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Math 151 WIR, Spring 2010, Benjamin
Aurispa
Math 151 Week in Review 10
Sections 4.5, 4.6, & 4.8
1. A bacteria culture starts with 2000 bacteria and quadruples every 25 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) Find the number of bacteria present after 2 hours.
(c) At what time are there 30,000 bacteria?
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
2. The half-life of a radioactive substance is 10 days. How much of a 30 g sample remains after 2 weeks?
3. 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.
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
4. A curve has the property that at every point the slope of the curve is 5 times the y-coordinate. If the
curve passes through the point (2, 4), find the equation of the curve.
5. A tank initially contains 300 kg of salt dissolved in 1500 L of water. Pure water is poured into the
tank at a rate of 10 L/min. The solution is thoroughly mixed and drains out of the tank at 10 L/min.
(a) Find a function that models the amount of salt in the tank after t minutes.
(b) When will the concentration of salt in the tank be 0.1 kg/L?
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
6. Evaluate the following.
√
(a) arcsin(−
2
2 )
(b) arccos(− 21 )
(c) tan−1
√1
3
(d) sin−1 (sin 5π
6 )
(e) cos(arccos 54 )
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(f) cos−1 (cos 5π
4 )
(g) tan(tan−1 18)
(h) arctan(tan 11π
6 )
(i) cos(arcsin(− 56 ))
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(j) sin(2 arctan 5)
(k) tan(cos−1 x)
7. Calculate the derivatives for the following functions.
√
(a) f (x) = x arcsin( 5x)
(b) g(x) = tan−1 (3x2 )
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
8. Find the equation of the tangent line to y = cos−1 ( x1 ) at the point where x = 2.
"
−1
9. Calculate lim sin
x→∞
x2 + 3
2x2 − 5
10. What is the domain of f (x) =
!
+ tan
−1
x2
4−x
arcsin(4x − 1)
?
arctan x
7
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Math 151 WIR, Spring 2010, 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
(ln x)2
x→∞
ex
(c) lim
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(d) lim x2 ln 4x
x→0+
(e) lim sin xtan x
x→0+
2
(f) lim (cos 3x)1/x
x→0+
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Math 151 WIR, Spring 2010, Benjamin
Aurispa
(g) lim (4 + e3x )2/x
x→∞
(h) lim (xe1/x − x)
x→∞
10