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Math32B_midterm1_solutions

# Math32B_midterm1_solutions - Friday 2/1 MATH 318 MIDTERM I...

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Unformatted text preview: Friday 2/1 MATH 318 MIDTERM I Last name : First name : Section : UCLA ED number : Signaﬁure : Instructions : fuii credit. wiii be given alliy if your solution is given with all the details. Notes and calculators are not ailowed. CREDIT : SCGRE : Probiem 1 :6 paints Probiem 2 :4 points Probiem 3 :10 points Probiem Li :9 points Probiew 5 :fi points E’mbiem 6 :7 points _, ‘5 I; 3 f. 5 1 u . :1? 9} TOTAL : 46 points L f j! 5 Problem 1. Consider the function ﬂz) m eﬁ. m a) Make precise the domain 13 and the wage R. 5%,}: b) \Vhat is Prove that f is invertible. i c) Find the inverSe function. E’mbiem 2. ' F'd Id. Zia) m 2 b) Find f de. a“; gasws';agw a: a? 5’ g {3 {is}: m, N f f g m ' n r a ﬁg @556??? M Q 2: ages? 4C Problem 3. ~ Censider m tanct ﬂ a) E’mve that. f’{:):) 2 Z + mm? 3:. b} Prove that f is invertibie. Let g m f”1 be the inverse function. (1) Make precise the domain and the range 0f g. 1 cl) Prove that g’(\$) 2 e) What is f; {Ll—Egan on the interval (—325, w w 2:“ N w f n x. , m N; Egg ’2 {a} ff‘ﬁf 3333: \f: 9??? g??? {My 5 W}: 1 \$372»; \$3332“ 2 yg‘x ,2 Ram " 52% ’35 55% Problem 4. Consider ftp) : 222111 \$. sgéi a} Fina? the limit 0f a: 0+ and +00. Make precise the Sign of ﬁx), depenéing on 9;. 4;? g b) On which interval f is increasing, decreasing ‘? {ﬁg (3) On which intervai f '13 concave up, concave down ? g (1) Graph f. E rig E5 w"; 943 M g, ; ﬂaws? E? if E: (“E 5:91 E if M :3 “if: [E g5 XV“ E m {3% mg @i aim W (ﬁgﬁf’m: fm {w \$5M 2m i my; WWW %/ ﬂag/22"»? if ? 2 {"6 {515713 {2655: 3 g {nu Kg}; 554?? . 3‘- Problem 5. Solve y’{9:) z y(x) w 1 for ail x, given y({)} 2 {l Problem 6. 21) Find I m ff imam. 1% b) We divide the intervai {438] into two intervals {4,6} semi {6,8} and apply Ehe midpoint ruEe, Say Ab, and the trapezoidal ruie, say T2. ‘What are the 2 estimates of I you get '2’ C) Prove that AIQSEnQSTg. 3 f f. yaw; 3 53% am ﬁérm: 5%} “LN: If; g If: {#32 a; fﬁw a} m a m2 wt m ’2 3’ :g 33‘ (7%” "£44: 4’4 E4? :ijiééix if? E,» g 3 ...
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