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