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(math013)[2010](f)midterm~yhong^_10039.pdf
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Math013 Calculus I, Fall 2010 Midterm Solution
1.i
5
1. ([8 pts]) Findall complex roots of the equation z=.
1+i
1.i (1.i)(1.i)1.2i+ i2 .2i
Solution Notethat = = = = .i.
1+i (1+i)(1.i)1.i2 2
1.i
z 5 == .i = cos(. )+isin(. )
1+i 22
If z = r(cos + isin ), then byDe Moivres Theorem,
55
z = r (cos5 + isin5)= cos(2k . )+isin(2k . )
22
where k =0, 1, 2, 3,....
Taking r =1 and5 =2k . 2 , for k =0, 1, 2, 3, 4, i.e.,

(4k .1)
=
10
we have the complex roots of the equation:

z1 = cos(. )+isin(. ) z2 = cos(3 )+isin(3)
10 10 10 10
z3 = cos(7 )+isin(7) z4 = cos(11 )+isin(11 )
10 10 10 10
z5 = cos(32 )+isin(32 )= .i
5x 2 +6
2. ([10 pts]) Given a functiony = f(x)= .
x2 +2x .8
(a) Find the domain of f. [5 pts]
Solution To have a well-de.ned square root, we must have
5x 2 +6 5x 2 +6
= 0 ,
x2 +2x .8(x +4)(x .2)
i.e.,
(x +4)(x .2) > 0

since 5x 2 +6 > 0 and the denominator can not be zero. The domain is : x< .4 or x> 2.
(b) Find all horizontal and vertical asymptotes of f. [5 pts]
5x 2 +6
Solution f(x)= . + as x . .4. or x . 2+ , so x = .4 and x = 2 are the vertical
(x +4)(x .2)
asymptotes of f.
As,

2
5x +6 5+ x6 2
lim =lim =5
2
x. x2 +2x .8 x. 1+ . 8
xx2
and also,
5x 2 +6 5+ x6 2
lim =lim =5
2 . 8
x.. x2 +2x .8 x.. 1+
xx2

there is only one horizontal asymptote: y = 5.
3. ([12 pts]) The number ofrabbits in a 200 day period is modeled by the following graph of a di.erentiable function q(t), 0 t 200.
q (rabbits)
2000

(40,1600) (120,1400)
1000
t 100 200 (days)
(a) What is the value of the derivative function q (t)when the number of rabbits is largest?
Solution q (80)=0,whenthelargestnumberof rabbitsisreached. [1pts]
(b) What does q (t)represent, and what is its unit? [2 pts]
Solution q (t)represents the rate of growth of the rabbit polulation.
Its unit is rabbits/day.

(c) Using thegivengraph,estimatethesizeof therabbitpopulation whenitsderivativeislargest,and also the size of the rabbit population when its derivative is smallest respectively. [4 pts]
Solution q (t)islargest(mostpositive) approximatelyinday40, with1600 rabbitsasthesize oftherabbit
population.
q (t)is smallest(most negative)approximately inday 120, with1400 rabbitsasthesizeof therabbitpopulation.
(Close enoughapproximations according to the given graph may also be acceptable answers.)

(d) Sketch roughly the graph of q (t). Show the scales in your axes. [5 pts]
q' (rabbits/day) 40
20
10 t 40 80 120 160 200 (days) -10
4. Choose coordinate axes and scales and then draw the graph of a function f on the grid satisfying all of the following conditions on the interval .1 <x< 5: [12 pts]
(a) f(0)=1, f (x)< 0 for .1 <x< 0, and lim f(x) =3
x0+
(b) f is continuous for 0 <x 2, f (x)> 0 for 0 <x< 1, and f(2) =2
f(2+h).f(2) 1
(c) lim = .