solution to the equation 1− 1 + w w 7 = 1. b Show that your solution in part a is purely imaginary. USE A SEPARATE BOOK CLEARLY MARKED QUESTION 3 3. i Suppose that y = hx is a continuous function over the real line with the property that lim hx = 2. x→∞ a. Draw a possible sketch of the graph of h. b. Using the formal definition of the limit explain why there exists a real number M with the property that x > M ⇒ hx > 1. c. Hence prove that ∫ 0 ∞ hx dx is a divergent improper integral. ii. The function f : R→ R is defined by fx = ex x . a. Find and classify the stationary point of f . b. Using LŇHopitalŇs rule evaluate lim fx. x→∞ c. What is the range of f? d. Sketch the graph of f . e. Define a function g : >1∞→ 0 e−[email protected] by the rule gx = fx Explain why the function g is invertible while f is not. f Let g−1 be the inverse function of g. Show that g−1x = xeg−1x g. On what interval is g−1 differentiable? h. Show that the derivative of the inverse of g satisfies >g−[email protected]′ = x1− g −1x g−1x . iii. Suppose that both the functions f : R→ R and g : R→ R are continuous n the closed interval >a [email protected] and differentiable on the open interval a b with ga 6= gb and g′c 6= 0. Let hx = fb− fagx− gb− gafx. By applying the Mean Value theorem to h prove that there exists c Ʉ a b such that f ′c fb− fa = g′c gb− ga . C. INTEGRALS∫ |k|∫ eax x 1 dx dx = = ln a 1 |x|+ eax + C BASIC = ln |kx| C = ln D. C∫ C∫ 1∫ sec2 sin cos ax dx ax ax ax = dx dx dx 1ln = = = aa −1 a 1 a 1 x a sin tan + cos ax+ C ax+ ax+ a 6= E. C∫ C∫ C∫ C∫ tan sec sinh cot cosh cosec2ax ax ax ax ax ax dx dx dx dx dx = = dx = = = a a 1 1 a 1 = a 1 a 1 ln ln ln cosh sinh −1 | | | sec sin a sec cot ax+ ax+ ax|+ ax+ ax|+ ax+ tan ax|+ F. C∫ C∫ a2 sech2ax cosech2ax dx + x2 = dx dx a 1 = tan−1 = a 1 tanh −1 a x a coth + ax+ ax+ G. a2 dx − x2 = a 1 tanh−1 x a + C x. < a = 1 coth−1 x + C a a |x| > a > 0 a2∫ C∫ C∫ dx√ dx√ dx√ x2 x2 a2 − + − x2 a2 a2 = = = = 2a 1 sinh−1 sin−1 cosh−1 ln ɈɈɈɈ a− a+ x a x a x a + x x + + ɈɈɈɈ+ C C x x > 2 a 6= > 0 END OF EXAMINATION 2019/10/14 15)14 第 1 ⻚页(共 1 ⻚页)


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