Name:
BC 2
Euler, Again
—
What we saw on the last sheet worked quite well when y’ was positive throughout the
interval in question. Let’s see if this area relationship still holds if y’ is both positive and
negative.
One more time. Sketchy, based on they’ graphed. Assumey(O)
(1)
Yç
=
0.
-
3
2
4
This time, consider the signed areas of the rectangles. Do the accumulated signed areas
agree with the values of y?
(2)
Consider y’ = 2x—x2 for 0 x 3.
Sketch a good graph ofy’.
(a)
(b) Using this graph, estimate the
value of f(2x_x2)dx.
Ar
2)1_J1()(t)
(c)
Use Euler on your calculator to get a final estimate for y and the signed area.
=0 O yrtJ OL
Skp
(d)
Now find a function f(x) with F(0)
=
0 where f’(x)
=
2x x2.
—
Efr y-x3
(e)
Evaluate f(3). What do you notice?
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IMSA
3-3O
EE3.1
S12
(3)
Letf(x) x2 + ion [0, 5].
Find F, where F(x)
(a)
f(x), so that F(0)
F(x
(Note
(c)
UseFto find f(x2+1)d.
(b)
‘i
J
(d)
tsc/v
)
0.
f(o) O
UseFto find f(x2+1)dx.
J2
(z)3Z
S
(x2[)
Use
=
parts (b) and (c) above to find
f2
)ox-f
2CC’
—.
—
7i
Define: LetA(x) = ff(t)dt, where a is some
constant in the connected domain of f,
be the signed area over [a, x], bounded
by the graph of y = f(x) and the x-axis.
Notes:
(a)
This really is a function of x. As x
varies, the amount of area enclosed also
varies.
(b)
This function and the area depend on the choice of a. In earlier
examples, a = 0.
(c)
In the diagram, x> a, but this is not necessary.
*‘f NQk: V
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di%j(0i4’l
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(w f 11?:(
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EE 3.2
S12
Let 1(t) = t sin(t).
(4)
LetA(x)
jkr
F/t
Ei
=
Sketchy
evoLy
f(t).
L f(t)dt.
FUse Euler with a small step to
,I approximate each of the following:
_,j/__•\
(sp S/7f
j
=
I
I
/
/
L
/
f2 L/c
(5)
Letj(t) andlet A(x)=J !dt forx>O.
(a)
FindA(x).
=tx
(r€//y
A() ifr/j
V
find A(e) using the function above. Then use Euler withy’ = f(t) with a small
(b)
step size to check your work. (Your two answers should be close, but they won’t be
exactly equal.)
A(e)
Jrq
IMSA
EE3.3
yeo2 CCI.
512
(6)
Let f(t)
(a)
=
sin(2t) + 1 and let A(x)
=
lox f(t)dt.
find A(x) analytically.
coOOC
) i
find A (4) analytically and then approximate this value with a calculator. Then
(b)
use Euler as you did above to see if the values are close to confirm your function A.
c3
E4tc
(c)
vif1
tx°-OI
I
‘I
find A(3) analytically and then approximate it on your calculator.
A(3 --co,
4(
(d)
L1xOl ,vcJ
Use these values to find f4f(t)dt.
—A(q) /1(3)
=c3—9o2O
(7)
Summarize what you’ve found on these pages about a function f, F, and Sf(x) dx
where f’(x)
=
f(x).
F S1i%ij
cofrpvle
IMSA
EE3.4
S12