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Math 151 WIR, Spring 2014, Benjamin
Aurispa
Math 151 Week in Review 10
Sections 4.5, 4.6, & 4.8
1. A bacteria culture starts with 2000 bacteria and grows at a rate proportional to its size. The population
has grown to 2400 after 20 minutes.
(a) Find a function that models the number of bacteria after t minutes, assuming the population
grows at a rate proportional to the number of bacteria.
(b) How many bacteria are present after 2 hours?
(c) When will there be 20,000 bacteria?
(d) At what rate is the population growing after 40 minutes?
2. The half-life of a radioactive substance is 10 days. How long will it take a sample of this substance to
be 1/3 its original size?
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
3. If a sample of a radioactive substance decays to 60% of its original amount after 4 hours, what is the
half-life of the substance?
4. An object with temperature 150◦ F is placed into a room with temperature 80◦ . After 20 minutes,
the temperature of the object is 120◦ F. Find a function that models the temperature of the object
after t minutes.
5. Suppose the rate of growth of a bacteria culture is always 5 times the current population. If there are
4000 bacteria after 2 minutes, find a function that models the population after t minutes.
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
6. Evaluate the following.
√ (a) arcsin −
2
2
(b) sin−1 (sin π3 )
(c) sin−1 (sin 5π
6 )
(d) sin(arcsin 14 )
(e) arcsin(sin 9π
8 )
(f) arccos − 21
(g) cos(arccos 54 )
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
(h) cos−1 (cos 5π
4 )
(i) arccos(cos 20π
11 )
(j) tan−1
√1
3
(k) tan(tan−1 18)
(l) arctan(tan 2π
3 )
(m) arctan(tan 17π
7 )
(n) cos(arcsin 56 )
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
(o) sin(2 arctan(−5))
(p) tan(cos−1 x)
7. Calculate the following limits:
−1
x2 + 3
2x2 − 5
−1
x2
4−x
(a) lim sin
x→∞
(b) lim tan
x→∞
!
!
8. What is the domain of f (x) = arcsin(4x − 1)?
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
9. Calculate the derivatives for the following functions.
√
(a) f (x) = x arcsin( x)
(b) g(x) = tan−1 (3x2 )
5
10. Find the equation of the tangent line to y = cos−1 ( x1 ) at the point where x = 2.
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
11. Calculate the following limits.
x2 + 3 x − 4
x→1 42x + ln x − 16
(a) lim
sin x − x
x→0
x3
(b) lim
(c) lim
x→1
1
1
−
ln x x − 1
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
(d) lim (xe1/x − x)
x→∞
(e) lim e−x (ln x)2
x→∞
(f) lim cot x ln(1 + 3x + 5x2 )
x→0+
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Math 151 WIR, Spring 2014, Benjamin
Aurispa
(g) lim
x→∞
1+
2
3
+
x3 x4
x3
(h) lim (4 + e3x )−2/x
x→∞
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