Sample EXAM 1 NAME:
Instructions : Graphing calculators are not allowed. Show you work
1. (15 pts) Consider the function g with graph given below:
z
1.5
1
0.5
Compute the following limits:
(a) limt_o- g(t) =
(b) limt-o+ g(t) =
--
(c) limt -+o g(t) =
h
(d) limt-i g(t) =
0
(e) limt -i g(t) = Q
(f) On what intervals is g continuous ? Z1 01 1 l I Z
L
2. (20 pts) Do the following two things: (i) compute the limits (write oc or -oo for infinite
limits); and (ii) write down the intervals on which each function is continuous.
(a) limxy, /2 x sin x
(h-LA
(b)
limx -4
-BIZ t Z ^
1-rz
a-IA
(xy?O(x4^ )
14
(c) limxy'+
cos(x)
I
X-7r
=
COS
-2
__
Y, -T1 0 =^^ - X
60S -M
cam-
l
00 )
(d) limx_l (x+2)(2x-3)
x-1
63
at x = 7r/4. (Recall
3. (10 pts) Compute the tangent line approximation of f (x) = i
sin x
that cos ( 7r/4) = sin (ir/4) = v/-2-/2.)
YK)
- (Si v\, X)-.2 (oS X
-(c
Z Si„?x
ov/
Z
4. (18 pts) Compute ax for y defined below:
(a) y = (3x + 4)3/2 - 3x-1/2 + X
& r ]^- (, V
Ok,f -Z
2-
a
(31"t-4 2+
77-
Z
(b) y = 2x + 1sin(2x2 + 2x)
z . 5 " ^ t2X A-z^ > -^
IV^
2X+.
cosC zx -^ 2 x^ (L42)
(2X 7)
2X}F
(C)
y
=
(3x
+^
)2
LA Y-)
) 106
3e
5. (15 pts ) Use the limit definition of the derivative, i.e.,
f,(x) = lio f(x + h - f(x)
zx+i
to compute the derivative of f (x) = x
2xt2k 1 x _ 2x+! +
x x+^
Z^x ^(2 1^^ --^2x+^^
U.
'2 X Y, -V(2 * -^ 1^^
w^ 0
I
76
2y
6. (12 pts) Use implicit differentiation to compute d-- for the folium of Descarte x3 + y3 =
6xy. Then find the equation of the tangent line to the folium at (3,3).
-2 2-
o1x
X Z^ _ lb
I
0
7. (10 pts) A tank has the shape of an inverted circular cone. If water is being pumped
into the tank at a rate of 2 m3/min, find the rate at which the water level is rising
when the water is 3 m deep. Note that V = 12 h3, where h is the height of the water
and V its volume.
6kt
Extra Credit: The figure shows a lamp located three units to the right of the y-axis
and a shadow created by the elliptical region x2 + 4y2 < 5. If the point (-5, 0) is on
the edge of the shadow, how far above the x-axis is the lamp?
Hint: Find the intersection point of the upper line and the ellipse.
Fic)tAIZE
^'"^ = ^^ (X+ 5
2x +
2
d w{
Tij "-Le tL- 6'to^' ^^ ^A q
2
` 2 5-i
5
0
VA
el-
ro5;^iw
rb -t