EXAM 2

BLAKE FARMAN

Lafayette College

Answer the questions in the spaces provided on the question sheets and turn them in atthe end of the exam period.

It is advised, although not required, that you check your answers. All supporting work isrequired. Unsupported or otherwise mysterious answers will not receive credit.

You may not use a calculator or any other electronic device, including cell phones, smartwatches, etc.

By writing your name on the line below, you indicate that you have read and understandthese directions.

Name:

Problem Points Earned Points Possible1 152 103 124 135 256 25

Total 100

Date: March 27, 2019.

1

Solutions

2 EXAM 2

Approximate Integration

Use the function f(x) = 2x3 � x+ 1 to answer Problems 1 and 2.

1. 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 estimate

Z 4

0f(x) dx

using the Trapezoidal Rule with an error less than 10�2.

f KKGE If41 12xf 11 12toBytheClosedIntervalMethod theabsolute maximum absolute minimum is thelargest smallestof

fca O f141 48The largestofthese two values in absolutevalue is 48 so on o43

48e fCHe48 IfHK48IETa 48,44793 41 44 e to44102E hZ

For Flor 442402 442.1042 4210 1610 160e Fi n

Therefore if we use at least 160 intervals the error will be smallerthanto 0.01

EXAM 2 3

2. 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 integral

Z 4

0f(x) dx.

I flxDt2flxzltf ox Xo a x at ox Xz at2ox b Lx bnd

fled23 Xt l a o b 4

ox bye 421 12 2Xo O X 012 2 Xz 01261 4

flxotfcot2co Ott If Xi f21 2123 211 218 1 16 1 15

fXz f 4 2.43 4 1 2 43 3 43 416 64

2664 3 1283 125

I fcxoltzfcxzdtfkd.z flxolt2f.CHtf KIt245 t 125 It 301 125 11561

4 EXAM 2

Differential Equations

Solve the initial value problems.

3. y0 = 2yx, y(0) = 1e

4. y0 = e�y(2x� 4), y(3) = 0.

dyd 2yx te teced t.eu tec

tydy 2xdx t.ec

ftydy f2xdx µ Eex2 e ex2

Inly1 21zx4tC x2tcy ex4c ecex2

y ecexZ

dyd eY 2 4 2Xej4 0 4191214 61314

eYdy 2 414 eo elntstel

JeYdy fCzxydx fxdx 4fdx 1 3te

113 3 31C

et 264 4xtC x 4xtc y c

cc EE E

EXAM 2 5

Parametric Equations

5. Eliminate the parameter to find a Cartesian equation for the curve

x = t+ 2, y = t2 + t� 6, �3 t 2

then sketch the curve and indicate with an arrow the direction in which the curve is traced as t increases.

Xa ft2 f X 2 f X 2 3 EX 2 2

y 12ft 6312 EX 21 2 E 2 2

Gt3 t 2 I exe 4

X 2 3 X Z Z

x 111 4y 3 4

y lxHKx41 on ft43is the partof the parabola between the

roots

ifC1,0 Yo

xXd

6 EXAM 2

Polar Coordinates

6. Sketch the rose r = sin(2✓), 0 ✓ 2⇡. Label the tips of the petals and draw arrows to indicate the

direction in which the curve is traced.

arsinkoTHI 5174,01

31,4

go Ying3 11 FIT11

ay

1,7144 1,174

x

1,3741,5144