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Unformatted text preview: Elman Math 31A Midterm 1 Name: TA: TO RECEIVE CREDIT FOR A PROBLEM YOU MUST SHOW YOUR WORK.
THERE ARE FIVE PROBLEMS. PROBLEMS 1,2, AND 3 HAVE TWO PARTS. PROBLEM 5 HAS THREE PARTS. ALL PROBLEMS COUNT EQUALLY. PUT YOUR ANSWER IN A BOX. WRITE YOUR NAME ON EACH PAGE. TOTAL 0 name: — 1. In each of the following two problems compute the limit if it exists. \/;1:2+4 ifx ¢ ~2,—1,2 a. 11333] ﬁx) if f(x) = e if m = _1.
3 if :1: = :E2.
hm x1 +11“ :Y‘F)
X4”!
. (V332 + 3 + 2)sin(1‘ — 1)
b. 11m ————————. 1—)1 $2+3—2 name: _...I 2. Do the following two problems:
a. Let
ﬂit) = I93  3
Where is f differentiable, i.e., where does the derivative of f(:c) exis
is the derivative of f (as) at each of these points? \ A; +5 {is rent mm 8, everLN“? “3 Mt xii
A“ d? a. " a “T. t "a \I f {Y “J IA 4? 1" ii. ﬁx a; (3%“) 63“?“ t ‘J ‘3‘; t: {5&3 MS em; in 1 .5.“ “~11 r qub (4161113 / b. F ind a. constant I: so that the function 32—1 ifxgl
f(x)= _ 22 k ifx>1 x+k is continuous at :c = 1. For this value of k where is f continuous? \.m 3V“ _ — O K: m 4" ""'“""wwlwu K i" 5 a <t ’ '
X“). yd if: 1“” l\ J
€66) Cf‘ﬁ'i“. (“veered/x?“ m a x .r" '75)
Sit—$3, f ‘2» nmnezJ
'l
3. Differentiate the following two functions "2 ‘€~
' . x z * _ 1' 1
aftx)=cosW—z2+1>. M: Van wm H3 aw Mr H r?“
.. 7' \ ,,. x . VXQW ‘ .9 ‘ ‘ . f” a a; gem: .. . /
iﬁcﬁzgairca $(.{r~}a\ n \‘5 x24 3H name: _ 4. Find all points on the graph of = m3  2x so that the tangent line to the graph at each of these points is parallel to the line 3: — y = 0. Find the normal lines to the
graph of this function at each of these points. [Recall that the normal line at a point on the graph is the line perpendicular to the tangent
line on the graph at the same point.] x"! : 0
+‘(x3723’x‘a‘ﬂg 131:1—
ll : 3);“ :1 . '5 ex}: A: \ X
“ x“: “(43% b” name: _ __
5. Answer the following three questions about the function Iii1 4' ‘17, _(x—no—2) ﬁx) _ (m — 1)(:1: — 3).
3/ a. J 3"": (a). Where is this function continuous?
y i l j X f ‘5 (b). Compute all the onesided limits where f (:r
are vertical asymptotes for those values. . 1M VL y 0 ..
:\* {I}; Q ) is not deﬁned and tell whether there
1 1 Mn X‘?V’(x"ii{¥"'33
,0 "
(462 "
[,0] Roll} .— 1 u"
XOI ” ’ /1 xO?) {xvi
A5
:.v€P*IG§7“ (c). Compute the limits of this function as 3: goes to :too. What are the horizontal asymptotes? L V) r I I
/ . n! 7. i v. 1 A. {/X W }
<v*fh;:;;Ww \in\ (5 ices; hortaswwg “ ‘“ :Y*L1.(.:;;;;2—2 ,  A. I l I I i I. ‘ “aw” . _ Pn ,.‘ n g 
{if Q: .1} i ii}; 13/3, x 57» f} \ u} 3 5’ _':;v ix. J
( E“ ii if “‘1' "“' '1} J
" O: xz‘J—éx‘i'i’
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 Fall '07
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