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Unformatted text preview: Math 123 Calculus l TakeHome Portion of Exam II Rules: Work alone {i.e. without the heip of others} but use any reference materials antlfor
technologyr you fincl useful. if the problems are unclear to you see your instructor [only]
liar clariﬁcation I. An outﬁelder throws a ball at 84 tniiese'hr = ill? ﬁr'sec at an angle :9 above the horizontal: _—'—'_ _'*.. "it
Me. may 9mg :13. W m isms; Museum '7 (Li I on tttl a. it~ the release point is 6.5 it above growtu, what angle ﬂ maximises the distance the
hull travels”? {ignore air resistance} Express your answer in degrees. b. How the does the ball travel in the honacntai direction, assuming the ﬂight is not
intpctlctl in any way? {Again i gnorc air resistance} c. The [1i ght ot" the ball (ignoring air resistance} thatr be described by what geometrical  a
cune. Fa i (Note: Don’t make this problem harder than it is  use the javelin handout discussed in
class and any,r formulas that appear on it to quickly work parts a. and b.} t I; it ‘
m we. 1) J 1 camsems) 2: 44.51? l §:?1Lr r"—————_n_———‘I—l a .2. a
= L?— J {2111+ 1(3'1)(£L§) : '51.. ffo.’??e. H I For the baseball purists, I realise that it‘sbettcr to "hit the cutoff titan" since the real goal is to get the ball
itt as cutest}! as possible. 2. Teams A and B play in the World Series. By convention the ﬁrst team to win 4 games
wins the WDI‘ld Series [it’s often referred to as a ‘best ofseven‘ series}. Suppose team A
beats team 33 in each individual garrth with probability 3:. Then the probability that team
A wins the lit'ort’eT Series in either a‘ or 5 games is given by {trust me} 5.x“ —4_r5
in this problem, your goal is to ﬁnd the value of the ﬁrst probability {3:} to make the second probability — the chance A wins the 1v‘r'orld Series in 4 or 5 games  equal to il.?5.
That is. you are attempting solve the problem fix} = 5x' — 4s" — tits = e
where .r is necessarily between [t and l .
a. What is fﬂl)? What is ﬁll“? it What theorem {give a name} guarantees that there is at least one I between ll and 1
making fix) = it? You will need to use part a. to answer this question. c. 1liltihy is therey'trst one I for which ﬁx} = ll? (Hint: Give an explanation wltieh uses the
derivative oi: f and the Fact that x. a probability. is between t] and 1.] d. Use Newton’s method to ﬁnd {approximately} the value oi" .‘t' making _f'{x]=tl. Continue using Newton‘s method {you get to choose the starting pointjl until subsequent
values agree to at least three decimal digits (Le. they agree to llﬂﬂl}. Please list the sequence of values I: .1’2 that you generate. @ $9): #555 {i (it’ll): {prod5‘ @ ﬂﬂdm '7C [IL5
{MW as. (0)13) (9 gECﬁJﬁE first): Slog—213mg: %x3(t¥)abj
(C t5 tacowag I The baseball fans among you may object to this assumption {because the outcome is affected by a number
or" variable factors including the particular pitehcrs playing a given clay1 the location of the game1 etc} but
let's keep things simple here and try a ﬁxed chance of winning as a ﬁrst approsiniation. 3 Since team It is going to win the series in 4 or 2'1 games with chance 035 this might suggest how to
choose your starting point. Should you start with an initial guess ofless than ELS or greater than [1.5. for
example? T)
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 Fall '06
 JOHNSON
 Calculus, World Series

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