EXAM 2
BLAKE FARMAN
Lafayette College
Answer the questions in the spaces provided on the question sheets and turnthem in at the end of the exam period.
It is advised, although not required, that you check your answers. Allsupporting work is required. Unsupported or otherwise mysterious answers
will not receive credit.You may not use a calculator or any other electronic device, including cell
phones, smart watches, etc.
Name: Solutions
Date: October 11, 2019.
1
2 EXAM 2
Inverse Trig Functions
1. Compute Zsec2(x)
1 + tan2(x)dx
a tan xdu Sec 4 1DX
I 71114 dx I du
arctancuttC
arctan tank te
EXAM 2 3
L’Hopital’s Rule
2. Computelimx!1
⇥ln(x2 � 4)� ln(x+ 2)
⇤
ftp.lnkiH lnlxtzD fnjnxlnfxx z1Since
h xi fig hisx2x o
we can let y Eff and rewrite theoriginal limit
EE kHx fishily 8
4 EXAM 2
Area and Volumes
3. Compute the area of the region bounded by f(x) = 4x3 � x and g(x) = 3x.
4. Compute the volume of the solid obtained by revolving the region bounded by thethree lines x = 0, y = 2� x, and y = x about the x-axis.
flugal 43X 3x 4 3 4 4 4211 4 1 114D
ry 4x34x A 14 3 4xdxtfyx4xdxu.oi.VE
x Who 2 49 2 21 XYO 1 do 1 24 0 Ct o
I 12 2 I
Y Cross Section4,2xi 4i am 4xx t
gAE'IIIii II
I 4th xIi i
U jfAlx1dx 4THDdxI4 1 6 zx4o 4th K
EXAM 2 5
Integration by Parts
5. Compute Zx2ex dx
U XZ exdu 2X du exdx
Liddy a x'ex fxeXdx a x v exdu dx du eXdX
x'ex 2 xeX fexdx
x2eX 2Xelt2e t
6 EXAM 2
Partial Fractions
6. Compute Zx+ 1
(2x+ 3)(3x+ 5)dx
234
5 2 37ps Xt I AGxts t B 2 3
Sx
EstIs Zz ACO t B 03 93 tf 13 2
E
Iz Zz I A EI TE tBco Az A I
f oyd zd gdX tzlnkxtsltz.tn 3xtslt
Cheek adxfzlnkxtdtzktsxtslt.cl I k 3 t E IesTexts t 3275
3 51 4622 3113 15
XII2 3Bxts
EXAM 2 7
Approximate Integration
Use the function f(x) = 2x3 � x+ 1 to answer Problems 7 and 8.
7. Use the inequality
|ET | K(b� a)3
12n2
where��f 00(x)
�� K on [a, b], to find the number of intervals needed to estimateZ 4
0
f(x) dx
using the Trapezoidal Rule with an error less than 10�2.
FYN 621
f CH DX
The range of RX on 6943 is 0,483 so
IfCxHE48implies
IEtle 4 493 4h41 itso
44102 14442 1602 eh
Therefore we needatleastl6lintervats
8 EXAM 2
8. Use the Trapezoid Rule
Tn =f(x0) + 2f(x1) + . . .+ 2f(xn�1) + f(xn)
2�x
with n = 2 intervals to estimate the value of the integralZ 4
0
f(x) dx.
fly 2 3 Xt1 DX 421 2 Xo o 4 2 Xz 4
fcxot fcolf.lyfCz 2C8 21 1 16 1 15
fXa f 41 2164 41 1 128 3 125
Tz 1121151 125.2 It 301 125 1560
EXAM 2 9
Improper Integrals
9. Compute the improper integralZ 1
1
1
x2 + 1dx
I dx fistI DX
himsoarctana ft
thin arctanCtI arctanal
Iz IT
10 EXAM 2
Differential Equations
10. Solve the inital value problem
y0 =2x
3y2, y(2) = 2.
y2dy 2xdX
y3 Etc
8 4 te C 4
y T
EXAM 2 11
Parametric Curves
11. Eliminate the parameter to find a Cartesian equation for the curve
x = t+ 3, y = t2 + 6t+ 8, �4 t �2
then sketch the curve and indicate with an arrow the direction in which the curve istraced as t increases.
tt6tt8 tt 4Ctt2Gt3 t 1 Gt3 l
Xtc x 1 2 I
t 4X 4 3 1 y O
X 213 1 y o
or yC1,01 1,01
Y x
wom
12 EXAM 2
Polar Coordinates
12. Sketch the rose r = sin(2✓), 0 ✓ 2⇡. Label the tips of the petals and draw arrowsto indicate the direction in which the curve is traced.
i iii ii
0 IT'Iz3I TSIy.FI If 2T
T z317 11744 HEH
Chitty
e
9 f1,5 C1,312
31Z
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