Math 19A Midterm 1 Spring 2006 Solutions

# Math 19A Midterm 1 Spring 2006 Solutions - MIDTERM 1 MATH...

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Unformatted text preview: MIDTERM 1 MATH 19 A 5/5/2006 Instructor: Frank Bauerle, Ph.D. Your Name: 30' Wng . ~4———-—.——————————._-——_—_—————————___._—_.—_ Your TA: —_—-__————.__—_.————__—_—..———-____-——.————_—— Max Your score Problem 1: 15 Problem 2: 10 Problem 3: 10 Problem 4: 20 Problem 5: 25 Problem 6: 10 Problem 7: 10 TOTAL: 100 Good Luck! 1. (15 points) (3.) Give an informal description of what it means that ”f(:c) is continuous at a: = hm Mo bunk) mt” UNI/{ad 0k§\/N‘{Jib)tﬁ at} \I ‘: Q . m Maﬁk a} ﬂy} Com L71: Wu jaw 03;, \qu V‘JWAOUJ’ {AK/bika \MLU prim ~. (b) Give a formal deﬁnition [using units) of ”f(:z:) is differentiable at x = a”. 5.4m Haw-Ml: {(0‘) Q/viS‘iS {1-) O . (Sc) 460*) i m .6 ’ PI {an X'ﬁ @‘gig (c) Use the limit deﬁnition of the derivative to compute the derivative of ﬁx) = 2:2 + l at; :c = 2. - ’1 p "L fiil) 7—: [MA )8.“ ﬂ (2“) :; “Vt/x bauq y»; 2 x-Z. we) Z ‘(L‘1 T. k ’ 1—91 QM) *— 4 ((1) Use the appropriate diﬁ'erentiation rules to compute the same derivative again to verify your answer in {Wklx {'(1) :‘i / 2. (10 points) Assume 5(1)) = t3 —— 1215 + 5 is the position of an object in a one—dimensional system. (a) What is the velocity of the object at time t = 0? I vﬂﬂt ’31” ~12 Vim-:42. (b) What is the speed of the object at time t = 0? Explain the difference to your answer in (a). ' Sferﬂ Ii +!l_ Ueiouig {MLQUJLM/h (c) At what time(s) is the object at rest? Sole we) : o 369—12.:0 3. (10 points) Use the given graph of f (3:) to carefully sketch the graph of f’(a:) in a new “d coordinate system. Hint Find/estimate f’(—1),f’(1),f'(0) and think about what it means for f(:c) when f’(\$) > 0 or when f'(I) < 0. 4. (20 points) Compute the following limits. Justify steps. You may use that lim 3m 3: = 3' ‘ z—HD a: . sing-“c .r_. I * SEMV ‘ " a.) 11m —— —~ [(Wx F‘ ' [VI-A §M ' 7‘ ”” 2x Z \L—MJ X r70 )1 [MU hwy} e t ' ‘ “H I O (QM/WW}TQ ) to ' b l+\$-\$3 __ i1 + J“ H V ()xlgnm H32 - [rm 7“ x m 00 \$990 1 ‘- A ier + I (LE-«m9 (om/s) tank (shuns, h .: 5'1 / '1 UV ‘EZQH‘K :0 :; QQC X/y:o :S‘eclo‘: :fﬁn/«W spam/r q Jgnm'ﬁ) "fan ‘ _ - ‘ . “A? Ti “ML” : w 3‘41”- mi :r-I—r—t (4‘70 [4 COYLl («~90 brief—th (d)t£n3+ﬂn_:l (I‘M gikt (QC (“ﬁg/d) 4:4)0'1‘ ‘(3 t —— W J? 1 {2w 0—“ {700* «(Tsark ‘L’ c, O I 3/0 5. (25 points) Compute the requested derivatives of the functions: {NO NEED TO SIM- PLIFY) (a)§:—fory=ﬁ Gig ‘ I 0H» 2,4): (b) y’ for y = \$2 +cosx +secw —-3 If: 274 ~9mz + Secxtqw (d) g]; for ﬁns): 23” tans: :3: :: 2Q7ltam£ + 26% 96’qu (e) —fory=(35—m4+x—2)(5\$9——13x2—\$3+:c—-12) -— z ( whim/iii) c sfr-szw-IZ) [Jr are ﬂab,” Q 45y3~2e7< =—3 Xlﬂ) el‘ 6. (10 points) Compute an equation of the tangent line to the curve 3; = at the point 3+1 ,5. ,l “f >L 13”?) 1C (wk—2M 6”". “e ;: w< 67‘ (NHL (1+1)?- 7. (10 points) Find all real number values for a so that the function f = as? - agar + 1 has a horizontal tangent line at a: = 1. “tic-mszmwl ’ PM: m—qz 2a —*0\ 3:0 Ctél‘CLBr—O 0\=-—Oj Ck:2 ...
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Math 19A Midterm 1 Spring 2006 Solutions - MIDTERM 1 MATH...

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