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Kouba
Worksheet l 7 1. Let y = (sin(XI2))x + 5". Compute y' at x=1c. 2 Assume that y isafunction of x and y3 + xy = 3y3‘. Compute
y" at the point (0, 3) . 3. Differentiate. a. y = tanx + arctanx b. y = sinW —— arooosxrxﬁ c. y=cot(sin(5x))+arcsec(cscx) d. y=ln(arctan(lnx)) e. y=log4(x53X) f. y=log3(x2+e“x) g. y=(x+1)5‘“<
3x2 h. logxy=eX i. (xy)x2=(tany)xy3 4. A rectangle is to be inscribed in the first quadrantvbelow the graph of
y = \l 4  _x . Determine the dimensions of the rectangle of a. maximum area .
b. maximum perimeter.
0. maximum sum of area and perimeter . 5. Evaluate the following limits. a. lim “+1 7” b. lim “3 "
n—v—oo n+2 “+4.00 1+n3 6. A baseball is fired horizontally from the top of a cliff, which is one
mile high, at 100 miles per hour. See diagram. a. How long does it take for the baseball to reach the ground ?
b. How far away from the base of the cliff does the baseball land ? c. What is the " vertical velocity " of the baseball as it strikes the
ground ? 7. Consider the function f(x) = x3  2x2 + 3/2. a. Sketch the graph of f. b. Use the IntermediateValue Theorem to prove that f(x) = O has a
solution r. 8. Prove that there is some number c , 3 < c < 4 , satisfying 403’ = In (9.57/91)
C4+1 ' HlNT: Consider the function f(x) = In (x4+1 ). 7 . For each of the following functions determine the xvalues for which
fis increasing, decreasing, concave up, and concave down. lndicate all maximum, minimum, and inflection points and intercepts. Neatly sketch
the graph off. 9)
<
H X8 b y = xlnx O
‘<
ll
(D
+
(D ‘0. Use L'Hopital‘s rule to evaluate the following limits. a. lim sinx X?O x
c lim X4'16
x41 ﬁ—wfi‘
e lim 9 “1‘2".1
9' “m xlnx+1—x x—~1 (x—1)2 1. lim 2" +2X Xq+oo 5x
. x 2
k. lim xe oos 6x
' X—ro €2X_1 m. "m arcsin x
X‘» o arctan 2x sin2x— x2 q. lim {sinx}“"
x—v 0* 8. lim (1+5/n)5"
Y1~§+oo u. lim x2 lnx X—v o+ . 2
b. llm X ‘1
x91 x—1
d. “m. tanx
X90 x+sinx
f_ "m xzsinx+xsinx
X+° x + 1— oosx
h. “m tanx
X E 1+secx
j. lim X3
Xar+oo 10x
L "m eX—1/x
X—"+°° 9" + 1/x
n. \im 1 2.
Xa’o 1oosx x2
. 1/
p. lim {lnx} x
Xv‘r'irOO
. 1/
r. hm (1+x) x
X90
. 1/
t. hm (1+n) n
h~—)+oo
. \lx/3
v. hm {tanx} Xao ll . With each of the following functions are given numbers x 1 (the initial
'xvalue) and x 2 (the ﬁnal xvalue). Compute the associated exact change in functional value, A f, and the differential of f (approximate
change in functional value), d f. a. f(x)=x3+x1
i. x1=1,x2=4
ii. x1=1,x2=2
ii. x1=1,x2=1.1
iv. x1=1,x2=1.01
b. f(x)=lnx
i. x1=e,x2=e+2
ii. x1=e,x2=e+.1
ii. x1=e,x2=e+.001
C. f(x) = esinx
i. x1=0,x2=1
ii. x1=0,x2=.001
iii. x1=0,x2=.00001 19.. Use differentials to estimate the following quantities. a. 4229 b (250.1)5
c. sin(1r—.O4) d. log10(9.9) 13 . Assume that the radius of a sphere is measured with a percentage
error of at most 2% . With what percentage error will the following
quantities be computed ? a. diameter of the sphere
b. volume of the sphere
0. surface area of the sphere ...
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 Winter '09
 Kouba
 Calculus, Derivative, Convex function, Concave function, j. lim, lim xe oos

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