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Math 19A MIDTERM EXAM 1 Version 1

4/26/2009 , Dr. Frank Bauerle, UCSC

Note: Show your work. In other words, just writing the answer, even if correct, may not be sufficient for full credit. Scientific calcula­tors are allowed, but no programmable and/or graphing calculators.

Your Name: _

Your TA's Name: _

Problem 1: out of 25 3Pk;l \~j!)') Problem 2: out of 10

Problem 3: out of 10 } CQ c ~1 \- C\. Problem 4: out of 10

Problem 5: out of 10 ] ~~f-hUJProblem 6: out of 10

Problem 7: out of 10 } JeK\M.~ Problem 8: out of 15

Total: out of 100

Good luck and have a good weekend!

1

1. (25 points) Compute the requested derivatives of the functions:

(a) ddy

for y = 3x 5

x - ',

J l.) "'" It) '('-/ cA'4.

eX (b) y' for y = -.­

smx

dz (c) dy for z = cosy + secy

IV\J t

(d) df dx for f(x) = e2(2cos(5) + 1)

l · ~f- '=-,0c\1

(e) l'(w) for f (w) = w 3

- 2w + w ~ w

'" t , ». L.­_u.J

'/ L .1+W ~

2

2. (10 points) Show that the curve given by y = 3x6 - x2

- x has at least two distinct zeroes. Hint: Use the intermediate value theorem.

~ -=- y... ( 3 / - 'f. - \ ') := ) ~ =oC) (.I C\."(eA'O.

_ 3 '3\~ )(=01 ) ~~ 3 "Ci l- C±/ - i --- .- -- <-·0 ~ lf If

(f )y~ 3> -- l - { =I

"tll {t Y~ '>i -,/-,e If (oL\ ~ ,\w:lv f ifl't C"~ \\] It­~ {bI,J 1 lOy ~ T VT /Lc,J J ~ C\ r 1A. Lt , I{.u [t1C

~;:. h-~ /C-=-i CU\J '(.=" l " =-) ~ kC\ J ct+ UL Q5+ ~! ~~J ~

3. (10 points ) For the given graph of y = j (x ), sketch the graph of j' (x ).

) ­

3

4. (10 points) If a rock is thrown upward on the planet Mars with a velocity of 21 mis, its height (in meters) after t seconds is approximately given by H(t) = 2lt - 1.86t2

.

(a) Find the velocity of the ball at time t = 2 seconds.

H, 'Ct-) = 2 /- .72 t-

r{ 1 tl')= 1 J - 7.YIf == ' 3.s; (b) Find the acceleration of the ball at time t = 2 seconds.

HII Ct) - --- 3.72 ~-----------~M-

( H"li - ) .7 2

5. (10 points) Compute the equation of the tangent line to the curve

when x = O. Give your answer in t he form y = mx + b. <) . '

f 'f.. . ') L ) i ( '#', , 1' L-J t /L 'I- ) ~ ;: e (TOv{ 'f.. + c'f - \ +- e. ~ '-L I '0 ' r \ _ ( 0 +- 0 -I) -+ \ . L l -t~,

Y IF0 ) _ \ + 8Pt{ C :::: vY\

y I'(= G e0 CtCtLi 0 + '? 0 2

- () -=- 1. t:

( == b)

4

6, (10 points)

(a) Use the limit definition of the derivative to compute the derivative of f( x) = ft1 ' I . f (~ +L) - f C'!- J (' i j - )C}) ~ I ((/\;\ ' \, -=- 1iA\ -i ''lt k fi ~l .~'f-

~ -/ 0 ttL JA -7 0 h.. (~ 1:4 -~ )

Ii'(AA h~O

'tiltN\ IA..- ')0

l ~+ {"1 i ~ )

flY t -VI 'it~)

\

I~)'3 t - :;- v ­L-l' !

(b) Use the appropriat e differenti ation rul es to compute the same derivative again to verify your answer above.

\ ~-

2

5

7. (10 points) Find the equat ions of the tangent lines to the cur~~...x - 8 that are parallel x + l

to .T - 9y = 9. Give your answers in the form y = mx + b.

.- q.­

(y i \) '2..

tjl()UA. l Zk.O- St''''f L1~ n in CJ~-~1-g

.- \ ( \==- ) v\ z: -'- '( - I := )

J q 9

.-­q f ~ .--- 'L

-- 5t fDtue ~ ~e-~ (~t' 'I

1 '=:) (j-r l ) ~ 81

- -+ c .-=: ") ''{ -+ ( ­

<: { == q ex 1- + 1 = -- 9 ::=) -i. > - 10=) = g

. , - - { :{VJ l-~ )( :=. - 10) ~ - -C\::

V'b -c > ~ ) y=­c,- v 1M "'~ ) rv,,,h H-I()\ 2 )

So s Iv(e VV\ zz- fJ ct\.

~ L1 _ L ·· C~ . O ) \ ~) ~ - '2-.:: ~ (-I +( G )(,UAC~ ,~ r,,,, i .. ) ) I

~ ) y- 'I, .z: it\ l 'f - 't-l ) z:') -y ::= 1 (C t l.- t- 2 ULU-~ r ~ '/ - 0 :.::: ~ ( y- - 8) \ q q

=) 'I=~)l-~l 6

-.1 c;AAiL 5\i it ( '··I-v.. _

8. (15 p ~int s) Use the appropriate limit laws and knowledge about special limit s such as sin x

lim - = 1 t o find t he following limits. If a limit does not exist ,explain why. x -----+O x

2x - x(a) lim-­

x -----+ 1 x - I

(b) lim sin (2x) 3 y-x -----+ o sin(3x)

$ r' IA (3 '/.) . . ~- l lA 7-'i {II 1,1;" --- ((\Ii-\ , C))

~ ll{ '{.

y-) t) ­~ ~ ( '-.~

~ i .

l'f ((c) lim t1 l \' vV\ -x -----+O x

J~ Cc\ Il y..-) 0 + '{)C1D(

{'{( :=0­ f L -'! \ YeO {'F ({Cw\ - -­

')(-)0- '{