Final Examination
Z
1. (a)
(8x + 12)
201-203-RE
p
x2 + 3x + 9 dx
Z
(x3 + 2x) cos(2x) dx
Z
x2 + 3x − 14
dx
(x + 1)(x + 5)2
(b)
(c)
Z
December 2016
7
2 − |x − 5| dx
(d)
2
t2 + t3/2 − 5t3 cot(t)
dt
t3
Z
cos(x)(sin2 (x) + tan(x))
(f)
dx
sin(x)
√
Z
x
√
dx
(g)
x+3
Z
(e)
2. Find f (x) given f 00 (x) =
3 − 3x 0
√ , f (1) = 4, and f (4) = 14.
2 x
dC
= 0.04x + 0.5, and if
dx
producing 40 units costs $13, find the average cost per unit of producing 60 units.
3. If the marginal cost of producing x units of dragon figurines is given by
4. Find the area of the region bounded by the graphs of f (x) = x3 − 6x2 + 9x and g(x) = x2 − 3x.
5. Given the demand function p1 (x) =
figurine company,
12
and the supply function p2 (x) = x + 2 of a unicorn
x+3
(a) Find the equilibrium point.
(b) Sketch and identify the regions representing consumer and producer surplus.
(c) Evaluate Consumer Surplus.
Z
6. Use Simpson’s Rule to estimate the value of
to 4 decimal places.
4p
x2 + 2 dx. Use n = 4, and round your answer
1
7. Determine whether y = xe2x is a solution to the differential equation y 00 − 2y 0 = 2y.
8. Solve the following differential equations.
(a) y 0 =
(b)
dy
dx
2x+sec2 x
2y
; y(0) = 5
= y(x + 2) ; y(0) = 2
9. In a city whose population is 200, 000 there is an outbreak of a flesh-eating zombie virus. When the
city health department begins its record keeping, there are 225 zombies. The number of zombies
N is increasing at a rate proportional to the square root number of zombies at time t in weeks.
One week later, there are 625 zombies.
(a) Write a differential equation representing the problem.
(b) Find the function N (t) for the number of zombies after t weeks.
(c) Find the number of zombies two weeks after the record keeping begins.
10. Evaluate the following limits:
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Final Examination
201-203-RE
December 2016
ln(x2 − 7)
x→∞ x2 + 4
x − sin(x)
(b) lim 3
x→0 x − 2x2
(a) lim
11. Determine whether the following improper integrals converge or diverge. If the integral converges,
find its value.
Z 5
x
√
dx
(a)
25 − x2
3
Z ∞
ex
(b)
dx
x
e +1
0
−2 1
6
13
12. Consider the sequence given by
,
, ,
,...
5 −15 45 −135
(a) Find the sixth term a6 .
(b) Find the general term an .
13. Determine if the following sequences converge or diverge. If the sequence converges, find its limit.
(−1)n (1 − n)
(a) an =
n2 + 3
(b) an =
(n + 1)!
n2 (n − 1)!
14. Genji wants to save for 10 years in order to buy a $30,000 ancestral samurai sword for his daughter
Zelda. If he can invest at an interest rate of 2.4% compounded monthly, what amount should he
deposit every month?
15. Determine if the following series converge or diverge. If the series converges, find its sum (when
possible).
∞
X
1
(a)
2
n + 2n
(b)
(c)
(d)
n=1
∞
X
n=1
∞
X
n=1
∞
X
n=1
3n
3n + n
2−n + 2n
3n
(n + 1)!3n
n!4n
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Final Examination
201-203-RE
December 2016
Answers
1. (a)
(b)
(c)
8 2
3/2 + c
3 (x + 3x + 9)
1 3
1
2
2 (x +2x) sin(2x)+ 4 (3x +2) cos(2x)−
3
3
4 x sin(2x) + 8 cos(2x) + c
1
− ln |x + 1| + 2 ln |x + 5| − x+5
+c
(b) y = 2e1/2x
√
dN
=k N
dt
(b) N (t) = ( k2 t + c)2 = (10t + 15)2
9. (a)
(d) 7/2
(e) ln |t| −
2
√
t
(c) 1225 zombies
− 5 ln | sin(t)| + c
1
2
sin2 x + x + c
√
√
√
(g) x − 6 x + 18 ln | x + 3| + c OR ( x +
√
√
3)2 − 12( x + 3) + 18 ln | x + 3| + c
(f)
2. f (x) =
2x3/2
−
2 +2x
2 5/2
5x
+ 2x +
10. (a) 0
(b) 0
11. (a) converges to 4
14
5
(b) diverges
3. C(x) = 0.02x + 0.5 − 39
x so C(60) = $1.05
Z 3
4. A =
x3 − 7x2 + 12x dx +
0
Z 4
71
−x3 + 7x2 − 12x =
6
3
12. (a) converges to 0
(b) converges to 1
13. (a) a6 =
(b) an =
5. (a) (1,3)
(b) graph
−33
(5)(3)5
(−1)n+1 (n2 − 3)
5(3)n−1
14. $221.01 once a month
(c) Consumer Surplus= 0.45
15. (a) converges to
6. 8.7242
3
4
(b) diverges by TFD (Test for Divergence)
7. No, it is not a solution
p
8. (a) y = x2 + tan(x) + 25
(c) converges to
11
5
(d) converges by the ratio test
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