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Calculus
6.56.5-6.7, 6.9 Review
o Find
Name:____________________________________
Mitchell
dy
.
dx
1) y = 7 5 x
4
2) y = log 3 5x
−3 x
3) y = x sin x
4) y = arctan(2x) + 2 arctan x + (arctan x)2
5) y = cos −1 (ln x)
6) y = arc cot 2x + 1
2
7) y = csc −1 (e x )
8) y = e 4 x sec −1 (4x)
o Integrate for problems 99-18.
9)
dx
∫ 4 + 25x
2
10)
2xdx
∫ 4 + 25x
2
1
11)
x 2 dx
∫ 4 + 25x 2
12)
∫
13)
25x 2 + 4
∫ x 2 dx
14)
∫x
15)
∫ cot(8x)dx
16)
∫
18)
sec 2 5x
∫ 9 + tan2 5x dx
17)
∫
csc 2 (2t)
1 − cot 2 (2t)
dt
4 − 25x 2
dx
dx
x8 − 9
sec(ln x)
dx
x
19) Find the
the area of the region bounded by the
x
graphs of y = 4
, x = 1 , and y = 0 .
x +1
20) If y > 0 and y = 4 when x = 0 , solve for y.
dy
= 7y
dx
21) The rate of change in the number of bacteria in a culture is proportional to the number present. The
culture had 10,000 bacteria initially. Ten minutes later, there were 50,000 bacteria present.
a) In terms of t only, find the
the number present at time t minutes where t ≥ 0 .
b) How many bacteria are there after 20 minutes?
c) At what time are there 20,000 bacteria?
22) A radioactive substance has a halfhalf-life of 5 days.
How long will it take for an amount A to
disintegrate
disintegrate to the extent that only 1% of A
remains?
23) Use your calculator to find the length of the arc
on y = sin−1 x from x = − 4π to x = 4π to the
nearest thousandth.
x2
(arc length =
∫
x1
dy
2
1 + ( dy
dx ) dx )
24) A balloon is released from ground level, 500
meters away from a person who observes its
vertical ascent. If the balloon rises at a constant
rate of 2 m/s, use inverse trig functions to find
the rate at which the angle of elevation of the
observer’s line of sight is changing at the instant
the balloon is 100 meters above the ground.
(disregard the observer’s height)
25) Consider the region bounded by y = 2 x , x = 0 ,
y = 0 , and x = 2 .
a) Calculate the volume of the solid formed
when the region is rotated about y = −1 .
b) Set up an integral that represents the
volume of the solid formed when the
region is rotated about x = 2 .
26) How long does it take to double your money if
it is deposited in an account at 12%
compounded
a) monthly?
b) continuously?
27) $1000 is put in a bank account for 5 years at an
annual rate of 6.5%. How much will be in the
account if it is compounded
compounded
a) annually?
b) monthly?
c) daily?
d) continuously?
o Find each limit, if it exists.
ln(2 − x)
28) lim
2x
x →0 1 + e
29) lim
e 2x − e −2x − 4x
x →0
x3
31) lim
30) lim
sin2 x
32) lim
x → 0 1 − cos x
x →0
x →0
sin 2x − tan 2x
x2
4 − x2 − 2
x
e x − ln(1 + x) − 1
33) lim
x →0
x2
ANSWERS TO 6.56.5-6.7,6.9 REVIEW
1) 7 5 x
4)
7)
10)
4
−3 x
2
1+ 4 x 2
(ln 7)(20x 3 − 3)
+ 2 arctan x (ln 2) ( 1+1x 2 )
+2 arctan x ( 1+1x 2 )
−2xe x
ex
2
1
25
2
2
e2x − 1
ln 4 + 25x 2 + C
13) 25x −
4
+C
x
16) ln sec(ln x) + tan(ln x) + C
19)
22)
π
8
1
5ln( 100
)
≈ 33.219 days
1
ln( 2 )
27 π
≈ 61.187
2ln 2
25) (a)
2
(b) ∫ 2 π(2 − x)(2 x )dx
0
28)
ln 2
2
31) 0
2)
1
1
or
2xln3
x ln9
5) −
1
2
x 1 − (ln x)
8)
sin x
3) y 
+ ln x(cos x) 
 x

6) −
1
(2x + 2) 2x + 1
9)
tan−1 ( 25 x) + C
1
10


1
4e 4 x  sec −1 (4x) +

4x 16x 2 − 1 

2
x − 125
tan−1 ( 25 x) + C
12)
1
5
sin−1 ( 25 x) + C
 x4 
sec −1   + C
 3 
15)
1
8
ln sin8x + C
17) − 21 sin−1 (cot 2t) + C
18)
1
15
20) y = 4e 7 x
21) (a) N = 10,000e 10 t
(b) 250,000
10 ln 2
min ≈ 4.307 min
(c)
ln5
1
24)
radians per second
260
11)
1
25
14)
1
12
23) 2.399
tan−1 ( 31 tan5x) + C
ln 5
t
26) (a) 69.66 months ≈ 5.805 yrs 27) (a) $1370.09
(b) 69.31 months ≈ 5.776 yrs
(b) $1382.82
(c) $1383.99
(d) $1384.03
29) 0
32) 2
30)
8
3
33) 1