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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